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Construction of strong Markov processes through excursions, and a related Martin boundary Salisbury, Thomas S. 1983

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OF STRONG MARKOV PROCESSES THROUGH EXCURSIONS, AND A RELATED MARTIN BOUNDARY By THOMAS S. SALISBURY B.Sc', McGill University, 1979 SUBMITTED IN PARTIAL'FULFILLMENT OF REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept this thesis as conforming to the required standard CONSTRUCTION A THESIS THE THE UNIVERSITY OF BRITISH COLUMBIA October 19831 ©Thomas Stephenson Salisbury In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publi c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. _ , , j- Mathematics Department of . The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Ock>W ( ^ , . 7 3 3 DE-6 ( 3 / 8 1 ) - i i -ABSTRACT For c e r t a i n Markov processes, K. I t o has d e f i n e d the Poisson p o i n t process of excursions away from a f i x e d p o i n t . The law of t h i s process i s determined by a c e r t a i n measure, c a l l e d i t s C h a r a c t e r i s t i c measure. He gives a l i s t of c o n d i t i o n s t h i s measure must obey. I add to these c o n d i t i o n s , o b t a i n i n g necessary and s u f f i c i e n t c o n d i t i o n s f o r a measure to a r i s e i n t h i s way. The main technique i s t o use a ' l a s t e x i t decomposition' r e l a t e d to those of Getoor and Sharpe. The more general problem of excursions away from a f i x e d s e t i s t r e a t e d u s i n g the E x i t system of B. Maisonneuve. This gives a u s e f u l technique f o r c o n s t r u c t i n g new Markov processes from o l d ones. For example, we o b t a i n a r i g o r o u s c o n s t r u c t i o n of the Skew Brownian motion of I t o and McKean, and another proof of r e s u l t s of P i t t e n g e r and Knight on e x c i s i o n of excursions. A r e l a t e d question i s that of determining whether an entrance p o i n t f o r a Markov process remains an entrance p o i n t f o r an h-transform of that process. Let E be an open subset of Euclidean space, w i t h a Green f u n c t i o n , and l e t X be harmonic measure on the M a r t i n boundary A of E. 2 I show t h a t , except f o r a \fk\ - n u l l - s e t of (x,y)eA , x i s an entrance p o i n t f o r Brownian motion conditioned to leave E at y. R.S. M a r t i n gave examples i n dimension 3 or h i g h e r , f o r which there e x i s t minimal a c c e s s i b l e M a r t i n boundary p o i n t s x^y f o r which t h i s c o n d i t i o n f a i l s . I give a s i m i l a r example i n dimension 2. The argument uses recent r e s u l t s of M. Cranston and T. McConnell, together w i t h S c h w a r z - C h r i s t o f f e l t r a n s f o r m a t i o n s . - i i i -Table of Contents Abstract i i Table of Contents i i i L i s t of f i g u r e s i v Acknowledgement v Pa r t 1. Excursions . 1 Section 1. I n t r o d u c t i o n 1 Section 2. Notation and Results 2 Section 3. Proof of Theorem 2 12 Sectio n 4. I n s u f f i c i e n c y of c o n d i t i o n s ( i i ) and ( i v ) 43 Section 5. Proof of P r o p o s i t i o n 1 48 Se c t i o n 6. Ray and Right Processes 51 Section 7. V a r i a t i o n s on Lemma 7 61 Sectio n 8. Regulation, Uniqueness and c o n s t r u c t i o n o f P o i n t Processes 67 Sectio n 9. Excursions away from a set 100 Section 10 A p p l i c a t i o n s 141 P a r t 2. M a r t i n Boundaries 165 Sectio n 11 I n t r o d u c t i o n 165 Section 12 h-transforms and Martin-Boundaries ..... 166 Se c t i o n 13 A M a r t i n Boundary i n the Plane ........ 189 Section 14 ^ Smallness. of, D'/D. 222 B i b l i o g r a p h y 229 - i v -L i s t of Figures F i g u r e 1 43 Figure 2 43 F i g u r e 3 44 Figure 4 192 F i g u r e 5 201 Figure 6 208 F i g u r e 7 208 Figure 8 221 - v -Acknowledgements Thanks are due to Diane Mackenzie, Ruth Ewasiuk, and Carole Samson f o r t h e i r much appreciated t y p i n g of my t h e s i s . I would a l s o l i k e to thank John Taylor f o r f i r s t e x c i t i n g me about P r o b a b i l i t y . I owe my parents so much more than I can mention; I thank them here f o r t h e i r advice and l o v e . John Walsh has,since I a r r i v e d given me every conceivable form of support and encouragement. I have learned from him, and been guided by him; i f I have become a p r o b a b i l i s t , i t i s he who has made me so. H i s i n s i g h t f u l comments throughout my work have been i n v a l u a b l e , and I thank him. L a s t l y , I must thank Kathy. Her patience and support throughout our stay i n Vancouver have made i t p o s s i b l e f o r t h i s t h e s i s to be. She makes i t a l l worthwhile. I thank her f o r her l o v e . - 1 -PART 1. EXCURSIONS  1. I n t r o d u c t i o n In the theory of Markov chains, i t i s e a s i l y shown that the 'excursiona' away from a f i x e d s t a t e are IID random v a r i a b l e s , w i t h values i n path space. This f a c t us used e x t e n s i v e l y i n the a n a l y s i s of recurrence, and s i m i l a r questions. In h i s c l a s s i c paper, "Poisson P o i n t Processes attached to Markov Processes" [^^], K. I t o obtains the corresponding property f o r the excursions of a continuous time standard process, away from a r e c u r r e n t p o i n t a. He c o n s t r u c t s a p o i n t process of e x c u r s i o n s , and shows t h a t i t i s a Poisson p o i n t process (PPP) . This decomposition, of the o r i g i n a l process i n t o i t s excursions, has proved u s e f u l i n a n a l y s i n g the o r i g i n a l standard process. For example, J.B. Walsh has i n [51] given a simple proof of the Ray-Knight theorems on the Markov property of l o c a l time, u s i n g the PPP of excursions of Brownian Motion away from 0. A d i f f e r e n t a p p l i c a t i o n l i e s i n the results; of P. Greenwood and E. P e r k i n s [2%y-who analyse domain of a t t r a c t i o n problems f o r random walk, u s i n g yet another type of excursion process. We w i l l be examining a converse problem; t h - t of c o n s t r u c t i n g a Strong Markov process from i t s excursions. In [32], I t o shows that the c h a r a c t e r i s t i c measure of the excursion process he c o n s t r u c t s , s a t i s f i e s c e r t a i n c o n d i t i o n s , ana then i n d i c a t e s how to reverse h i s c o n s t r u c t i o n provided those c o n d i t i o n s h o l d . That i s , given a PPP whose c h a r a c t e r i s t i c s measure obeys tnese c o n d i t i o n s , i t should be p o s s i b l e to c o n s t r u c t a process whose excursion process i s that PPP. I t o doesn't s t a t e t h i s converse r e s u l t as a formal theorem, and hence doesn't w r i t e out a proof. - 2 -Our f i r s t goal w i l l be to make h i s argument r i g o r o u s . This i s perhaps more i n t e r e s t i n g i n that the converse i s not t r u e as s t a t e d above. In s e c t i o n 4, examples w i l l be given showing that unless we strengthen two of h i s c o n d i t i o n s , the process constructed from the PPP may f a i l to be strong Markov or r i g h t continuous. This strengthening gives us necessary and s u f f i c i e n t c o n d i t i o n s f o r the process obtained to be a r i g h t process. The argument f o r s u f f i c i e n c y i s presented i n s e c t i o n 3, and that f o r n e c e s s i t y i n s e c t i o n 5. S e c t i o n 6 w i l l examine the Ray property and 'hypotheses d r o i t e s ' of the constructed process. Section 2 w i l l give n o t a t i o n and a statement of the r e s u l t s . Section 7 contains a v a r i a t i o n on the main lemma used i n the proof of s u f f i c i e n c y . S e c t i o n 9 w i l l t r e a t a g e n e r a l i z a t i o n of I t o ' s r e s u l t and of our converse, to the s i t u a t i o n of excursions away from a set ( r a t h e r than a p o i n t ). This i s done usi n g the E x i t System of B. Maisonneuve ( g e n e r a l i z i n g the excursion measure of Tjto's PPP ). Section 8 contains some t e c h n i c a l f a c t s about the excursion process i n t h i s new s i t u a t i o n . Section 10 gives a p p l i c a t i o n s to skew Brownian motion, and to the r e s u l t s of F. Knight and 0. P i t t e n g e r on e x c i s i o n of excursions. 2. Notation and Results E w i l l be a seperable metric space, and E^ the a - f i e l d of i t s B o r e l subsets. E w i l l be the u n i v e r s a l completion of E^ . U w i l l be the set of r i g h t - c o n t i n u o u s E-valued paths, and, f o r a e E, U w i l l be the set of a l l such paths u , - 3 -s a t i s f y i n g u(0)=a . ( V t > 0 w i l l be the coordinate process on U ; W t(u) = u ( t ) . For t>0, U° w i l l be the smallest a - f i e l d such that f o r every s e [ 0 , t ] , W i s a measurable f u n c t i o n from (U,(J°) t o (E,E ) s t 0^, > (£,! U w i l l be the u n i v e r s a l completion of U°, and U t h a t of U°. t t 0 0 (II,P) w i l l be the measurable space of (U,U)-valued p o i n t f u n c t i o n s . That i s , we a d j o i n a p o i n t 6 to U, and l e t II be the set o f f u n c t i o n s p:[0,°°) -*• Uu{6} such t h a t p ( t ) = 6 except f o r countably many t . P i s the a - f i e l d on n generated by the fu n c t i o n s p B- N(A,p), where M(A,p) i s the number of times t such t h a t ( t , p ( t ) ) e A c [0, 0 0) x U. Here A belongs t o the product B a U of the B o r e l f i e l d S on [0,"»), and of U. ^ v t ^ t > 0 w i l 1 b e t h e coordinate process on II; y t0?) = p ( t ) -For A e 8 a U, we define the r e s t r i c t i o n of p e II t o A t o be: (p(t) , i f ( t , p ( t ) ) e A 6 , otherwise S p e c i a l cases of t h i s w i l l be the k i l l i n g o p e r a t o r s : a s ( p ) = P | [ 0 , s ] x U k s ( p ) = P | [ 0 , s ) X U For t>0, P w i l l be the sub a - f i e l d a t _ 1(P) , of P. We define the s h i f t operators of II t o be: 0 s (p) = p ( t + s ) , f o r t>0 Q > - 9 s ( p ) | ( 0 , " ) The same n o t a t i o n w i l l be used f o r the corresponding s h i f t operators on U. P n ( t ) = A - 4 -For u e tf we de f i n e the h i t t i n g time and debut of {a} c E t o be a (u) = i n f {t > 0; u(t ) = a} T (u) = i n f {t > 0; u(t ) = a}. I t i s w e l l known t h a t both and are ) stopping times. A Poisson p o i n t process on a p r o b a b i l i t y space ( f t , F , P ) , w i t h values i n (U,U) i s a measurable f u n c t i o n Y : ( f t ,F) (II,P) , together w i t h a f i l t r a t i o n (F of (ft,F) ( F t c F c F f o r t < s ) , such t u2U t S t h a t : (a) cx t(Y) e F f o r t > 0. (b) ©°(Y) i s independent of F^ and has the same law as Y|,„ t t |(0,°°) f o r t > 0. (c) There e x i s t sets A^ e l i , A^ t U such t h a t N([0,t] x A^Y) < « a.s. f o r each k , t . The s p e c i a l r o l e of time t = 0 i n (b) w i l l be u s e f u l when we consider the p o i n t process of excursions away from a p o i n t a e E, of a Markov process t a k i n g values i n E. Time t = 0 corresponds to the f i r s t e x c u r s i o n , which i s e x c e p t i o n a l i n t h a t we w i l l want t o all o w i t to s t a r t i n any i n i t i a l d i s t r i b u t i o n . In c o n t r a s t , the other excursions w i l l s t a r t i n a manner d i c t a t e d by the t r a n s i t i o n p r o b a b i l i t i e s o f the given Markov process. Under c o n d i t i o n s (a), (b), (c) above, there e x i s t s a a - f i n i t e measure n on u (the c h a r a c t e r i s t i c measure of Y) such t h a t E[N((0,t) x A,Y)] = t«n(A) f o r A e U, t > 0 . This measure determines the law of Yl , .. In f a c t , i f I (o,°°) ( s . , t . ) x A. are d i s j o i n t and 0 < s. < t . , f o r i = 1 .. k, then the i i i i i - 5 -N ( ( s . , t . ) X A . , Y ) are independent Poisson random v a r i a b l e s , of means (t.-s.)«n(A.). 1 1 1 In the theory of r i g h t processes, i t i s customary to equip a r i g h t continuous, E-valued process X based on (ft,F), w i t h laws p 3 on (£2,F), one f o r each b e E. We w i l l reserve the n o t a t i o n P*3 f o r laws on the c a n o n i c a l space (U ,U). Late r on we w i l l work w i t h a r i g h t process b -b X, and the P w i l l be the image laws under X of the P . In g e n e r a l , on a f i x e d measurable space (ft,F), l e t ^ Xt^t>0 b e a r i g h t continuous process, based on Q, w i t h values i n E. We w i l l say b th a t (X t,F t,y,P ) has the strong Markov property a t T i f the f o l l o w i n g s i t u a t i o n h o l d s ; ^t^t>0 ^S a f i l t r a t i o n o f (^ ,F) , X ^ " ( ^ ) <= F , _ b u i s a a - f i n i t e measure on (ft,F), (P )^ E i s a f a m i l y of p r o b a b i l i t y measures on (U,U) such t h a t b |-> p (A) i s E-measurable f o r each A e U, T i s a stopping time f o r the f i l t r a t i o n ^ t + ^ ' l j ( xT + . e A ' T < ° ° ' B ) = p X T ( A ) d y f o r A e U, B e F T+ Bn{T«»} Except i n Lemma 6 below, we w i l l always take u t o be a p r o b a b i l i t y measure. For (X f c) r i g h t continuous w i t h values i n E, we w i l l say t h a t b b (X t,F t,y,P ) i s strong Markov i f (X t,F t,y P ) has the strong Markov property a t each (F^) stopping time. For aeE, ^t^t>0 * S 3 -*-oca-*-at a , i f L i s continuous, nondecreasing, and adapted to (F ), w i t h s e t of in c r e a s e e x a c t l y {t;X t = a}, such t h a t f o r every (F ) s t o p p i n g time T w i t h X T = a, ( X . + T L . + T " L T ) i s i n d e P e n d e n t of F. , w i t h the same law as (X , L ). t-p *+o (X), '+a (x) a a lme - 6 - -J Ito performed the following construction (Actually, he considered only the case of a standard process, but as pointed out to me by J. Pitman, h i s arguments apply without change. Henceforth s i m i l a r q u a l i f i c a t i o n s w i l l be omitted.); Let P be a p r o b a b i l i t y measure on (ft,F) under which F i s complete, and suppose the f o l l o w i n g conditions hold: (2.1) (Xfc) i s r i g h t continuous with values i n E, (X t ,F f c,P,P ) i s strong Markov, and each F contains a l l the P-null sets of F. (2.2) X i s recurrent at a point a e E. (P (o <°°) = 1 f o r b e E) . a (2.3) If P a ( a =0) = 1, then there i s a l o c a l time (L ) f o r X at a , a t which i s canonical i n the sense that i t i s normalized to make -a_(X) f». E[e a ] = E[ e _ t dL ] (Condition (2.3) holds i f X i s a r i g h t process). I f p a ( a = 0) = 1, l e t S(s) be the r i g h t continuous inverse l o c a l time: S(s) = i n f { t > 0;L > s}, S(O-) = 0. Let X(S(s-)+t), i f 0 < t < S(s) - S(s-) a , i f t > S(s) - S(s-) > 0 Y (t) = s Y 6 , i f S(s) - S(s-) = 0. s Then Ito shows that Y i s a (U ,U)-valued PPP with respect to P and the f i l t r a t i o n of Y . I f , on the contrary, p a ( a = 0) = 0, then X v i s i t s a at a d i s c r e t e set of times. In t h i s case, l e t S(k) be the k'th h i t t i n g time of a; S(0) = 0, S(k+1) = i n f { t > S(k),-X = a}. Let (x(S(k) + t) , i f 0 < t < S(k+1) - S(k) a , i f t > S(k+1) - S(k). Y (t) = k - 7 -Then I t o shows t h a t under P , the Y , k > 1 are HD, (U,U)-valued, F-measurable random v a r i a b l e s . Let n be the c h a r a c t e r i s t i c measure of Y i n the f i r s t case, and the common d i s t r i b u t i o n of the Y ,k > 1 i n the second. We c a l l n K. the e x c u r s i o n measure of X from a. I t o i s concerned w i t h c l a s s i f y i n g a l l processes t h a t agree up t i l l the de"but o f a p o i n t a e E. S p e c i f i c a l l y , suppose t h a t (2.4) (p b) i s a f a m i l y of p r o b a b i l i t y measures on (U , U ) such t h a t o c b f o r each c e E, the coordinate process (W, , U , P , P ) i s strong . t t o o Markov. ( Note t h a t here, c i s f i x e d , and b ranges over E ) (2.5) P b { u ; x (u) < 0 0, and u ( t ) = a f o r t > T (u)} = 1 f o r each o a a b e E. The problem i s to c l a s s i f y a l l f a m i l i e s ( P ) of p r o b a b i l i t y measures on (U , U ) f o r which there e x i s t s ( X ^ , F T , P , P ^ ) on some p r o b a b i l i t y b space, which i s a r e c u r r e n t extension of ( P q ) i n the sense t h a t (2.1), (2.2), (2.3) h o l d , and b b (2.6) P {U;U(«A t (u)) e A} = P (A) f o r each b e E, A e ii. a o I t o achieves t h i s c l a s s i f i c a t i o n i n terms of the exc u r s i o n b measure n of X from a. He shows t h a t the P and n determine o the P B , and then d e r i v e s a the f o l l o w i n g l i s t of c o n d i t i o n s that n obeys Theorem 1 (Ito) Let ( X T , F T , P , P B ) s a t i s f y (2.1), (2.2), (2.3). . Let n be the e x c u r s i o n measure of X from a, and d e f i n e ' Q ( ^ ) t 0 be P b { u : u ( « A T (u)) e A}. Then the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : Si ( i ) ..n ' i s concentrated cn {u;0 < cr (u) < <a, u ( t ) = a f o r t > a (u) } . a - 8 -( i i ) n{u,-u(0) i V } < 0 0 f o r every open neighborhood V of a. ( i i i ) (1 - e ) dn < 1 (iv) n{u,-g (u) > t , u e A, 8 (u) e M} 3L TI P u ( t ) (M)n(du) o An{a > t } 3, f o r t > 0, A £ l i t , M e U (v) n{u;u(0) e B, u e M} = {u;u(0) e B} P U ( 0 ) (M)n(du) o ibr M e U, and B e E such t h a t a I B. ( v i ) E i t h e r (a) n i s a p r o b a b i l i t y measure concentrated on U a = {u;u(0) = a} ( d i s c r e t e v i s i t i n g case); or (b) n i s f i n i t e , n(U a) = 0, and (exponential h o l d i n g case); (1-exp(-a )) dn < 1 a or (c) n i s i n f i n i t e and n(U a) = 0 or oo (instantaneous case). The main r e s u l t o f t h i s paper i s t h a t i f c o n d i t i o n s ( i i ) and (vi) are strengthened, we o b t a i n c o n d i t i o n s t h a t are necessary and s u f f i c i e n t f o r a a - f i n i t e p o s i t i v e Treasure n t o a r i s e as the e x c u r s i o n measure of a r e c u r r e n t extension of a f a m i l y (P^) s a t i s f y i n g (2.4) and (2.5). The c o n d i t i o n s are: ( i i ' ) n{u;u leaves V } < 0 0 f o r every open neighbourhood V of a. ( v i 1 ) E i t h e r (a) n i s a p r o b a b i l i t y measure concentrated on U 3. I f n > n' > 0 and n' s a t i s f i e s ( i v ) , then n 1 i s a m u l t i p l e of n ; or (b) as i n ( v i b) ; or (c) n i s i n f i n i t e . I f n > n' > 0 and n' s a t i s f i e s (iv) , then n' (U 3) = 0 or <=°. - 9 -The statement o f n e c e s s i t y i s ' P r o p o s i t i o n 1 under the c o n d i t i o n s of Theorem 1, c o n d i t i o n s ( i i 1 ) and ( v i 1 ) a l s o h o l d . A strong form of s u f f i c i e n c y i s : Theorem 2 Assume t h a t (n,F,P) i s complete, and t h a t (P^) s a t i s f i e s (2.4) and (2.5). a) I f ( y t ' ^ i s a P P P w i t n v a l u e s i n (U,U) and w i t h c h a r a c t e r i s t i c measure n, such t h a t : (F f c) i s r i g h t continuous, and each F c o n t a i n s a l l the P - n u l l s e t s of F 5 P(Y e M) = o P Y ° ( 0 ) (M) dP for M e U 5 o ( i ) , ( i i ' ) , ( i i i ) , ( i v ) , (v) and e i t h e r (b) o r (c) of ( v i ' ) hold., Then there i s a r i g h t continuous strong Markov process (X t , G t,P,P b) such t h a t : Y i s the PPP constructed from X as above, P-a.s. ; 3D fo (X. , G. ,P,P ) i s a r e c u r r e n t extension of (P ) 5 t t o G s<t +) = F t ; (Gfc) i s r i g h t continuous, and each G contains a l l the P - n u l l sets of F. b) I f (F ^ i s a n i n c r e a s i n g f a m i l y of sub p;-fields of F, each c o n t a i n i n g a l l the P - n u l l s e t s o f F, and f o r each k > 0, Y i s a measurable f u n c t i o n from ( f i , F k ) to (0,U) such t h a t : For k S 1 the Y, have a common d i s t r i b u t i o n n ; k P(Y eM) = o - 1 0 -P U ( o ) (M ) P ( Y edu) + n(M ) P ( Y eU*) o o o u\u a for M e U 5 a(Y^;i>k) i s independent of F^ for k > 0 ; ( i ) , ( i i " ) , (iv) and (a) of (vi') hold. Then there i s a strong Markov r i g h t continuous process ( X f c , G t P,Pb) such that : • The Y are the excursion random variables constructed k from X as above (discrete v i s i t i n g case), P-a.s. ; ID b (X ,G , P , P ) i s a recurrent extension of ( P ) ; t t o G S ( k + l ) - c Fk c GS(k+l) . k > o ; . (Gfc) i s r i g h t continuous, and each G contains a l l the P - n u l l sets of F • This r e s u l t can be used to give a rigorous construction of processes such as skew Brownian motion (see J . Walsh [5Q]). Similar r e s u l t s i n s p e c i a l cases, obtained by d i f f e r e n t techniques can be found i n Blumenthal [4] and S. Watanabe [31],[54]. The key to the proof of T heorem 2 given here is. to f i n d an expression f o r conditioning Y ^ on the s t r i c t past F^ , f o r T an (F ) stopping time. This i s done i n Lemma 7, which i s r e l a t e d to s i m i l a r r e s u l t s of M. Weil [53|f [54j a s follows. A version of Lemma 7 can be proven f o r (Y »F ) now a cadlag Markov process, and T an (F ) stopping time such that LTTTf c u J[q + S 0 0 I , where S i s a terminal time for the natural f i l t r a t i o n of Y (the smallest r i g h t continuous f i l t r a t i o n to which Y i s adapted). - 11 -P(Y Te«|F T_) i s i d e n t i f i e d i n terms of P(Yg€'IF g_), and Weil's r e s u l t s i d e n t i f y the l a t t e r i n terms o f a Levy system f o r Y (assuming t h a t Y i s standard.) See s e c t i o n 7. A f t e r completion of t h i s work, i t was pointed out to me that p a r t s of i t have appeared b e f o r e ; In h i s Thesis [34 ] , S. Kabbaj obtained a r e s u l t s i m i l a r to Theorem 2. He shows t h a t under I t o ' s c o n d i t i o n s , and w i t h the minimal f i l t r a t i o n t o which the r e c o n s t r u c t e d process (X f c) i s adapted, (X ) i s s t r o n g Markov a t a l l ) stepping times. His proof uses t t a weaker form of Lemma 7 below, and r e l i e s h e a v i l y on the powerful t o o l s o f ma r t i n g a l e theory. The proof presented here works f o r o T a (G ) stopping time, i s more elementary, and a p p l i e s i n g r e a t e r g e n e r a l i t y ( we assume no compactness c o n d i t i o n s on El ). A l s o , when we are given a r i g h t process (X f c) , and l e t (X ) be i t s e x c u r s i o n process, Lemma 7 i s s t i l l of i n t e r e s t . In t h i s c ontext, i t i s c l o s e l y r e l a t e d to Getoor and Sharpe's l a s t e x i t decompositions, and a very s i m i l a r r e s u l t has appeared i n Getoor and Sharpe [25]. Some of the methods used i n the proof of Lemma 7 appear, both i n [25] and i n Pitman E45] . P a r t of the i n t e r e s t of the r e s u l t s presented here, however, l i e s i n that they apply i n a more elementary context than the theory of r i g h t processes. As pointed out i n s e c t i o n 6, the proof of Theorem 2 can be g r e a t l y s i m p l i f i e d i n the case that (P^) comes from a r i g h t process. I would l i k e to thank Yves LeJan f o r b r i n g i n g [34] to my a t t e n t i o n . - 12 -3. Proof of Theorem 2 Part (a) i s shown f i r s t , and then the arguments are modified to show part ( b ) . In part ( a ) , conditions ( i ) and ( i i i ) are used i n the construction; the l a t t e r so that the normalization of l o c a l time agrees with (2.3). Condition ( i i ' ) appears i n Lemma 3, in the proof of the right continuity of paths. Condition (vi'b/ also appears i n t h i s lemma, and i s used to make the " inverse l o c a l time " s t r i c t l y increasing, so that l o c a l time w i l l be continuous. Conditions (iv) and (v) are put i n a more convenient form i n Lemma 6, which, together with Lemma 7, y i e l d s C o r o l l a r y 2. Lemma 7 i s also used with conditions (vi'b) and (vi'c) to give C o r o l l a r y 1. Note that these two lemmas e s s e n t i a l l y show the strong Markov property. In part (b), the conditions are put to the same uses, except that we use condition (vi'a) instead of conditions (vi'b) and (vi'c) . We s t a r t the proof of part (a), by constructing X as an e x p l i c i t measurable function of Y. Put -o m = 1 - (1-e a ) dn S (s,p) = ms + £ a ( p ( r ) ) , f o r s £ 0, p e II r<s a (with the convention that a (&) = 0 ) , and a S +(s,p) = l i m S (r , p ) . r i s Then S (*,p), S +(*,p) are nondecreasing, and r e s p e c t i v e l y l e f t and r i g h t continuous, with values i n [O, 0 0]. I f S (s,p) < 0 0. , - 13 -then S T(s,p) = S (s,p) + o a ( p ( s ) ) . s (s,«) e Pg_ since ,±t i s l e f t -continuous , and S (s,p) = S (s,a (p)) . Thus also S + ( s , 0 e P . S o ' Put £ f c(p) = i n f {s > o; °° > S +(s) > t} , with the usual convention that inf ( 0 ) = +°°. Then i s a (P ) t s+ stopping time, and s"(£t(-),«) < t < S"+(£t(.) ,.) (with the convention that S+(°°,p) = °° ). Put (y£ (t - S~{1 ,-)) , i f y, 4 6, Z > a , otherwise . We w i l l show next that x i s measurable from Vp to E° . Since t t+ 0 — (u,r) H- u ( r ) i s measurable from U»B to E , and S (£ (•),') e P/ * t since S i s pred i c t a b l e , we need only show that y f e ? p We state t h i s i n a more general form, to be us e f u l l a t e r , as ; Lemma 1. Let (F ) be a f i l t r a t i o n of a measurable space ( f t , F ) , and l e t Y be a function which i s measurable from ( f i , F t ) to (H,P t ) , for every t > 0. Let R be an (F ) stopping time such that f o r each e > o, {R < °°} <= {a (Y ) > e f o r only f i n i t e l y many times t any compact time set}. Then Y e F_ . R R Remark: In the present s i t u a t i o n , we apply the lemma with Y the i d e n t i t y map, and with F = P J r t t + i n - 14 -proof: Let A e il, s > o. Then {Y £ A, R < s} = {Y e A, R < s, a (Y ) = 0 } u H R 3. R U u {Y £ A, R < s, cr (Y ) > -} . R a R m m>l Let B , n = 1,2, ... be an open base i n the space [ o , s ] . We can n w r i t e {a (Y ) = o, R < s} = a R = n u [ { R e B } n { N ( B X{CJ > -} ,Y) = 0 } ] ^. ^. n n a m m>l n>l and {Y e A, R < s, a (Y ) > -} = R a R m = n [ {R < s} n ( { R i B } u {N(B X [ A n{ o > - ],Y) > 1})] n n a m n>l Thus {Y e A, R < s ) e F , as r e q u i r e d . 0 R S Put M = {p)S (s,p) < 0 0 f o r each s > o, S (s,p) -»- 0 0 as s •+ °°, S (*,p) i s s t r i c t l y i n c r e a s i n g , and f o r each open neighbourhood V of a, p(s) leaves V f o r only f i n i t e l y many times s i n any compact s e t of times} . Thus on M, t^<°° f o r every t , t I—»• Z ' i s continuous, and y (t - s " ( s f O ) i f S~(s, •) < t < S + ( s , •) - 15 -Lemma 2 t H- X i s r i g h t continuous on M. proof: F i x some element p of M. By d e f i n i t i o n of U, x.(p) i s r i g h t continuous on each i n t e r v a l " [S (s,p), S + ( s , p ) ) . I f t l i e s i n no such i n t e r v a l , then i t must equal S +(s,p) f o r some s, and hence x^_ (p) = a. Let V be an open neighbourhood of a. Since peM , there i s an s' > s such that y^(p) remains i n V for any r e ( s , s ' ) . Thus x (p) £ V f o r any q e ( S + ( s , p ) , S ~ ( s 1 , D ) ) . But bv d e f i n i t i o n of M, q S (s',p) i s s t r i c t l y greater than S + ( s , p ) , so that as V was a r b i t r a r y , x (p) must be r i g h t continuous at t . • (The same argument would show that i f we had taken U to be the space of cadlag paths, then on M, x, would be cadlag as well.) For convenience, we separate out the co n t r i b u t i o n of p(o); we have shown that there i s a measurable function F: (U * H, U 0 P) — y {13,U) such that x. = F(y ,y|, . ) ( t ) on M. t Jo J (o,°°) P u t S (s) = S (s,Y) + + S (s) = S (s,Y) L t - V Y ) X t = x t(Y) Since Y i s , by the hypotheses of Theorem 2, a measurable function from (fi,F ) t 0 (H,P ), f o r each s, the above r e s u l t s imply that S and S + are adapted to ( F ) , and that f o r each t > o, L^ _ i s an ( F ) stopping time and X i s measurable from F T to E (as F i s L t L t complete ). - 16 -Lemma 3. P(Y e M) = 1. proof: For f € 8 w i t h f ( o ) = o, I f ( a ( Y ) ) = o<t<s f ( r ) N ( { a e dr} x ( 0,s),Y) a (o,°°) i s F-measurable, and has ex p e c t a t i o n s | f ( r ) n (CT e dr) = s foCT dn. a J a (o,°°) In p a r t i c u l a r , E t I CT < Y J 1 * s CT dn o<t<s a Z . J . a CTa(Yt)<l { C T a " l } s T , -i a . dn 1-e" 1 (1-m) < » A l s o , there are only f i n i t e l y many times t e [0,s] with a (Y )>1 , Si t 1 f ~ C Ta n (CT > 1) < r- (1-e a ) dn < °° , 1-e J so t h a t S (s) < 0 0 a.s., f o r each s. I f n ( U ) = o (so t h a t a i s a t r a p ) , then m = 1 so t h a t S ( s ) = s a.s. I f n ( U ) 7* o, then by ( i ) there i s an £ > £> such t h a t n(a > E) > o. Thus, there are a.s. i n f i n i t e l y many times s cl such t h a t CT (V ) > E, so t h a t i n e i t h e r case, S (s) -»• 0 0 as s 0 0 a s a.s. I f m > o, i t i s c l e a r t h a t S i s s t r i c t l y i n c r e a s i n g . I f m - 17 -i then by (vib) i t f o l l o w s t h a t n(U) = oo, and hence t h a t {s;a ( Y ) > o) a s i s dense i n [o,°°) , a.s. The l a s t c o n d i t i o n i s obtained from ( i i 1 ) , l e t t i n g V run through a countable base of open neighborhoods of a. • Appealing t o the completeness o f F , we w i l l without l o s s of g e n e r a l i t y assume t h a t Y e M s u r e l y , hence t h a t X i s r i g h t continuous and by Lemma 1, t h a t Y e F f o r every ( F ) stopping time R. H. R S I f we are given a f i l t r a t i o n (I/. , a o " - f i e l d 1/ c \J t t>o o- o and a random v a r i a b l e R w i t h values i n [o,°°] , r e c a l l t h a t 1/ i s R-d e f i n e d t o be the f i l t r a t i o n generated by 1/ and by sets of the form A n {R > s} , f o r A e F and s > o. In our case, l e t F be generated s o-by a l l the P - n u l l s e t s of F . For R a random time, and r > o, put = F R _ v a ( Y R ( s ) ; o < s < r) Put L r° - w t " t n ( t - s " ( L f c ) ) -G ° = G - F o- o- o-Lemma 4 (a) (G ) i s r i g h t continuous, i n c r e a s i n g , and each G^  contains a l l u t the P - n u l l s e t s of F . X e Gfc f o r each t > o. - 18 -(b) I f T i s a (G ) stopping time then L i s an (F ) stopping "C T s time, and f c G_ c G_ c F L T - T- T L T (c) S + ( s ) i s a (G f c) stopping time, f o r s > o. I f T i s any (G f c) stopping time such t h a t S + ( L ) = T then G = F T T L T (d) I f R i s an t'F^) stopping time, and T i s a (G f c) stopping time such t h a t T < S + ( L T ) on {T < «}, put V = T - S (R) , i f L = R < T i otherwise Then V i s an (H ) stopping time. proof: (a) We know t h a t each L f c i s an ( F g ) stopping time. Thus, i f t < t ' then G ° n { L t < L f c l} c t f ^ n {L f c <L t,} c F n {L < L } (as Y e F_ ) t z fc L t L t c F c G ° and G ° n {L f c = L t,} i s , by monotone c l a s s arguments, and Prop 18 of [2], generated by L t L f o- t and by - 19 -t - L H n {t-S(L )>r} n {L =L } c H fc' n {t-s~(L. ,) >r} n {L =L'} •tr t t t 3T "C t t c t f ( t - s - J t t . t , ) ) - c G f * Thus, G° c G°, . For s > t we can w r i t e X as a measurable f u n c t i o n o f L , t s k L (Y), and V s - (y • ) ( l e t s s s k T (Y) (r) , i f r < L L s s Y = i r k s - S - ( L ) ( Y L } ' l f r = L s s s , i f r > L Then X = x (Y ).) t t • As i n [7] Prop. 25, the f i r s t two are measurable and the L s~ l a t t e r l i e s i n H. .. V S - S (li )) -s (b) For r > t > 0 we have t h a t L F c G c G° c H r c F . L — t r °° L t r By d e f i n i t i o n , L + L, as t + r , hence • r t t r>t r so t h a t G <= F . t L t I f now T i s a (Gfc) stopping time, then by the r i g h t c o n t i n u i t y of L, {L. < X} = u {T < q} n {L < X} . t ^ q - 20 -But {T < q} e G c F. { L T < X > ^ FX-so t h a t s i n c e L i s an (F ) stopping time, F u r t h e r , i f A £ G ^ , then A n {T < q} e G c F q L q so t h a t A n {T < q} n {L < X} £ F. . q X Taking the union over q e $, we get t h a t A n { L < X } e F . , L A hence t h a t A e F L T We w i l l prove (c) before showing the remainder of (b); t h a t (c) Since S and S + are s t r i c t l y i n c r e a s i n g , and S (L f c) < t < S + ( L t ) , we see t h a t hence S + ( s ) i s a (G ) stopping time. S i m i l a r l y , i f S + ( L ) = T, then { L T < V = { T < t } ' and hence i f A e F , then by [ 2 ] Prop. 16, T A n {T < t } = A n {L < L } e F. That i s , A s G T . s i n c e F. we get t h a t G = F T L T D G T by the p a r t of (b) already shown, To f i n i s h the proof of (b), l e t A £ F = G + S T ( s ) • Then s 21 -A n { L T > s} = A n {T > S + ( s ) } e G T by [7], Prop. 16 again. { V < h} = {T < S (R) + h, L < R < °°}\{L < R} T T = (T < ( S ~ ( R ) + h) A S + ( R ) ) \ { L T < R} , as T < S + ( L T > on {T < °°) . Because T i s a stopping time, { L • < R} e F c tf* T R- h and {T < ( S ~ ( R ) + h) A S + ( R ) } € G° , . , „ + , „ . , [ ( S (R) + h) A S ( R ) ] - . We must t h e r e f o r e show t h a t the l a t t e r f i e l d l i e s i n tf, . I t i s h O R generated by G = F <=: tf , and by o- o- h G° n { ( S - ( R ) + h) A s+(r) > t } , f o r t > 0. By monotone c l a s s arguments, the l a t t e r i s generated by tf n {(s~(R) + h) A S + ( R ) > t} o-and by L tfrfc n { t - S ~ ( L ) > r} n {(s"(R) + h) A S + ( R ) > t } , f o r r > 0. a (Y ) i s an (tf ) stopping time, s i n c e a i s a (U ) a J K s T" a s • stopping time , each a - f i e l d tfg i s complete, and Y" 1^ U ) c tfR . (t - S ~ ( R ) ) v o i s a l s o an (tfR ) stopping R s s s+ time, s i n c e S~(R) e F R _ as S~ i s p r e d i c t a b l e . Thus - 22 -{ ( S ~ ( R ) + h) A S + ( R ) > t} = {a (Y ) A h > (t - S ( R ) ) V 0} 3. R £ H ( a (Y ) A h ) - C h cl R A l s o , FT n{L J_ < R} c F n {L < R} L T _ t R- t by [7] f r ^ P 18. Therefore L t + H n {(s (R) + h) A s (R) > t} o-c F„ n {(s"(R) + h) A s + ( R ) > t} c H R , R- n as { ( S ~ ( R ) + h) A S + ( R ) > t} c {L f c < R} . We argue as i n p a r t (a) t o see t h a t L f/ r n { t - S ( L F C ) > r} n { ( S ~ ( R ) + h) A S + ( R ) > t} n {L f c < R} C F N n {(s"(R) + h) A s + ( R ) > t } c H J \ R- h and L H fc n { t - S (L f c) > r} n { ( S ~ ( R ) + h) A s + ( R ) > t} n {L f c = R} c tfR n {a (Y ) A h > t - s~(R) > r} r a R C C (Y ) A h) - " H l • a R 1 2 Lemma 5. Let (f2^,F t), (P^/F^.) be r i g h t continuous f i l t e r e d spaces. - 1 2 1 Let Z: fi^ -* ^ 2 be such t h a t Z Ffc F f o r every t . Then f o r 1 2 every (Ffc) stopping time T^, there i s an (Ffc) stopping time such t h a t T^ o Z = T^, - 23 -proof: For r e q>, l e t B = {T < r } , and f i n d A e F 2 such that r x r r B = Z - 1 A . Set r r A. = u A . * r<t r re<g Thus, p u t t i n g T 2(u) = inf{t;co £ A }, we have that {T < t} = A e F 2 «i t t A l s o , -1 ~ -1 Z A = u z A = {T < t } , f o r every t > 0 , t r<t r 1 1 re© so t h a t T 2 o Z = T . • The f o l l o w i n g lemma would be much s i m p l i f i e d i f . i n s t e a d of the c o n d i t i o n s of Theorem 2, we had assumed t h a t the coordinate process (Wfc) was a r i g h t process under (P*3) . ( I t would f o l l o w as usu a l from the r i g h t c o n t i n u i t y of the U a f ( W ) at R ,(U a) being the re s o l v a n t f o r ( P Q ) •) 3D Lemma 6. Let (P Q) s a t i s f y (2.4) and (2.5), and suppose t h a t n i s a a - f i n i t e p o s i t i v e measure on (U,U) s a t i s f y i n g ( i ) , ( i v ) and ( v ) . Let R be a (^t+) stopping time such t h a t n ( u a , R=0) = 0. Then the coordinate process (W t,U t,n,P Q) i s strong Markov a t R. Proof: By r e p l a c i n g R by the (tl ) stopping times f R on B R B = ( 0 0 o f f B f o r B £ U , we see t h a t i t s u f f i c e s t o show t h a t f o r A £ U, n O ^ A , R <-) = j" P ^ ( R ( U ) ) (A) n (du) {R<°°} Let h > 0. By ( i v ) , - 24 -(3.1) f(u,0,u) n (du) n {a >h} , , a = [ f(u,v) P^ ( h ; (dv) ] n (du) , {a >h} 3. for f of the form 1 where B e U , C e U. Thus, this holds for every f e U, 8 U, f > 0. h Put \ ' Uh 0 U t • Since' n is o-finite, i t follows as in Theorem 7.3 of Blumenthal and Getoor [5], that there i s a stopping time R such that n(R jt R) = o. Define Z:D^ UxU by Z(u) = (u,ehu) . -1 -A o Then Z {U^) = U ^ t ^ + .(Note that this might f a i l for the universal completion U^ t +^ + .) Thus by Lemma 5, there i s a (^ t + ) stopping time R with R(u,0,u) = (R(u) - h , i f R(u) > h ( oo otherwise. Let f = 1{(u,v); R(u,v) < oo,0. . , (v) e A}. R(u,v) Since R i s a (^ t + ) stopping time, i t i s immediate that R(u,') is a (LT) stopping time, for each u e U _ since also (w , U ,pc p b) was t+ t t' o o assumed to be strong Markov fox: c£E»- we can use (3.1) twice to. get that n(o > h, h < R < °=, ©~^A) 3 K f(u,0_u) n (du) n ic >h) a {a >h} a - 25 -0 u(R(u,e, u)) (a >h} 3. ( A ) 1 { R < ° ° } ( U > \ U ) n ( d u ) {a >h} a u(R(u)) (u) n(clu) o l A ; i{h<R<»} L e t t i n g h f 0, we get t h a t -1. n(O<R<oo,0 A) = R {0<R<°°} From (v) , we see that u(R(u)) , P n (du) o (3.2) f (u(0))g(u)n(du) = u(o) f (u(o)) E q (g) n(du) u\u u\u° for nonnegative f e E, g e U. Let the set of branch p o i n t s be E. = {b e E; P b(W fi b) > 0} e E. br o o c b c The strong Markov propert y of (W.,_,U ,P ,P ) shows t h a t P (W eE, ) t t o o o o br f o r every c e E. Thus, n (R = 0, U \u A,A) = 0 P ^ ( 0 ) (R = 0,A) n (du) u\u c u\u [/ P V ( 0 ) (A) P U ( 0 ) ( d v ) ] n (du) a {R=0}° ° (SMP) P ^ ( 0 ) ( A ) P U ( 0 ) ( R = 0) n (du) u o' u\u u\u P U ( o ) (A) 1 (u) n (du) {R=o} (by 3.2)) Since n({R=0} n U ) = 0 by h y p o t h e s i s , we can see t h a t (3.1) holds . • - 2 6 -The h e a r t o f the proof of Theorem 2 l i e s i n the next r e s u l t , whose proof we defer u n t i l l a t e r . and Lemma 7. Let T be a (Gfc) stopping time such t h a t L > 0 T < S + ( L m ) on {T < »}. Let H be the a - f i e l d F S U on Q. x U . T t L T - t Then there i s an ^n't+) stopping time R such t h a t (3.3) P{oj;n{u;R(oi,u) < «} < °°} = 1 (3.4) R(w,Y T (CO)) = ( T - S ~ ( L ) ) (w) , i f T(w) < «> • " m T • T (3.5) R(cu,u) = «> f o r every u e U, i f T(w) = °° (3.6) P(Y_ e A, T - S ~ ( L J e B, T < 0 0 J F ) (w) L_ T L_— T T = n{u;ueA,R(cu,u) e B,R(w ,u) < °°} n { u ; R ( a J f U ) < m } f o r P-a.e.w, where A e U, B e 8, and we take the convention t h a t 0 «° 0 = ° = C o r o l l a r y 1. Let T be a (Gfc) stopping time such t h a t X T = a. Then T = S + ( L T ) a.s. c b proof. By the strong Markov propert y of ^ W-fc'^t' Po' Po^ C 6 E ' a n <^ the hypothesis that p 0 ^ T a < 0 0' w t = a f ° r e v e r Y fc - T a ) = 1# w e get t > O = 1, we get that ^(o^ =0) =1, and hence that P^ (W0=a,aa>0) = 0 c e E. Since a l s o P(Y e M) o Py o ( o ) (M) dP, i t f o l l o w s t h a t S + ( 0 ) = 0 o a.s. on {X =0}, hence t h a t T = S + ( L ) a.s. on {L = o} . Thus p u t t i n g o T T B = {L > 0 and T < S + ( L ) } , T T we may repl a c e T by Tfi, to get t h a t > 0 and T < S + ( L T ) on {T < - } . We need t o show t h a t T.= °° a.s. - 27 -Apply Lemma 7 t o T, t o get an stopping time R. For CJ e ft, l e t H(oi) = (u;u(o) = a, R(w,u) = 0}. Since R i s an stopping time, i t i s immediate t h a t R(w,») i s a (U t +) stopping time, hence t h a t H(co) e U q + f o r each w e f t . Since a l s o H(w) c u a , we have t h a t n * n l H ( . ) * °» and t h a t n l H ( w ) s a t i s f i e s ( i v ) and (v) . i f n(U)<°°, then n(H ( c o ) ) <n(U a ) = 0 by, ( v i ' b ) . I f n(U)=«, then n(H(co)) » Q or 0 0 f o r each co, by ( v i ' c ) ( t h i s i 8 ' t h e only place we use t h i s c o n d i t i o n ! ) . But n(H ( c o ) ) < n{u;R(co,u) < °°}, which i s i t s e l f f i n i t e f o r P-a.e. co. Thus i n e i t h e r case, n(H ) = 0 a.s., so P(T < «) = E [ j j | ^ - . ] = 0 n(R<°°) o by ( 3 . 6 ) ( r e c a l l that 0 / 0 = 0 ) . • Using these three r e s u l t s , we w i l l show C o r o l l a r y 2. Let T be a (G^_) stopping time, and A e U. Then — X P[Y T ('+T-S ( L m ) ) € A, T < °°] = E[P T ( A ) ,T < °°] LT T O proof: Replacing T by v a r i o u s (Gfc) stopping times T^ , i t s u f f i c e s to t r e a t s e v e r a l d i s t i n c t cases, namely t h a t X^ = a on {T < °°},that X T and L T > 0 on {T < •}, and t h a t X T 4 a and L = 0 on {T < »} In the f i r s t case, C o r o l l a r y 1 shows t h a t the c o n c l u s i o n i s t r i v i a l . In the second case, a l s o T < s + ( L T ) on {T < «}, s o t h a t we can apply Lemma 7 t o o b t a i n R as i n t h a t r e s u l t . I t f o l l o w s t h a t i f f e F L _ 8 U i s bounded, then T - 28 -(3.7) E [ f ( • , Y (•)), T < » ] L T r _ ± f ( w , u ) n (du) C n { u ; R(a ; , u ) < .} J 1 P(dcj) { R ( C U , ' ) < ~} Take f = 1 { (u ,u ) ;R(OJ,U) < oo,0 u € A} ' R (w,u) t o o b t a i n t h a t P ( Y T ( • + T - S ~ ( L m ) ) e A , T < oo) J-1™ T T n(0~ 1 (A ) ,R < oo) = E [ n ( R < oo) 1 • S i n c e R i s an W ) s t o p p i n g t i m e , i t i s immedia te t h a t R (w, * ) i s a ) s t o p p i n g t i m e , f o r e a c h u e S3. A l s o , s i n c e X m f a on (T < o°} we g e t t h a t n(u A,R (w , * ) = o) = 0 f o r P - a . e . w . T h u s , by Lemma 6 n i e : ) . (A), R(OJ,.) < ») = f p U ( R ( w ' u ) ) ( A ) n ( d u ) R(cu, #) J o { R ( 6 J , « ) < 0 ° } f o r P - a . e . o j . T h e r e f o r e by ( 3 . 7 ) a g a i n , P ( Y T ( ' + T - S " ( L )) e A, T < oo) T T Y_ (R(.,Y_(-))) - L i L i = E t P (A) , T < °°] o X T = E [ P (A) , T < °°] . o In t h e t h i r d c a s e , t h a t L T = 0 and ^ a on {T < o o } , t h e n a l s o T < S + ( L T ) on {T < o o } . Thus Lemma 4, p a r t (d) a p p l i e s , t o - 29 -show t h a t T i s an s t o p p i n g t i m e . S i n c e (H° ) i s a c o m p l e t i o n o f t h e n a t u r a l r i g h t c o n t i n u o u s f i l t r a t i o n o f t h e p r o c e s s X , . , ^ * -ACT (x) ' a » i t i s e a s y t o u s e t h e h y p o t h e s e s t h a t P (Y eA) = P ^ ° ^ (A) P (Y edu) o J 0 o c b f o r A e ( J , and t h a t f o r c e E t h e c o o r d i n a t e p r o c e s s (W ^ ,LT , P , P ) t t o o i s s t r o n g M a r k o v , t o c o n c l u d e t h a t t h e p r o c e s s ( X t A C T (X)' Hl' ^ a i s s t r o n g M a r k o v . T h i s s u f f i c e s . Q The r e m a i n d e r o f Theorem 2 p a r t (a) now f o l l o w s e a s i l y ; F o r A e U and b e E p u t P b ( A ) 1, ( F ( u , v ) ) P ( y | £ dv) P b ( d u ) A ( o , ° ° ) o S i n c e b t—»• P (A) i s E - m e a s u r a b l e f o r A e U , t h e same h o l d s f o r o b p b ( A ) . I f T i s any ( G ) s t o p p i n g t i m e , t h e n by t h e s t r o n g Markov p r o p e r t y o f P P P ' s (see I t o [32]Theorem 5.1), 0° (Y) i s T i n d e p e n d e n t o f F T , w i t h t h e same law as Y I , . . L T (of») By r e p l a c i n g T w i t h t h e (G^) s t o p p i n g t i m e s on B TB = o f f B , where B e G , i t f o l l o w s f r o m C o r o l l a r y 2 t h a t f o r A e U , P(Y_ ( -+T-S ( L m ) ) e A | G ) = P (A) on {T < » } . T T o S i n c e a l s o G^ c F L , Y L ( . + T - S ~ ( L T ) ) e , and T T T X - + T = F ( Y L ( ' + T " S ( L T ) } ' Ql Y ) ' T T 30 -t h i s y i e l d s the strong Markov property of (X t,G t,P,P b) a t T. By d e f i n i t i o n of (P^), (X t,G t,P,P b) i s a r e c u r r e n t extension of ( P ^ ) . The remaining p o i n t s have already been d e a l t w i t h , except f o r showing t h a t Y i s the PPP const r u c t e d from X as i n I t o [32] . Since P 3 ( a = 0) = 1, t h i s w i l l f o l l o w provided (L ) s a t i s f i e s (2.3) The s et of inc r e a s e of (L f c) i s e x a c t l y { f c; x t = aK s i n c e X = only i f t = S + ( s ) f o r some s. The n o r m a l i z a t i o n - a a ( x ) E[e ] = E [ r°° - t e dL t] o f o l l o w s e a s i l y from the PPP nature of Y, and Theorem 4.5 of I t o D2] F i n a l l y , we can w r i t e < X. + T< L. + T- L T> = ( F ( Y L (•+T-S"(L)),0°Y)^(0°Y)) , T T T so t h a t by C o r o l l a r y 1, i f T i s any (Gfc) stopping time w i t h X T = a then ( X < + T , L < + T - L T ) i s independent of GT w i t h a law not depending on the choice of T, as r e q u i r e d . Thus, except f o r the proof of Lemma 7, the proof of p a r t (a) i s complete. proof o f Lemma 7; Choose 6,+0 w i t h n(a >6\ ) >0 f o r each k. For k a k q e 5, q > o, l e t S = i n f {s > q; a (Y ) > 6, }. q a s k The S^ are ^Fg) stopping times. By completeness of F, we may assume without l o s s of g e n e r a l i t y t h a t N({a > 6 } * [0,s], Y(w)) < °° cX K. f o r every aj,k and s, and t h a t N({CT > 6 } x [o,°°) ,Y(w)) = <*> f o r a jc every OJ and k. Thus, each S^  i s s u r e l y f i n i t e , and ( w r i t i n g tvj f o r the graph {(t,w);0 < t < <*>, t= V(w)}) - 31 -* ( V { a (Y ) > 6 > D C U m I S a > a LT k q e $ + A l s o , f o r any k,k',q,q' we have t h a t S k < S k,, i f k > k' and q < q', q q> { S k = S k!} £ F k i f k < k' q q' S£-Put R q = ( T - S - ( V ) k ^ T q t t « t ' q = F s * 8 U t q-By Lemma 4 (d), R k i s an (h^.^) stopping time. Use Lemma 5 w i t h ft. = ft, = f f ' q , ft = ft x U, F^ = ' q, Z(ui) = (to,Y k ( w ) ) , 1 t t+ ^ t . L .T" ~k cr ~k to o b t a i n an (tf ' ) stopping time R w i t h t+ q ~k k R (to,Y v(u))) = R (w) f o r each w e f t , q S£ q Let ^ Rk' (u>,u), i f k 1 > k, q' 6 £+, s^(«).'- S k!(w), and f o r every k M i k', q" e Q, k" k' such that S „ (CJ) =5 , (w), a l s o .1 q k" k' Rn„Cw,u) » R^,(w,u). R (<i),u) q , i f f o r every k 1 > k, q 1 e $ t h i s f a i l s to h o l d . - 32 -Thus, f o r every k, k' e 2 + and q, q' e <g+ we have t h a t ~k ^k' k k 1 R = R . on {S = S , } x U , and q q' q q' ~k ' - k R (u,Y_k(u)) = (T - S (L m))((o) f o r w e {L = S } . q S q T T q * k "• k cr We now show t h a t R i s an (H stopping time. Let t > 0, and q t+ choose an open base B^B,,, ... of the space Co,t). Then {R k < t} = u n f({S k," r S k} x U) u q k'>k k">k' q q q'€$ + q"e£ + u (({R k," e B.} n ( S k l ' = S k} x u) n 1>1 ]>i q" j q" q n ({R k] e B.} n {S k] = s k} x U))] q D q q k * k Since {S , = S } e F v whenever k ' > k , i t w i l l thus s u f f i c e t o Sq show t h a t H k' , q' n ({S k[ = S k} x u) c / £ ' q t q q- t f o r every k' > k, q' e J>+. This h o l d s , s i n c e by monotone c l a s s arguments, (F ck- 8 U.) n ({S k' = S k} x u) s„,- t q' q ( F o k , n {sk,' = s k}) 8 U S*,- q" q t ( F k n {S k i = S k}) 8 U V q t c F c k 8 U , . S q ~ t - 33 -Put R = A ( R k ) / T Jc, k,q ^ T = V X U ' ~k k Then R = R on {L = S } x u. We w i l l show t h a t R i s an (H ) q T q t+ stopping time. Let t > 0, k e Z-+. Then {R < t} n {a (Y T ) > 6, > x u = U [{R k < t } n { s k = L m } x u] q«B + q q As befor e , H k , q n { S k = Lj x u t q T = H. n { S k = L m} x u. t q T Since a l s o { S k = L T> = {q < L T ^ S k} n ( o M Y ^ ) >6 k} , we get t h a t f o r each k, {R < t} n {a Y T ) > 6. } x u a L„ k T " "k 0 { a a ( Y L > " V X U ' T f o r some s e t I i e H . Using t h a t a (Y ) > 0 on {T < °°}, i t i s k t a L T easy t o see th a t i n f a c t {R < t } = [ n u H„ ]\{T = «,} x u e H , k>l £>k K-as r e q u i r e d . P r o p e r t i e s (3.4) and (3.5) are now immediate. (3.3) w i l l f o l l o w - 34 -from (3.5), (3.6), and the convention t h a t *>/<*> = 0. Thus a l l t h a t remains i s t o show (3.6). Let A e F . By the Markov property o f Y at r , and Theorem 4 . 4 A of I t o [32], P(Y k e B, S > r,A) S q q P(A,S > r ) P ( Y k q S (q-r) vo e B) D , k • n ( B ' C T a > V  = P ( A ' S q > r ) n(o > 6 J f o r B £ U . That i s , f o r f = 1, . A £ F „ > _ , B e U we have A x B SK q E [ f ( ' ' V ( ) ) ] = n 7 o - l ^ T q a k f (to,u)n (du) )P (dw) { a >5. } a k Therefore t h i s holds f o r f e F k 8 U , f > 0. Take f(tu,u) = 1 ( co)l ( u ) l (R ( c o,u))l . C A B q , k , ^ {R < «} q (w,u) , where C e F k , A £ U , B £ 5. Then q P(C,Y £ A, T - S"(L ) £ B, L = S k) = E [f (• ,Y k (• ) ) ] L T T T q S£ k k r n(A,R (u>,*) £ B, R (u,*) < ° ° , a > 6 , ) 1 q q_ a k_ n ( a > 6 . ) a k P(du) P ( L T = S a | F S k - ) ( a ) )  H q k k n(A,R (CJ,')£B,R ( c j , « ) < ° ° , a >6 , ) q q a k k n (R (co, • ) <°° ,a >6 , ) q a k P(du) Cn{L=S*} T q k k n(A,R (u,*)eB,R (u ) , « ) < ° ° , a >6 , ) 3 g a k k n(R (OJ,« ) < ° ° , a ><5, ) q a k P(dco) - 35 -Enumerate <D as q, , a , 1 ^ 2 , and l e t C e F V Then there are C e F v such t h a t q S K -q Thus c n { L =s k) = C k n { L = s k} T q q T q P ( C , Y T e A, T - S ( L ) e B, a (Y_ ) > S. , T < °°) LT T a LT K I P ( C k Y L £ A, T - S ( L T ) £ B, L T = S k S k r S k f o r i < j ) j T 4 j i H j = 1 n(A,Rg. (a), •) £ B,Rg. (<D, • ) <°°/0- >6fc) ^• j a k C k n{s k = L } n { S k ^ S k f o r i < j ) D q j T q ± q_. P(dto) , k k , as {S S } £ F_k q j q i S q 3 - I n (A,R((JJ , •) £ B, R(u,') < c o a >6 ) a, jc n (R(w, •) <°°,a >5, ) a k C n { S ^ = L T > n { S * fi S * f o r i < j} P (dto) n(A,R(o),')£B,R(a), ' )< O T ,a >6 ) a k P(do) . N(R(OJ, •) <°°,CT >6, ) a k C n { d (Y )>6, } a L T k I f A i s chosen so t h a t n (A) < °°, then the integ r a n d converges boundedly t o n (A, R (a), •) £B, R (to, •) <°°) n (R(u>, •) < °°) - 36 -Thus P ( C , Y T e A , T - S ~ ( L ) e B, T < °°) T n(A,R(ui,Q e B, R(o), - ) < °°) n (R(w, •) < °°) P(du) whenever n (A) < ° ° . S i n c e n i s a - f i n i t e , t h i s h o l d s f o r e v e r y A £ U, w h i c h y i e l d s (3.6). • P a r t (a) i s p r o v e n . The p r o o f o f p a r t (b) i s s i m i l a r . F o r U 0 ' U 1 ' U 2 ' "'" € U ' p U t i<k k > 0 F ( u , u l f . . .) (t) = U k ( t - S k ( U 0 ' U l ' - J ) ' l f V W ' ^ < \ + l ( u 0 ' U V ) , i f t>S ( u n , u . , . . ) oo 0 1 P u t X. = F (Y , Y. , . .) ( t ) , S(k) = S, (Y , Y 1 , . . ) t 0 1 k o J. F o r A £ Uf p u t V F ( u ' v i ' V P (A) = U U : X J x . . . ) ) P ( ( Y , Y 2 , . . ) £ d ( v x , v , . . ) ) P q ( d u ) , i f b^a l A ( P ( u f v l f v 2 f . U U x U x . . .) )P ( (Y ,Y , . . ) £ d ( v l f v 2 , . .) )n (du) , i f b=a . We l e t F_]_ be t h e s e t o f P - n u l l s e t s o f F . F o r R a random v a r i a b l e w i t h v a l u e s i n {-1,0,1,2,...}, F i s d e f i n e d t o be t h e a - f i e l d R g e n e r a t e d by s e t s i n F^ n {R = r } , f o r -1 < r < » . F o r R a non n e g a t i v e i n t e g e r v a l u e d random t i m e , we p u t - 37 -C= FR-1< " t = FR-1 V ° ( Y R ( S ) ; For r > 0, l e t L = in f { k ; S ( k + l ) > r } . We put r Since n(u\u a ) = 0, we have t h a t S(k)-><», a.s., and t h a t f o r each k > 1, S(k) < °°, S ( k - l ) < S ( k ) , and X . . = a, a.s. Delete a S Ik) n u l l s e t of P. t o ensure t h a t these statements h o l d s u r e l y . The d i s c r e t e - t i m e analogue to Lemma 4 i s ; Lemma 4' (a) r^-9nt continuous, i n c r e a s i n g , and each G f c contains a l l the P - n u l l sets o f F . X e G f o r each t > 0. (b) I f T i s a (G ) stopping time, then L i s an (F ) stopping time, and F , c G c F . I f f u r t h e r X f a on {T < <*>}, then LT T T F c G . V 1 T~ (c) S(k) i s a (G^) stopping time, f o r k > 0. I f T i s a (Gfc) stopping time such t h a t X T = a on (T < <»}, then 1 => ^T-* T (d) I f R i s an (F ) stopping time, and T i s a (G ) stopping time, put v = T - S(R) on {L = R < °°} T °° otherwise P Then V i s an (H t +) stopping time. proof; The proofs of p a r t (a), and t h a t f o r T a (Gfc) stopping time, L i s an ( F ) stopping and G c F , are as i n Lemma 4. F u r t h e r , T K T - 38 -FL _! " (T < t} = F n { L - 1 < L - 1} N { T < t} T T T C F L -1 n { L T " 1 ~ L t " 1 } n { T K t } = F L f c-1 " (T < t} c Gt, so t h a t F„ , c G . L -1 T T For k > 0 we have t h a t (S(k) < t} = (k < L. } e F T . <= G , t XJ^~ J. t so t h a t S(k) i s a (Gfc) stopping time. Applying the previous computation, we see t h a t F k C G S ( k + I ) " Let T be a (G^) stopping time such t h a t ^ a on {T < »}, and l e t A e F, . Then k A n ( L T - 1 > k} = A n { L T > k + 1} = A n {T > S(k + 1)} £ G S ( k + 1) " { T > S ( k + 1 ) } £ GT- • Now l e t T be a (G ) stopping time such t h a t X = a on {T < °°}. Then t T {T >r} = {L^ > 1^}. By monotone c l a s s arguments, G^  i s generated by and by T G n {T > r} c F_ n { L M - 1 > L } c F r L T r L -1' r T f o r r > 0. Thus p a r t (c) i s shown. Pa r t (d) f o l l o w s as i n Lemma 4. • - 3 9 -The d i s c r e t e time v e r s i o n o f Lemma 7 i s Lemma 7' Let T be a (Gfc) stopping time such t h a t X Q = a {L = 0, T < «}. Let H = F 8 U . Then there i s an (H ) stopping time R such t h a t on ( 3.8} R(U),Y T ( t o ) ) = (T - S ( L J ) (co) i f T (o i ) < < T ( 3.9) R((o,u) = °° f o r every u e U i f T(to) = 0 0 (3.10) P(Y T e A, T - S ( L J £ B, T < 001 F ) (<o) L_ T L - X T T _ n{u;u e A, R(u),u) £ B, R(to,u) < »} n{u;R(a),u) < o=} f o r P-a.e. <u, , where A £ U, B e 8, and we take the convention t h a t 0/0 = 0. proof: For k > 0, l e t R k = (T - S(L )) { L T = k > = F 8 11 t k-1 t By Lemma 4'(d), R k i s an stopping time. Use Lemma 5 w i t h ^1 = n ' F t = Q2 = Q * U ' F t = ^t+ 3 1 1 5 Z ( u ) = ( t l ) ' Y k ( w ) ) ' t o o b t a i n an (h^.+) stopping time R such t h a t f o r every to £ ft, ^k k R (<o,Y ( t o ) ) = R (to) . Let R = A (R k) k { L T = k} x u . Thus f o r t > 0, - 40 -{R < t} = u {R k < t} n {L = k} x u . k t By monotone c l a s s arguments, n { L t = k} x u = ( F k - 1 n { L t = k}) 8 U t c F. L -1 8 U t = tff T so t h a t R i s an (^t+) stopping time. P r o p e r t i e s (3.8) and (3.9) are immediate. For k > 1, Y i s independent of F w i t h law n, so that E [ f ( - , Y (•))] = ( f(u,u) n (du) P (du) K J J f o r f = 1 , A e F, - , B e U. Thus t h i s holds f o r f e F, , 8 U, A XB K-X k-1 f > 0. As i n the proof o f Lemma 7 i t f o l l o w s from t h i s t h a t f o r k > 1, C e F k _ x F A e U, and B e 8, we have t h a t P(C,Y L e A, T - S(L T ) e B, L^ = k, T < °°) Cn{L T=k} k k n(A,R ( c D , » ) e B, R ( t o , •) < o ° ) n (R k ( u , •) < 0 0 P(dto) We w i l l show t h i s f o r k = 0 as w e l l , as then the argument of Lemma 7 w i l l give (3.10). Let E b r = { b e E : P " ( W n = b ) * 1 } • 0 0 .c _b Since ^ w t'^' P 0' P 0) i s strong Markov f o r each c e E, we get t h a t P (W e E ) = 0, f o r every c e E. o o br 1 By h y p o t h e s i s , P(Y € M) = o PU ( o ) (M) P(Yedu) + n(M) P(Y e U 3 ) , , o o U\U - 41 -so t h a t as a i and n (u \u ) = 0, we see t h a t P (Y (o) e E ) = 0 o br Thus, P(Y eMnU ) = o Pu ( o ) ( M n U a ) P(Y edu) + n(M) P(Y eU a) o o o U\U' = n(M) P(Y eU ) o Since F ^ c o n s i s t s of sets of P-measure 0, i t f o l l o w s t h a t f o r f e F 0 U, f > 0, E [ f ( - , Y (•)), Y eU ] = E[ o o f(«,u) n(du)] P(Y eU ), o Taking f as before and using the hypothesis t h a t X q = a i f L^ we see t h a t f o r C e F ^ , A e U , B e 8 , P(C,Y e A, T - S(L ) e B, L = 0, T < °°) L T T T = E(C,n(A,R°eB,R0<°°)] P(Y eTja) o E(C yn(A,R 06B,R°<°°)] E[n (R°<°°) ] P ( L T = 0) n(A,R°(to,Q £ B, R°(OJ,')<°°) P(du)) Cn{L =0} T n (R (a),«)<°°) • Once the f o l l o w i n g r e s u l t i s e s t a b l i s h e d , the remainder of p a r t (b) f o l l o w s immediately, as before. C o r o l l a r y 2' Let T be a (G^) stopping time, and A e U. Then P(Y T (• + T - S(L )) e A, T < °°) Lfc T = E[P T (A), X M ^ a, T < °°] + n(A) P ( X M = a, T < °°) . O T T - 42 -proof: As b e f o r e , i t s u f f i c e s t o t r e a t three cases; t h a t on {T < <»}, r e s p e c t i v e l y = a, ? a and L^ > 0, or X T f a and L^ = 0. The second case i s handled as b e f o r e , using Lemmas 7' and 6. In the t h i r d case, we use Lemma 4' (d) to see t h a t T i s an (h^!+) stopping time. Since (^ +) a completion of the n a t u r a l r i g h t continuous f i l t r a t i o n of the process X , ,, and * . AT (X ) cl P(Y eM) = o U \ lf P Q ( 0 ) (M) P ( Y Q e d u ) , f o r M e U, M n = 0 i t i s simple t o use the strong Markov property of the processes c b (W . U . ,P ,P ), f o r c e E, to conclude t h a t the process t t o o a i s s t r o n g Markov. This s u f f i c e s . In the t h i r d case, we apply Lemma 7' t o o b t a i n an R s a t i s f y i n g the c o n c l u s i o n s of t h a t r e s u l t . Put H(u) = {u;R(u,u) = 0}. Then f o r A £ U, P(Y £ A, T < °°) = L T n(H(u) nA) n(H(w)) , as T - S ( L T ) = 0 on {T < <*>}. Since R i s an ^' t +^ stopping time, a l s o R(OJ,») i s a (^,) stopping time f o r each u e Q, so that H(k') e li . Thus n | H ^ s a t i s f i e s ( i v ) , so t h a t by ( v i ' a ) , i t must be a m u l t i p l e of n. Since n(H(ui) ) = n | (H(co)) , we must i n f a c t have nj . = n or 0, so th a t n(H(w)) = 1 or 0 H vet)  f o r each u>. Thus - 43 -P(Y_ e A, T < °°) = n(A) P(T < °°,n(H) = 1 ) , f o r A e U. t P u t t i n g A = U, we have t h a t n(H) = 1 a.s. on {T < °°}, so th a t P(Y e A, T < °°) = n(A) P(T < °°) , L T as r e q u i r e d . • 4. I n s u f f i c i e n c y of c o n d i t i o n s ( i i ) and (vi) F i r s t , we present some examples t o show t h a t the c o n d i t i o n s ( v i ) and ( i i ) do not s u f f i c e . Example 1 : ( v i a) i s not s u f f i c i e n t : Let P Q correspond to uniform motion as i n d i c a t e d , w i t h a b s o r p t i o n at a. There are two excursion measures n^, n^ » fo-corresponding t o strong Markov processes F i g . 1 " l + n 2 v i s i t i n g a d i s c r e t e l y . s a t i s f i e s ( v i a ) , and gives a Markov process which v i s i t s a d i s c r e t e l y , but i s not strong Markov. Example 2: ( v i c) i s not s u f f i c i e n t : Consider a B e s s e l process on (o,°°) (so t h a t 0 i s an entrance, non-exit p o i n t ) , and make the p o i n t 1 absorbing. Wrap (0,1] around to make a c i r c l e E, and l e t P q correspond t o the r e s u l t i n g process on E. Let a = { l } . P q corresponds t o a continuous process t h a t i s absorbed a t a, but which approaches a only from the counter clockwise d i r e c t i o n (^ j^) F i g - 2 3 i We have s t r o n g Markov continuous r e c u r r e n t extensions w i t h a - 44 -instantaneous, corresponding to making the Bes s e l process (slowly) r e f l e c t i n g a t 1, w i t h various delay c o e f f i c i e n t s . Let the ex c u r s i o n measure w i t h delay c o e f f i c i e n t m be n , so t h a t m m = 1 - (1 - e ° a) dn m There are a l s o continuous 'recurrent extensions' which are not s t r o n g Markov, corresponding to stopping the o r i g i n a l process a t a, h o l d i n g i t there an exponential time, and then making i t enter E\{a} i n the counterclockwise d i r e c t i o n ( t h i s i s p o s s i b l e , as 0 i s an entrance p o i n t f o r the Bes s e l p r o c e s s ) . For m e (0,1), t h i s gives an 'excursion measure" n 1 such t h a t m -a. m = 1 - (1 - e a ) dn m (m determines the mean of the h o l d i n g time a t a ) . Though c o n d i t i o n ( v i c) f a i l s f o r the n' , i t w i l l h o l d f o r any m measure n + n' f o r which m + q > 1 and m q e (0,1) m q * These measures are r u l e d out by ( v i ' c ) Example 3: ( i i ) does not s u f f i c e : Let E be the subset of 3R2 d e s c r i b e d i n p o l a r coordinates as u { ( r cos 0, r s i n 0}; r = R cos 0 > 0} Re (1,2] Let a be the o r i g i n , and l e t ( P Q ) correspond to uniform clockwise motion around the c i r c l e s r = R cos 0, a t speed On-Rd-R)) \ w i t h absorption a t a. F i g . 3 - 45 -For y any a - f i n i t e measure on E\{a}, n = s a t i s f i e s ( i ) , ( i v ) , ( v ) , and ( v i ' c ) . Let n be the ex c u r s i o n measure of one dimensional o Brownian motion, from 0, and l e t M(u) = max { |u(t) |; 0 < t < <*>}, u e U. Let f:E -* (0,1] x (0,2ir) be given by f (r cos 0, r s i n 0) = ( — - 1 , ^ - 0 ) . cos 0 2 Then f o r y(A) = n ((a ,M) e f ( A ) ) , we get p r o p e r t i e s ( i i ) and ( i i i ) O 3L f o r f r e e , but ( i i ' ) f a i l s , so t h a t the r e s u l t i n g process i s not r i g h t continuous. We can rep l a c e ( i i ' ) by the more appealing c o n d i t i o n ( i i ) , p r o vided we assume some r e g u l a r i t y of (P ); o P r o p o s i t i o n 2 (a) Suppose t h a t (P b) s a t i s f i e s (2.4) and (2.5), and t h a t n s a t i s f i e s ( i ) , ( i i i ) , ( i v) and (v). Suppose a l s o t h a t (4.1) f o r every open neighbourhood V of a, there i s an open neighborhood V of a, V c V , such that P b((W ) leaves V) o t SUP — < <»> beV' E b ( l - e " a a ) o (with the convention t h a t -^ = 0 ) . Then ( i i ) holds i f and only i f ( i i ' ) does. (b) Conversely, suppose t h a t f o r every i n i t i a l measure u w i t h u{a} = 0 and E ^ t l - exp(-a )] < 1, the measure i s the ex c u r s i o n o a o measure of a r i g h t continuous process. Then (4.1) holds . proof: (a) Let V,V be as above, and assume t h a t ( i i ) h o l d s . Then - 4 6 -n((W )leaves V) < n(W i V ) + n(W I V , (W ) leaves V) t o o t The f i r s t term i s f i n i t e by ( i i ) , and the second term i s n(W Q e V'\{a},(W ) leaves V) + l i m n(W = a, W e V f o r t e [ 0 , 6 ] , a > 6, ( w j leaves V) 5+o o t a t P ^ ( 0 ) ( ( W t ) leaves V)n(du) + l i m {W eV\{a}} 64-° P o ( 6 ) ( ( W t ) leaves V) n(du) {W0=a,Wt€V» f o r t e [ 0 , 6 ] , a >6} < sup P Q((W f c) leaves V) b e V E b ( l - e " a a ) b^a ° E ^ ( o ) ( 1 _ e - 0 a , n ( d u ) + {W eV'\{a}} o + l i m i n f 6+0 E «<«> ( 1 . e " ° a ) n(du) {W =a,W eV f o r t e [ 0 , 6 ] , 0 > 6} sup P ((Wfc) leaves V) beV E b ( l - e ~ C a ) b*a ° ~ [ -a. (1-e a ) dn + l i m i n f 6 +0 {W eV'\{a}} o ( l - e - a a - 6 ) dn {WQ=a,W eV f o r t e [ 0 , 6 ] , a > 6 sup P ((W ) leaves V) f b e V E b ( l - e °a) b^a 0 (1-e a ) dn < (b) Assume (4.1) f a i l s f o r some open neighborhood V of a. Then there are b e E\{a}, b a such t h a t k k P k((W ) leaves V) o t  E b k(l- e-° a) o By p a s s i n g to a subsequence, we may assume t h a t a > k f o r each k. - 47 -Let X, = ( k 2 E bk ( l - e ' ^ ) ) " 1 K. O k=2 k (where e i s the p o i n t mass concentrated a t b) ( l - e - a a ) = I k" 2 < 1, k=2 Then wh i l e P^((W t) leaves V) = £ ^ a ^ E ^ k d - e " ^ ) k=2 ^ I * k=2 -1 = 00 • The corresponding c o n d i t i o n on (p b) under which ( v i ' ) may be o re p l a c e d by ( v i ) i s t h a t the c l a s s o f p o s i t i v e measures n s a t i s f y i n g ( i ) , ( i i i ) , ( i v) and n(U\U a) = 0 c o n s i s t s o f e i t h e r m u l t i p l e s of a s i n g l e p r o b a b i l i t y measure, or c o n s i s t s completely o f i n f i n i t e measures. I t i s perhaps worth mentioning t h a t though t h i s c o n d i t i o n f a i l s f o r example 3, (4.1) i s not i n gene r a l a consequence o f t h i s c o n d i t i o n . As example we can replace the space E of example 3 by E' = E n { (x,y) ; y < v^2x} . In t h i s case there are no measures s a t i s f y i n g ( i ) and ( i v ) , and concentrated on U , sin c e no path r = R cos 0, R e (1,2] l i e s e n t i r e l y w i t h i n E 1 , whereas by P r o p o s i t i o n 2 ( b ) , there do e x i s t n s a t i s f y i n g ( i ) , ( i i ) , ( i i i ) , ( i v ) , ( v ) , ( v i ' ) but not ( i i 1 ) . - 48 -5. Proof of P r o p o s i t i o n 1 C o n d i t i o n ( i i 1 ) i s c l e a r l y necessary. Assume t h a t a i s instantaneous (n(U) = «>) , but ( v i ' c ) f a i l s . We w i l l f i n d a s e t H e U such t h a t o < n (H) < °°, and H c u an {a > o}. Let H° e U° o+ a o+ s a t i s f y H c H and n(H\H° )=0 . By completeness of each F , we o b t a i n e a s i l y that f o r h > 0 , the (U.lf?) -valued process (X ) i s h r t+* p r o g r e s s i v e l y measurable, f o r the f i l t r a t i o n (F , ) n . Thus, t+h t O T = i n f { t > o; e H°} i s an (F^ , stopping time, f o r every h > 0 , so t h a t i t i s i n t+h+ t>o J f a c t an ^^ +^ stopping time. T i s f i n i t e almost s u r e l y s i n c e 0. 0 0 n(H ) > 0 , and X T + . e H a.s., s i n c e n (H ) < °°. This c o n t r a d i c t s the strong Markov pr o p e r t y of (X f c,F t,P,p b) , as a i s instantaneous, y e t X = a and a (Y ) > o. T a L„, T S i m i l a r l y , i n the case t h a t X v i s i t s a d i s c r e t e l y , but (vi'a) f a i l s , we w i l l o b t a i n a c o n t r a d i c t i o n t o the s t r o n g Markov property of X by f i n d i n g a s e t H e U such t h a t H c u&n {o > o) and o+ a o < n(H) < 1. In both cases, the argument would be s i m p l i f i e d i f we had assumed t h a t X was a r i g h t process, and we had the Ray-Knight c o m p a c t i f i -c a t i o n at our d i s p o s a l . In the f i r s t case, l e t D be the s e t of dyadic r a t i o n a l numbers. For e, 6 > o l e t B . = {beE; P (a > 6) > e} e E e, 6 o a H = u {W = a, W e B . f o r t e D n (o , n ) , a >0 } . e , 5 n > Q o t e , 6 a - 4 9 -We w i l l show t h a t f o r some e,6 > o we have o < n(H J < °°, e, o so t h a t H . s a t i s f i e s the above c o n d i t i o n s , e, o For e and 6 f i x e d , l e t x (u) = i n f { t > o; t e D and u ( t ) 4. B „}. e ,6 Then T i s a ^^+^ stopping time. For p e D n (o,°°) , we have {x > p } e ^ p + / so t h a t by Lemma 6, n(a > S) > n(x > p , a > p + 6) a a P u ( p ) (o > 6) n(du) o a {x>p } > e n (T > p ) . L e t t i n g P 4- o, p e D we ob t a i n t h a t n (a > 6) > en (H „) . Thus a e ,6 n(H ) < oo f o r every e,6 > o. E f 0 Conversely, s i n c e ( v i ' c ) f a i l s , there i s some measure n' s a t i s -f y i n g (iv) and (v) such t h a t n > n' > o, and o < n' (U a) < °°. F i x e 6 > o such t h a t n(H J = o. Then a l s o n'(H J = 0. Thus, f o r n > o, n' (W = a, W e B . f o r every t e (o,n) such t h a t o t e, 6 -k t = j 2 f o r some j ) + o as k That i s , [ 2 k n - l ] l ± n ' ( W o = a ' W j 2 " k * B e,6, W i 2 - k e B e,6 f o r 1 * 1 <J > tn' (U 3) as k -*• - 50 -Thus, f o r n e ( 0 , 6 ) , n' (W = a, (j > 25) O cl [ 2k n - l = l i m I k->~ j = l W i 2 " k e B e , 6 f o r 1 < i < j ) [ 2 k n - l ] = l i m I k-*=° j.=l P U ( : j 2 } (0 > 26 - 12 k ) n(du) Wi2-k € B | e , 6 f o r 1 < i < j } < (1-e) n 1 (U a) . a Since n'(U ) e (o,»), t h i s cannot happen f o r every e,6>o, so t h a t indeed n(H ) > o f o r some e,6. In the second case, suppose t h a t n i s a p r o b a b i l i t y measure concentrated on U a, y e t (vi'a) f a i l s . Then we may f i n d V- L»y 2 e (0,1) and p r o b a b i l i t y measures n^ and n^, each concentrated on u a and s a t i s -f y i n g ( i ) , ( i v ) and (v) , such t h a t n 1 r and n = V^11^ + U 2 n 2 * Since n^ 7* n^, we o b t a i n from Lemma 6 t h a t there i s an open neighbourhood V of a, a s e t A e E w i t h A <= E\V; and numbers X.,,X0 such t h a t where x (u) = i n f { t > o; u(t) ji V}. Let V be an open neighborhood of a, w i t h V' c v. Let l' " 2 1~ B = {b e V ; P (W e A) > X,} . o x 1 Let D be the dyadic r a t i o n a l s , and put - 51 -H = u {W = a, a > o, W. e B f o r every t e D n (o,n)} o a t ri>o Then as before we o b t a i n t h a t X_ > n (W e A) > X,n (H) , £ 2 T 1 2 so t h a t < ^ 2 ^ 1 < ^' n e n c e n(H) = y in 1(H) + y 2n 2(H) < u + u 2 = 1. Conversely, i f n(H) = 0 then a l s o n^(H) = 0, so t h a t as b e f o r e , X. < n (W e A) < X Tn (U a) = X, , 1 I T 1 1 1 which i s impossible. • 6. Ray and Right Processes Let (P ) s a t i s f y (2.4), and put U a f ( b ) = E b [ e ^ f (Wfc) d t ] , f o r f e fc, a > o. (P ) i s s a i d t o be Ray i f E i s compact, and i f ct f o r each a > 0, TJ f i s continuous whenever f i s . P r o p o s i t i o n 3 Let (p b) be Ray, and s a t i s f y (2.5). Let (p b) s a t i s f y (2.4) and (2.6). Then (P ) i s Ray. proof: U a f (b) = E b [ 3 e a t f (Wfc) dt] + E b [ e " a T a ] U a f (a) U a f (b) = E b [ o o 3- —CIT 1 b —a. T e f(W f c) dt] + ± E Q [e a] f ( a ) Thus U a f ( b ) = U a f(b) + E b [e a T a ] ( u a f ( a ) - a 1 f ( a ) ) , o o and so we need only show t h a t E [exp(-ax )] i s continuous i n b. o a - 52 -Using p a r t i t i o n s of u n i t y , choose continuous f u n c t i o n s f on E n such t h a t 0 < f < 1 and n E\{a) n=l ct Each U i s continuous, so t h a t by dominated convergence so i s U o 1 E \ { a } ' w h e n e v e r ct > 0. But T , hr —T u"1*, r n (b) = E b [ o E\{a} o L P r o p o s i t i o n 4 Let (P ) be Ray, and s a t i s f y (2.5). Suppose a l s o t h a t p b ( U a ) < 1 f o r each b r a. Let n be a p o s i t i v e measure on (U,U) s a t i s f y i n g ( i ) , (iv) and (v) , and suppose t h a t n (a > 6*) < <*> f o r every c l 6 > o. Then n s a t i s f i e s ( i i ' ) . I f n (U 3) > o then n a l s o s a t i s f i e s ( v i ' c ) . Remark: The c o n d i t i o n t h a t n(o > 6) < °° f o r 6 > o w i l l be s a t i s f i e d a provided ( i i i ) h o l d s . In p a r t i c u l a r , i n the above s i t u a t i o n , there are no ' d i s c r e t e - v i s i t i n g ' . extensions of ( P b ) . o proof: Suppose n(U 3) > o. I f n i s f i n i t e , or ( v i ' c ) f a i l s , then there i s a f i n i t e non zero measure n' concentrated on U 3 which s a t i s f i e s ( i v ) , so t h a t u (A) = n' (Wfc e A) ID ID d e f i n e s a bounded system of entrance laws f o r (P ). Because (P ) i s o o Ray, there i s a f i n i t e measure U q on E such t h a t u f c = P Q 1 ° ( W e ( • ) ) . Thus n' = P^°, which i s i m p o s s i b l e , as n' i s concentrated on U 3. Thus ( v i ' c ) h o l d s , and (vi'a) i s vacuous. a . 1 - E [e a ] e - a t dt] = 2 !_ D a - 53 -To show ( i i ' ) , observe as i n P r o p o s i t i o n 3, t h a t E [ l - e x p ( - a ) ] O 3. i s continuous i n b, hence V = (b; E b [ l - e ~ a a ] < e ) , e > o e o define a nested f a m i l y of open neighborhoods of a. Since P b (ua) < 1 f o r b ? a, t h e i r i n t e r s e c t i o n i s {a}. Thus i f W i s any other open neighborhood of a, {W} u {E\V ; e > o} e forms an open cover of E. Since E i s compact, t h i s shows t h a t the V E form a base of open neighborhoods of a. Thus, i t s u f f i c e s t o show tha t n ((W ) leaves V ) < °° t e f o r each e > o. F i x e > o, and l e t x(u) = i n f { t > o; u ( t ) i V } . e Then W i V on {T < °°}, s i n c e V i s open. A l s o , f o r b e E\V and X £ £ E 6 > o we have t h a t E < E b (1 - e"° a) < 1 - e" 6 + P b (a > <5) , o o a so t h a t by lemma 6, 0 0 > n (a > 6) ^ n(T < °°, a > 8 + x) a a U ( T ( U ) ) P (a £ 5) n (du) o a {T<°°} > (e - (1 - e " 6 ) ) n (x < «>) . Choosing 6 s m a l l , we o b t a i n t h a t n (x < °°) < °° • - 54 -We could use t h i s r e s u l t t o v e r i f y the c o n d i t i o n s of Theorem 2 f o r the measures n considered i n [ 4 ] . b ci The c o n d i t i o n t h a t P (U ) < 1 f o r each b ^ a r u l e s out the o f o l l o w i n g pathology: Example 4 : Let E = [ 0 , 1 ] x { 0 , 1 } , a = ( 0 , 0 ) . Make a absorbing, and on [ 0 , 1 ] x { 1 } l e t ( pk Q) correspond t o Brownian motion, r e f l e c t i n g at ( 1 , 1 ) , w i t h ( 0 , 1 ) a branch p o i n t t o a. On ( 0 , 1 ] x { 0 } , l e t ( P Q ) correspond t o a Brownian motion r e f l e c t e d a t ( 1 , 0 ) , except t h a t there i s a jump from ( x , 0 ) t o ( x , l ) a t r a t e g ( x ) , where g(x) ->• 0 0 f a s t enough as x H so t h a t P B ((W ) h i t s [ 0 , 1 ] x {1}) = 1 o t f o r every b e E\{a} . Then we can f i n d u on E\{a} such t h a t n = P ^ s a t i s i f e s ( i ) , ( i i ) , ( i i i ) , ( i v ) , ( v ) , and ( v i ' ) , but not ( i i ' ) , o even though ( P B ) i s Ray. We now con s i d e r Right processes; Let E be a U-space, t h a t i s , a u n i v e r s a l l y measurable subset of some compact m e t r i c space. Let E be the a - f i e l d of i t s u n i v e r s a l l y measurable subsets. Let (ft,F) be a -b measurable space, and l e t ( P ) be a f a m i l y o f p r o b a b i l i t y beE measures on (J2,F), such t h a t P ' ( B ) i s E-measurable f o r each B e F. Let (Ffc) be a f i l t r a t i o n of (fi,F) which i s r i g h t continuous, and s a t i s f i e s F = n t u t where f o r each f i n i t e p o s i t i v e measure u on (ft,F), F^  denotes the a - f i e l d obtained from F by a d j o i n i n g a l l the p^- n u l l seta of the A l l P completion of F. These c o n d i t i o n s w i l l be assumed throughout the - 55 -remainder of the s e c t i o n . I f f u r t h e r X i s a r i g h t continuous process w i t h values i n E, which i s adapted t o (F ) • Ab .. • P (X = b) = 1 f o r each b e E (6.1) ^ o P b (x, _ e B [ F. ) = p X f c (X e B) , P b-a.s., f o r t+h t h each b e E, B e U and t , h > o (6.2) f (X.) i s P - a.s. r i g h t continuous, f o r each a > o, f i n i t e p o s i t i v e measure u, and a - e x c e s s i v e f u n c t i o n f. Then (X^F, Ffc,P ) i s c a l l e d a r i g h t process. Under c o n d i t i o n (6.1), we w i l l w r i t e p b(B) = P b(x.eB) , f o r b e E, B e U. F o l l o w i n g Sharpe [4'6] we c a l l a process (Z ) , adapted t o (F ), n e a r l y o p t i o n a l i f i t i s o p t i o n a l w i t h respect t o each f i l t r a t i o n (F^) . A f u n c t i o n f € E i s n e a r l y B o r e l i f f o r each u there e x i s t B o r e l f u n c t i o n s f ^ and f ^ on E such t h a t f < f < f ^ , and P U (f (Xt> r f 2 (X f c) f o r some t > o) = 0 . R e c a l l t h a t i n the presence o f (6.1), c o n d i t i o n (6.2) i s e q u i v a l e n t t o the c o n d i t i o n s (6.3) (X t,F t,P ,P ) i s st r o n g Markov f o r each c e E, (6.4) f ( X . ) i s n e a r l y o p t i o n a l , f o r each a > o and each ct-excessive f u n c t i o n f. A stro n g e r c o n d i t i o n than (6.4) i s (6.5) For every a > o, each ct-excessive f u n c t i o n i s n e a r l y B o r e l . - 56 -This p r o p e r t y has the advantage of being i n v a r i a n t under choices o f r i g h t continuous r e a l i z a t i o n s , hence i s a property of the t r a n s i t i o n b Ab b laws (P ). That i s , i f (X f c,F,F t,P ) s a t i s f i e s (6.1), and the (P ) are the t r a n s i t i o n laws of some r i g h t process w i t h a l l a-excessive "b f u n c t i o n s n e a r l y B o r e l , f o r a > o, then i n f a c t (X t,F,F f c,P ) has n e a r l y B o r e l a-excessive f u n c t i o n s f o r a > o as w e l l , so t h a t i t i s a r i g h t process. I t i s unknown (see Sharpe [4'6]) whether (6.4) (and hence (6.2)) are i n v a r i a n t i n the same manner. Suppose now t h a t Y i s a measurable f u n c t i o n (ft, F) -»- (II,P) such ~b t h a t w i t h respect t o each P , (Y f c,F t) i s a PPP w i t h c h a r a c t e r i s t i c measure n not depending on b. Define (p b) by b -b P (B) = P (Y e B) , o o f o r B £ U, and suppose t h a t ( p b j s a t i s f i e s (2.4), (2.5) and P b(W = b) = 1 , o o f o r each b e E. Suppose f i n a l l y t h a t n s a t i s f i e s ( i ) , ( i i ' ) , ( i i i ) , ( i v ) , ( v ) , and ( v i ' ) . I n the proof of theorem 2, we used the complete-ness of our f i l t r a t i o n t o d i s c a r d c e r t a i n subsets of ft. Under the present c o n d i t i o n s on (F^), these se t s w i l l s t i l l l i e i n F , so t h a t t o the proof of theorem 2 w i l l give a s i n g l e process ^ X t ' G t ^ a f a m i l Y b ^ c b of laws (P ), such t h a t f o r every c e E, (X f c,G t,P ,P ) i s st r o n g Markov, and P° (X e B) = P°(B) "b f o r each B e U. We can ask when (Xfc,F,Gt,P ) i s a r i g h t process, - 57 -By the above d i s c u s s i o n , the f o l l o w i n g gives a s u f f i c i e n t c o n d i t i o n f o r t h i s t o h o l d . P r o p o s i t i o n 5 Suppose t h a t f o r each a > o, the a-excessive f u n c t i o n s f o r (p b) are n e a r l y B o r e l . Then so are those f o r ( P b ) . p r o o f : Let f be a-excessive f o r (P*5) , f o r some a > o. Then f i s a-excessive f o r (P^) , so that by hypothesis, i t i s _3lso n e a r l y B o r e l f o r (P^) . „ . , , , . 0 Using (iv) and (v) we 1 2 1 2 can f i n d B o r e l f u n c t i o n s f and f w i t h f S f < f and n f f ^ W j f f 2(W ) f o r some t > 0) = 0 . t t I t f o l l o w s t h a t f i s n e a r l y B o r e l f o r (P ) , as r e q u i r e d . • U n t i l the q u e s t i o n i s s e t t l e d , of whether ( 6 . 4 ) i s i n v a r i a n t under choices of r i g h t continuous r e a l i z a t i o n s , i t w i l l be impossible t o show t h a t ( X ,F,G. , P B ) i s i n general a r i g h t process, even when the (p b) L. L. O a r i s e from one. We can show, however, t h a t t h i s i s the only o b s t r u c t i o n t o a proof of t h i s r e s u l t ; Assume t h a t the ( P B ) a r i s e from a r i g h t process. Then there i s a compact m e t r i c space E (the Ray-Knight c o m p a c t i f i c a t i o n ) , c o n t a i n i n g E as a u n i v e r s a l l y measurable subset, such t h a t the r e s o l v a n t b of (P q) extends t o a Ray r e s o l v a n t on E, s e p a r a t i n g p o i n t s thereon. F u r t h e r , any r i g h t process on E w i t h t r a n s i t i o n laws ( P ^ ) , i s almost o s u r e l y r i g h t continuous i n the topology of E. Replace U by the set U = {u; u i s r i g h t continuous: [0,°°) E, i n the E-topology, and i s r i g h t continuous i n the E topology at a l l times t such t h a t u ( t ) ^ a.}. - 58 -U and (U^) w i l l be as b e f o r e , w i t h U r e p l a c i n g U. The ( P Q ) ~b induce laws (P ) b e E on t h i s new U. Suppose we are given n and Y s a t i s f y i n g the c o n d i t i o n preceding P r o p o s i t i o n 5, but f o r our new o b j e c t s U and (P^). Then the same argument a p p l i e s t o a give a U valued process (X^), a f i l t r a t i o n (G^), and laws (P^) such t h a t A c b ~c b f o r each c e E, (X t , G f c,P ,P ) i s s t r o n g Markov, and P (X.eB) = P (B) f o r B e U. P r o p o s i t i o n 6 ( X t , F , G t , p b ) i s a r i g h t process. p r o o f : As i n Sharpe [46] w e need only show th a t U a f ( X t ) i s n e a r l y o p t i o n a l whenever a > o and f i s p o s i t i v e , bounded and E-measurable. Since E i s u n i v e r s a l l y measurable i n E, we need only check t h i s f o r f the r e s t r i c t i o n t o E of a p o s i t i v e , bounded, E-Borel f u n c t i o n , and hence by monotone c l a s s arguments, only f o r f the r e s t r i c t i o n t o E of a p o s i t i v e , E-continuous f u n c t i o n . The argument of P r o p o s i t i o n 3 shows t h a t f o r such f , U f i s the r e s t r i c t i o n t o E of an E-continuous / \ f u n c t i o n , so t h a t i n f a c t , we need only show t h a t X i s E - r i g h t continuous, P^ 1 - a.s., fo r y f i n i t e and p o s i t i v e . By the proof of P r o p o s i t i o n 4, there i s a base of E-neighbourhoods of a, such t h a t (6.6) n((VJ) leaves V ) < °° t e ** b ^ f o r each e > o. (Note t h a t s i n c e "the r e s o l v a n t of ' ( P ) on E o ^ *" b 3. separates the p o i n t s of E, the c o n d i t i o n t h a t P (U ) < 1 f o r each b e E i s s a t i s f i e d ) . In case ( v i ' b ) , t h i s s u f f i c e s , as i n Theorem 2. Suppose t h a t ( v i ' c ) h o l d s . Then {W = a, W i V f o r some sequence t, \ o} e U , o t, e k o+ k so t h a t by ( v i ' c ) , i t has n-measure zero. Thus - 59 -(6.7) n(W = a but W u. a i n E as t + o) = 0. o t r As i n Theorem 2, (6.6) and (6.7) s u f f i c e t o show the r e s u l t i n case ( v i ' c ) . Suppose f i n a l l y , t h a t ( v i ' a ) h o l d s . We use entrance laws, as i n the proof of P r o p o s i t i o n 4 t o conclude the analogue of (6.7); t h a t (6.8) There i s a p o i n t C € E\E such t h a t n(W = a, W^  4- c i n E as t \ o) = 0. o t Since c f a, (Xfc) w i l l hot be E - r i g h t continuous, however, the U a f ( X t ) w i l l be, s i n c e a i s an E-branch p o i n t t o c, so t h a t U a f ( a ) = u a f ( c ) • We can r e i n t e r p r e t ( v i ' ) i n terms of the Ray Knight c o m p a c t i f i c a t i o n ; b ^b P r o p o s i t i o n 7 Let (P ), E, and (P ) be as above. Let n be a = o o p o s i t i v e measure on (U,U) s a t i s f y i n g ( i ) , ( i v ) , and (v) (f o r E,U, and (P )) , and suppose t h a t n (a > 6) <- 0 0 f o r every § > o. Then (a) n (E - l i m W does not e x i s t ) = 0 t+o (b) Suppose n i s a p r o b a b i l i t y measure concentrated on U a = {u € U; u(o) = a}. Then (vi'a) i s e q u i v a l e n t t o (6.8). (c) Suppose n i s an i n f i n i t e measure, and n(u a) > o. Then ( v i ' c ) i s e q u i v a l e n t t o (6.7). proof: (a) C l e a r l y n (W r a, E - l i m W W ) = 0 , t-l-o so t h a t without l o s s of g e n e r a l i t y , we may assume th a t ntDXD3') = 0. As i n the proof of P r o p o s i t i o n 4, there i s a base of open E-neighborhoods V_ of a, such t h a t - 60 -n( (W ) leaves V ) < » f o r each e > o. F i x e > o, and l e t B = { f o r every 6 > o, W i V f o r some s e (o,<5)}. e s e Then u (A) = n (W e A, B ) t t e d e f i n e s a bounded system of entrance laws f o r ( P ^ ) , so t h a t , as u s u a l , there i s a f i n i t e p o s i t i v e measure u on E w i t h o p. (A) = P P° (W. € A) , t o t f o r t > o. Thus n i = P U° , lB o e so t h a t - • n (B , E - l i m W does not e x i s t ) = o, E t+o t f o r every e > o. But U\ u B = {E - l i m = a} . e , | t e>o t+o so t h a t (a) i s proven. (b) I t was shown i n the proof of P r o p o s i t i o n 6, t h a t (vi'a) i m p l i e s (6.8). Conversely, i f (6.8) holds f o r n, then whenever n > n' > o, i t a l s o holds f o r n'. Thus i f n* s a t i s i f e s (iv) as w e l l , a l s o n 1 = n" (U) P C = n' (u) n. o (c) : I t was shown i n the proof of P r o p o s i t i o n 6, t h a t ( v i ' c ) i m p l i e s (6.7). The converse i s an immediate consequence of p a r t ( a ) , and P r o p o s i t i o n 4. • - 61 -Note t h a t the proof of Theorem 2 could be shortened under the co n d i t i o n s of P r o p o s i t i o n 6; I t i s w e l l known t h a t a Markov process w i t h a Ray r e s o l v a n t , i s i n f a c t strong Markov. Thus, by the proof of P r o p o s i t i o n 3, i f we s t a r t w i t h a (P b) which i s Ray, then i t s u f f i c e s t o check the Markov property of the const r u c t e d process (X f c), a t d e t e r m i n i s t i c times t . This i s simp l e r because no exc u r s i o n can s t a r t a t t w i t h p o s i t i v e p r o b a b i l i t y , but as w e l l , because an e x p l i c i t k expression can be given f o r the times R . I f i n s t e a d , only the q c o n d i t i o n s of P r o p o s i t i o n 6 are v e r i f i e d , we c o n s t r u c t a r i g h t continuous process (Xfc) as i n Theorem 2. P r o p o s i t i o n 7(a) shows th a t (X f c) has an E - r i g h t continuous m o d i f i c a t i o n (X f c), t o which the argument j u s t given a p p l i e s , showing t h a t (X f c) i s strong Markov. Then (6.7) and (6.8), which are eq u i v a l e n t t o (vi'a) and ( v i ' c ) , are seen t o be e x a c t l y the c o n d i t i o n s r e q u i r e d t o i n f e r the strong Markov property of ( X t ) , from t h a t o f (X f c). 7. V a r i a t i o n s on Lemma 7 We t u r n t o the v a r i a t i o n s of Lemma 7 a l l u d e d t o i n s e c t i o n 2. Let E be a t o p o l o g i c a l space and E i t s B o r e l f i e l d . R e c a l l t h a t a f u n c t i o n K(x,dy) i s c a l l e d a k e r n e l , i f f o r each x e E, K(x,«) i s a p r o b a b i l i t y measure on ( E , E ) , and i f K(« ,A) e E whenever A e E . P r o p o s i t i o n 8 Let (ft,F,P) be a p r o b a b i l i t y space. Let (Y^) be a cadlag process w i t h values i n E and suppose i t i s adapted t o a f i l t r a t i o n ( F ) of ( f t , F ) . Let K(x,dy) be a k e r n e l on E. Let Q be a countable ordered s e t , and f o r each q e Q, l e t S be an (F. ) q t stopping time. Suppose t h a t the f o l l o w i n g c o n d i t i o n s h o l d : 62 I f q < q' then S < S , and {s < S ,} e F q q q q' s _ q P (Y g e A | F g ) = K(Y , A) a.s., f o r q e Q, A e E . q q~ q" Let T be an (F ) stopping time such t h a t [[TD c u I S 1 , qeQ q {T = S } e F g _ v a (Y g ), f o r qeQ q q Then there i s an F measurable random subset H of E (that i s , { (io,e) ; e e H(to)} e F T _ 8 E ) , such t h a t P (Y, e A | F ) = K ( V - H n A ) T T- ; ; a.S. K (Y T_,H) f o r any A e E . proof: Since (T c s } e F s _ v a (Y g ) , q- q there i s a f u n c t i o n f : ft x E-> {0,1}, measurable w i t h respect t o F g _ 8 E , w i t h q Put {T = S } = {(JO; f (o^ Y,, (co)) = 1} q q Sq f = V [f A 1, - ] q r (S = S } x E J r q r Then f e P 0 E , s i n c e q Sq-- 63 -{ f r A = S } x E ' 1 } = { f r = l } n { S q - S } x E q r H •«-e ( F _ 0 E ) n ({S = S } x E ) = ( F _ n {S = s }) 0 E sr " q r = ( F _ n {s = s }) 0 E S - q r q c F s _ 0 E , q as {S = S } e F . A l s o , f = f o n { s = S } x E , and q r S - q r q r {T = S } = { U;f- (tO,Y (u>)) = 1} . q q q Put f = V [f A l r - ] a q {S = T} x E J Because f = f on {S = S } X E , i t f o l l o w s t h a t f = f on q r q r q {S^ = T> x E . We w i l l show th a t f e F T 0 E . This f o l l o w s as above, once we show t h a t {S^ = T} e ^ j - * T o s n o w t h i s , observe t h a t s i n c e the are i n c r e a s i n g i n q e Q , and CTJ <= u US U , q * we can w r i t e {s = T} = u [{s > T} n {s = s } n n {s , < T } ] . Q r r q . r H r<q H r' <r But {S > T} n n {s ,< T} e F , r r' T-r' <r and {S = S } n {S < T} e F _ n {S < T} c F n {S < T} , r T- r so t h a t f o r r < q, { s r > T} n { s r = sa> n n { s r , < T} q r'<r r ' e ( F T _ n { s r < T}) n ( { S r > T} n n {s , < T } ) r' <r r = F n ({s > T} n n {s , < T } ) r' <r as r e q u i r e d . Let H (co) = {e; f (co,e) = 1} q q H((o) = e; f(co,e) = 1} . We have assumed t h a t E [g(-,xs (•))] = [ g(co,x) K (X g (co), dx)] P(dco) q-f o r g of the form l A x f i, A e F f i _ , B e E . This t h e r e f o r e extends t o general g e F s E , g > 0 . Take q 5 = ^XB * f q ' where A e F , B e E . Then - 65 -P(A, X T e B, = T) = E (g(',X g (•))] r [ I f (<o,x) K (X (co) , dx)] P(dto) A BS -q K (X„ (co) , B n H (co)) f S - q P(S =T\¥ )(co) q — — P(dco) J q S " K(X (co) , H (co) A q S - q A n K(X„ B n H ) a~' q S =T} S q ' q q Enumerate Q as q,,q_, , and l e t A e F . Then there are A e F„ such t h a t q S q -A n { T = S } = A n { T = s } q q q Thus P(A, X £ B) = I P (A , X e B, S = T, S ^ S f o r i < j ) q. T 3 3 q. q. i J " I I K ' ( X g _ , B n H )/K(X g _ H ) dP q.i j V j A n {S = T, S f S f o r i < j } q. q. q. q. 3 D i 3 = 1 K (X T_, B n H)/K (X T_, H) dp A f l { S = T, S ? S ; f o r i < j } q^ q, q, = E [A, K ( X m H n B 1-J. ] K(X T_^H) • - 66 -A s i m i l a r r e s u l t holds i n d i s c r e t e time. For c l a r i t y , the proof w i l l be given w i t h a d i s c r e t e s t a t e space, but the general v e r s i o n can e a s i l y be obtained by modifying the preceding proof. P r o p o s i t i o n 9 Let E be a countable s e t , and (X ) • an E-valued ' n n>l Markov chain w i t h t r a n s i t i o n p r o b a b i l i t i e s P(x;dy). Let T be a stopping time w i t h respect t o the n a t u r a l f i l t r a t i o n of ^ x n^- Then there i s an F_ , measurable random subset H of E such t h a t T - l P (X • B n H) * < X T e * l W " P (x T_ i ; H) ^ ^ r B C E. proof: Atoms of F , are of the form T - l A = { T = n , X. = a ... , X . = a .} . 1 1 ' n-1 n-1 Since T i s a stopping time w i t h respect t o the n a t u r a l f i l t r a t i o n , i t f o l l o w s t h a t f o r a e E, {T = n, X^ = a^ .. X V l = a n - l } n { X - a } = 9 or {x = a , .. , X = a X = a} •L 1 n-1 n-1, n Put H(u>) = {a ; A n { X T = a} f 0} , where A i s the atom of F T c o n t a i n i n g to. Thus, f o r every atom A o f F m . , T - l P (A, X m = a) = P (X = a. ... X . = a .) P ( X m . ; Hn{a}), on A T 1 1 n-1 n-1 T - l so t h a t P(A, X T = a) p ( x T _ x ; Hn{a}) P(A) _ P(X ;H) ' ° n A • - 67 -8. Regulation, Uniqueness and Construction of Point Processes This section deals with some general results that w i l l be needed later, when we discuss point processes of excursions away from sets (as opposed to points, as in the preceding sections). In general, these processes w i l l not be PPP's, but w i l l instead have property (8.7) below. The results we w i l l discuss w i l l be analogous to those found in Ito [32], in the PPP case. Our point processes w i l l be connected to a Markov process (X^) (the extraneous " ~ " is for consistency with the next section). Throughout this section, we could have assumed that (X^) was a right process with nearly Borel excessive functions (actually, we only need that the version of (X £) on the canonical space be a right process). In applications i t w i l l in fact be a time change of such a process, and Gzyl [28] has shown that such objects are right processes (the nearly Borel proviso follows from his proof) . Being unaware of this u n t i l just before completing the f i n a l draft, I have not made use of Gzyl's results. The relevant properties of right processes w i l l thus be specified, and proved directly when needed. The net effect i s that we show an apparently stronger form of Corollary 6, than i s needed for applications, and we work harder than need be in Theorem 3. The two relevant properties are called 'regulation' and 'strong regulation' in the following. The latter plays a role only in uniqueness results, while the former i s also used for existensc. See Weidenfeld [55] for an analysis of similar ideas appearing in a general study of time changes. (A) Hypotheses. F i r s t , some general definitions: Given two general measurable spaces (fi^,F^) i=l,2 , an (fi^,F^;^>F 0) kernel is a function n : J2 x F 2 -»• [0,»] such that n(io1 , •) is a measure on (^ 9 ,F 9 ) for fixed w eft , - 68 -and n(« jdu^) i s F ^ measurable f o r f i x e d du>2 e . I f ( f t , F ) i s a measurable space, and ( F ) i s a f i l t r a t i o n t h e r e o f , r e c a l l that the 'usual c o n d i t i o n s ' (with respect to a measure v ) s t a t e that F i s v-complete; F Q _ c o n s i s t s of the v - n u l l sets of F together w i t h t h e i r complements; ( F ) i s r i g h t continuous; and FQ F Q_ . I f now a a e A i s a f a m i l y of measures on ( f t , F ) , we l e t F be the v -completion SL SL of F . We l e t F N c o n s i s t of a l l the v - n u l l sets of F , together U— a w i t h t h e i r complements, and then put F a = F v F a  r t • t+ V ' This l e t s us form the augmented f i e l d s ( f o l l o w i n g Sharpe [46] ); I f F = F and f = f , we say that ( F , F F C ) i s augmented (with respect to the v ) . We w i l l i n any case use the n o t a t i o n ¥a, F 3 , F Q _ l a t e r , without f u r t h e r e x p l a n a t i o n . In t h i s context, we w i l l as usual say that a property holds almost s u r e l y ( a . s . ) , i f i t holds v -a.s. f o r each a . Note that ( F A , F 3 , v ) s a t i s f i e s the 'usual c o n d i t i o n ' , T— Si f o r each a . We w i l l now f i x a U-space F , together w i t h a m e t r i c d thereon. We d e f i n e objects rF,0 r F T TF F T TF E , fc , U , u , W , e t c . . . - 69 -as i n s e c t i o n 2 (we w i l l l a t e r have a second space E , and reserve the n o t a t i o n E^ , U , e t c . . . f o r the corresponding o b j e c t s , w i t h respect to E.) Now l e t Q X(du) be an F F F (F,E*; t f , 0*) Q k e r n e l , such that f o r every c e F , Q i s a p r o b a b i l i t y measure, and CwJ, U\, 0°, Q X) - - F F i s strong Markov. Let U , be the augmentations of U , under the Qy , u p o s i t i v e and c r - f i n i t e . R e c a l l that (K t) i s c a l l e d a p e r f e c t a d d i t i v e f u n c t i o n a l (PAF) , adapted to (U^) , i f i t i s r i g h t continuous and nondecreasing i n t £ 0 , i s adapted to (^t) »\ and s a t i s f i e s K = K + K o 9 f o r every t,s > 0 . I t i s c a l l e d pure jump i f Kfc i s the sum of the jumps of K . at times s < t . Regardless of whether t h i s h o l d s , we may always f i n d (U t) stopping times = , such that { t > 0 ; Kfc_ ^ Kt> = { T ] c ; k > 1} n [0,») . Since F i s a U-space, i t f o l l o w s from the proof of (13.4) of Getoor [23] , that f o r every p , F u (8.1) { ( t , u ) ; W t_(u) does not e x i s t } i s Q - i n d i s t i n g u i s h a b l e from the union of the graphs of countably many p r e d i c t a b l e (U ) stopping times. - 70 --u F Thus, f o r any t o t a l l y I n a c c e s s i b l e (U£) stopping time x , W^ _ e x i s t s Q y-a.s. Our main d e f i n i t i o n can now be s t a t e d . We say that a pure jump 0 x PAF (K^) regu l a t e s (Q ) , i f there e x i s t s a s t r i c t l y p o s i t i v e F F b e- E ® E such that the f o l l o w i n g c o n d i t i o n s h o l d f o r each y : (8.2) < » a.s., f o r each t ; K° - u (8.3) x, = x. i s (U ) - t o t a l l y i n a c c e s s i b l e , i £ & t and WF ± WF , K° = K° +b(W F ,WF ) Q y-a.s. on {x, «*>}. T k " T k T k T k " T k ~ T k k (8.4) I f x i s any ( U ^ ) - t o t a l l y i n a c c e s s i b l e stopping time, such that WF + WF Q y-a.s. on {x:< °°} , then K° ± K° Q y-a.s.. X- T T- T on {x < °°} . (Q ) i s s a i d to be s t r o n g l y regulated i f there e x i s t s a r e g u l a t o r (K^) f o r i t , and the f o l l o w i n g a d d i t i o n a l c o n d i t i o n h o l d s . (8.5) I f T i s any (Uy) t o t a l l y i n a c c e s s i b l e stopping time, then WF ^ WF Q y-a.s. on {T < °°} . T- T (so that the jumps of (K^) are e x a c t l y the t o t a l l y i n a c c e s s i b l e times w i t h respect to the n a t u r a l f i l t r a t i o n ) . Note that i f (WF, U, Ut, QX) i s a r i g h t process, then (7.6) of Getoor [23] i d e n t i f i e s the t o t a l l y F i n a c c e s s i b l e times x. as times at which W exists-, - t -l i e s i n F , and does - 7 1 -F ~F • F not equal W (here W denotes the l e f t l i m i t at t , of W i n x t -the topology of. i t s Ray-Knight c o m p a c t i f i c a t i o n . ) At such times, ~F F W = W , so that (3.3) of Benveniste et Jacod becomes the statement T— T — that (Q ) i s s t r o n g l y r e g u l a t e d . Thus, F - - x C o r o l l a r y 3 I f (Wt, U, U t» Q )• i s a r i g h t process then i t i s s t r o n g l y r e g u l a t e d . 0 x Lemma 8 Let (K ) r e g u l a t e (Q ) . Then there i s a continuous PAF (K t) , and an (F, E F ; F, E F ) k e r n e l N(x,dy) such that N(x,F) < 1 f o r each x , and the f o l l o w i n g two c o n d i t i o n s hold f o r each y : (K f c) i s a v e r s i o n of the ( U y , Q y)-dual p r e d i c t a b l e p r o j e c t i o n of (K^) ; and f o r every p o s i t i v e ( 0 ^ ) . p r e d i c t a b l e process (Z.) , and t t t F F every p o s i t i v e f e E ® E , Q J F F f) Z f(W ,W )dK U ] s s- s s 0 Z s F(W F y)N(W F,dy)dK ] S S o proof: Theorem 2 of Benveniste et Jacod [2] a p p l i e s to construct (K f c) . I t i s continuous s i n c e the jumps of (K^) are t o t a l l y i n a c c e s s i b l e . Mokobodski's absolute c o n t i n u i t y argument ((2.5) of [2] ) works i n our case, so that as i n Benveniste et Jacod F [2] , we may construct the N(x,g) , g e E . We make them i n t o a k e r n e l using the ' c l a s s i c a l argument' at the end of Maisonneuve [38] • . (B) Uniqueness. Turning now to p o i n t processes l e t (V,V^) be a - 72 separable measurable space and l e t 1/ • be i t s u n i v e r s a l completion. In a p p l i c a t i o n s , (V,l/^) w i l l be (U,iP) , where E i s a separable m e t r i c space, so that no confusion w i l l a r i s e i f we c a l l the corresponding space of p o i n t f u n c t i o n s , e t c . . . by the names II , P , e t c . . . , that were used i n s e c t i o n 2. Let n(x,dv) be an (F, E ; V, f ) k e r n e l s a t i s f y i n g the (unnecessarily strong) c o n d i t i o n t h a t : (8.6) There e x i s t Vfc e 1/, \ + v such that sup n(x, V^) < » f o r each k . x x F F F Let Q (du) be an (F, E ; U , U ) k e r n e l c o n s i s t i n g of p r o b a b i l i t y measures, and l e t (ft, F, P) be a complete p r o b a b i l i t y space, w i t h a f i l t r a t i o n (F t) s a t i s f y i n g the 'usual c o n d i t i o n s ' . Let (X t) be a r i g h t continuous process w i t h values i n F and adapted to (F t) » such that (x t , F t, P, cf) i s strong Markov. We w i l l consider p o i n t processes adapted to (F ) ( f o r a poi n t process, t h i s means that Y : ft ^ II- s a t i s f i e s Y ^(P t) c Ffc) and s a t i s f y i n g the c o n d i t i o n ; (8.7) For every p o s i t i v e (F t) p r e d i c t a b l e process CZfc) , and every p o s i t i v e f e V , EC I Z f ( Y )] = E[ ^ Q S n S S S>0 Y ^6 s Z s f (v)n(X g,dv)ds ] We w i l l determine to what degree (8.7) determines the j o i n t law of of (X., Y.) , consider the strong Markov property of the p a i r (X., Y.) , and then show how to construct such processes Of ) . I t turns out' that s o l u t i o n s to (8.7) do not have unique c o n d i t i o n a l laws. The reason t h i s i s so i s that (8.7) does not t e l l us how Y behaves at the t o t a l l y A i n a c c e s s i b l e times f o r the minimal f i l t r a t i o n of (^t) • We w i l l soon impose c o n d i t i o n s on the (Q ) that give us i n f o r m a t i o n about the form of these times. To o b t a i n uniqueness, we a l s o f i x a f u n c t i o n G , F F measurable from (F*V, E &> I/) to (F, E ) , and we w i l l impose the f u r t h e r c o n d i t i o n that (8.8) Xfc = G(X t_, Y t) f o r every t > 0 such that Y ± 6 , a.s.. (Note that by (8.7), Y T = <5 a.s. f o r any (F ) p r e d i c t a b l e time T , so A that by (8.1), Xfc_ e x i s t s f o r every time t such that 1 ^ 6 , P-a.s.). Let GQ = {(x,v) ; G(x,v) = x} Then f o r p o s i t i v e f e 1/ , g e E , x -> 1" . ' (x,v)g(G(x,v))F(v)n(x,dv) r c i s E - measurable, so that by (8.6), V.58 of D e l l a c h e r i e et Meyer [14] , and the s e p e r a b i l i t y of \P , there e x i s t s a p o s i t i v e F F E s E measurable f u n c t i o n n ( - , •; f ) : F x F -»- [0,1] , f o r each f e \l w i t h sup | f (v) | < 1 , such that - 74 1 . (x,v) 1 (G(x,v))f(v)n(x,dv) r > c A  G 0 l A ^ { x } ( G ( x , v ) ) n ( x , G ( x , v ) ;f)n(x,dv) f o r each A e f . [Note: i f V i s i t s e l f a U-space, and V® i s i t s B o r e l f i e l d , then the ' c l a s s i c a l argument' i n the l a s t s e c t i o n of Maisonneuve [38] shows F F that we can f i n d (FxF, E &E ; V, I / ) k e r n e l s n(x,y; dv) w i t h n(x, x; V) = 0 and n(x, y; V) < 1 , such that n(x,y;f) = f(v)n(x,y;dv) w i l l work i n the above. This i s the case when we take (V,l/^) = (\J,iP) provided E i s a U-space; see p.217 of Maisonneuve et Meyer [39] ]. Lemma _9 Let (Y f c) be adapted to (F t) , and s a t i s f y (8.7) Then f o r every p o s i t i v e ( f t ) p r e d i c t a b l e process (^t) > a n d p o s i t i v e F F h e E ®E we have that E[ I Z h ( i X ) ] = E[ s>0 Y ± 8 s s s-, s h(X s,v)n ( x s,dv)ds] 0 F 0 proof: Let h(x,v) = g ( x ) f ( v ) w i t h g e E ' p o s i t i v e and bounded and f e \I p o s i t i v e . The argument of (13.4) of Getoor [23] y i e l d s even more than (8.1), namely that g ( X t - ) ; L { X t _ e x i s t s } i s (Ffc) p r e d i c t a b l e . Thus by (8.7), - 75 -E[ I Z sh(X s_ X g ) ] = EC s>0 V6 h ( X s - , v ) l { X e x i s t s } n ( X s , d ^ d s ] s-But because F i s a metr i c space and (X f c) i s r i g h t continuous, (X f c) can have only countably many d i s c o n t i n u i t i e s . Thus, the r i g h t hand s i d e equals EC h(X ,v)n(X ,dv)ds] . s s o By monotone c l a s s and completion arguments, t h i s extends to general F p o s i t i v e h £ E 8 1/ The f i n a l c o n d i t i o n that we w i l l assume, r e a l l y belongs i n a d i s c u s s i o n of the problem of c o n s t r u c t i n g 0^) a s above. I t i s : F F (8.9) For every x e F , t > 0 , and p o s i t i v e f e E » E , we have that q s e ( 0 , t ] S ' S {K U ^K U} s- s > E*C 1. -(W F,v)f(W F,G(W F,v))n(W F,dv)ds ] U C o S o fa 0 Go Observe that by (8.8), t h i s would be a s p e c i a l case of Lemma 9, provided we assumed that f o r each x e F , there e x i s t ft, F , ( F f c ), P» (X f c ) , (Y ) as above, w i t h Q X the image law of P under X . x N Lemma 10 Let n s a t i s f y (8.6, 7, 8, and 9), and suppose that (Q ) i s r egulated by (K^) . Then 76 -F F (a) R e c a l l that (8.3) gave us a f u n c t i o n b e E a E . There e x i s t s an E F a E F measurable f u n c t i o n J : F x F •> [0,1] , such that f o r F F u each j i , t > 0 , and p o s i t i v e g e E a E , we have that Q -a.s., 1 c(W F,v)g(W F,G(W F v))n(W F dv)ds J(W F,y)g(W F ,y) b(W F,y) (WF y)N(W F dy)dK "Wo} s s (b) If e > 0 and r > 0 , put R = i n f { t > r ; K°(X.) > K°_(X.) + e} . Then the i n f i s a t t a i n e d , and i f J i s any f u n c t i o n as i n ( a ) , and f e \l i s p o s i t i v e and bounded by 1, then E[f(Y R-) , Y R t 6 | F R_ v a(X t;t>0)3 = JCX^.X^nCX^.X^f) , P-a.s. on {R<-} . (c) If r > 0 and k > 1 , set T = i n f { t > r ; Y ^ V ^ X T_=X T> Then the inflmum i s a t t a i n e d , and f o r each p o s i t i v e f eiV , E [ f ( Y T ) , T < o o | F T_ vo (X t;t>0)] / 1 G ( X r v ) . f ( v ) n ( X T , d v ) \ 0 / 1 (X v)n(X dv) V, G 0 T T k P-a.s. on {T < °°} - 77 -(d) I f i n f a c t (Q ) i s s t r o n g l y r e g u l a t e d , then P(T» t F v a(X ;s>0)) = exp(-J r v; 1_ (X v)n(X -,dv)ds) G0 s s f o r every t > r . proof: Suppose that f e E i s p o s i t i v e , and E X [ f(W F)dK ] = 0 f o r each x . x J s s 0 Then by Lemma 8 , E X [ Q J f C < _ ) d K ° ] * E X C J 0 f(W r)dK ] = 0 s s f o r each x . Let G 1 = {x; 1 (x,v)n(x,dv) > 0} J0 Thus by (8.9) we have that E r o f(W )1 . (W ,v)n(W ,dv)ds] s c s s G0 0 , s- s and hence E X C 0 f ( W * ) l _ (wf)ds] = 0 s s - 78 -As remarked i n the proof of Lemma 8, (2.5) of Benveniste et Jacod [2] a p p l i e s to our s i t u a t i o n , and hence there i s a p o s i t i v e h e E such that f o r every V , t t 0 I:. (W r)ds = G 1 S h(W )dK , Q y-a.s. s s o As before, we use V.58 of D e l l a c h e r i e et Meyer [14] to o b t a i n F F E &< E measurable d e n s i t i e s J^(x,y) and J 2 ( x , y ) , f o r n^x.dy) = 1 ^ n ( x , { v ; G(x,v) e dy}) , and f o r N(x,dy) , each w i t h respect to n^(x,dy) + N(x,dy) . Let J(x,y) = h ( x ) b ( x , y ) J 1 ( x , y ) J 2 (x,y) 1^ . j ^ (x,y) By (8.9) , 0 f o r every x , and hence s i m i l a r l y , 0 1- n(WI>v)g(WF,G(WF,v))n(WF,dv)ds] . C o S o S 0 J(W ,y)g(w!,y) b(wsyy) F F f o r every y , every p o s i t i v e g e E ®' E , and every p o s i t i v e (U1^) p r e d i c t a b l e process (^t) - 79 -(b) Since Y e F , and X ^ i s c o n d i t i o n a l l y independent of F R given X R , i t w i l l s u f f i c e to show that the r i g h t hand side of (a) equals E C f ( Y R ) , Y R * 6 , R < » | F R_va(X R)] . Let R Q = i n f { t > 0; K° > K ° _ + e } . Let C e;F , t e r , and l e t g e E be p o s i t i v e . Then EEC,t<R<°°", Y R*5 , f C Y ^ g C ^ ) ] = EEC, I f(Y )g(G(X ,Y )) se(t,R] S S Y H s 1{(x,v);G(x,v)/x,b(x,G(x,v))>e} ( Xs--' Ys ) ] R = EEC, f(v)g(G(X„,v)) i { ( x , v ) ; G ( x , v ) ^ x , b ( x , G ( x , v ) ) > £ } ( X s ' v ) n ( X s ' d v ) d s ] Xt = EEC,t <R, E Q E R„ H (x,z);x^Z,b(x,Z)>e}(WI'G(WI'v)) g(G(WF v))n(W F,G(wf,v);f)n(wf,dv)ds]] o s s s x R-EEC,t < R, EQ Ej 0 ,J(W ,y)g(y)n(W*,y;f) b(WF,y) 1 { ( x , z ) ; x ^ , b ( x , z ) > £ } ( ^ ^ N ^ s ' d y > d K s ] ; ] R F F F F F X f o W ,W*)g(W*)n(W*_ ,w\-f) = EEC, t < R, E E S - S 5 ? 5 - 80 -• L , s , w -,(WF ,W F ) dK° ] ] { ( x , z ) ; x f z , b ( x , z ) > e j s- s s ECC,t<R, E. 'CJCW? ,WF )g(W F )n(W F ,WF ; f ) ] ] Q R o " R o R o V R o = E[C,t<R<», JCX^.X^gCX^nCX^.X^f)] , by Lemma 9 par t ( a ) , and (8.3), as req u i r e d (c) We argue as above, and use that X T = X T_ e , to see that i t s u f f i c e s to show that the r i g h t hand s i d e of (c) equals E [ f ( Y T ) , T < 0 0 F T_] . Let C e F , t > r . Then by Lemma 9, a p p l i e d t w i c e , E[C,t<T<~, f ( Y T ) ] = EEC, l f ( Y s ) l ( Y s ) l ( X s _ , Y s ) ] s e ( t , T ] k 0 Y +& s = EEC, t v, 1. (X.v)f(v)n(X ,dv)ds] G Q s s 1_ (X ,v ) f ( v ) n ( X _,dv) se( t , T ] k f Y /6 s 1 (X ,v)n(X ,dv) G 0 S " S = E[C,t<T<°°, f 1 (X ,v ) f ( v ) n ( X dv) G 0 T " ' l G o ( X T _ , v ) n ( X T _ , d v ) as r e q u i r e d . (d) For t > r , l e t T be a v e r s i o n of - 81 -P(T>t | F v a ( X s ; s > 0) which i s r i g h t continuous, decreasing, and equals 1 at t = r . Let z = z 1 • z 2 ( i + r) , 1 2 F where Z e F and Z e U i s a f u n c t i o n of the form r I f f . o WF . 1 1 l t 1=1 i Write (°Z t) f ° r a r i g h t continuous v e r s i o n of the (P, F t )-martingale E[Z I F f c] . Write ( Z ^) f o r the r i g h t continuous process that i s a v e r s i o n of the Q y-martingale E^CZ 2|U y] , f o r each y ( I t e x i s t s by Lemma 1 of Benveniste et Jacod [3] ). Then a d i r e c t computation shows that % = z 1 • °Z? (X .) t t - r r+-= Z 1 • °Z 2(X.) f o r every t > r , P-a.s. X Because (Q ) i s s t r o n g l y r e g u l a t e d , i t f o l l o w s that f o r each y , { ( t , u ) ; °Z 2_(u) * °Z 2(u), K°_(u) = K°(u)l i s Q ^ - i n d i s t i n g u i s h a b l e from the union of the graphs of countably many (U y) a c c e s s i b l e stopping times. Since T i s t o t a l l y i n a c c e s s i b l e f o r (F ) , and = , i t f o l l o w s that °Z2_(X.) = °Z 2(X.) P-a.s., - 82 -and hence a l s o that °Z T_ = P-a.s. . But V.10 of D e l l a c h e r i e [12] shows that Z = °Z t t-i s the (P,F f c) p r e d i c t a b l e p r o j e c t i o n of Z , so i n f a c t , z T •= EC z | F T ] Thus, f o r p o s i t i v e bounded g:eiB , E[Z g ( t ) d T t ] = -E yCZg(T), r. < T < 0 0] -E[Z g(T), R < T < «3 -EC I Z s g ( s ) l ( s ) l ( X s _ , Y s ) ] se(r,TJ k 0 V6 = -ECZ Bis) r V, 1 (X v . ) n ( X dv)ds] b Q s s oo t = ECZ( g(s) r r V, 1- (X ,v)n(X ,dv)dsdT \JQ S . S t-- T r o _ j 8(s>: 1_ (X v)n(X ,dv)ds)] . u 0 Taking g i n a countable dense subset of CpCCr,0 0)) , we see that O  t g ( t ) d T t = g(s) r r V, l G ^ ( X s , v ) n ( X s , d v ) d s d T t - 83 -- T g(s) r V, 1 (X ,v)n(X dv)ds u 0 f o r every bounded p o s i t i v e g e 8 , P-a.s. . We put t g(t) = l r _ .. n ( t ) exp( L r » E 0 ' rV, 1 (X v)n(X dv)ds) , G Q s s f o r tg > r , to o b t a i n that f 0 exp( •r V, 1 (X ,v)n(X dv)ds)dT. G Q s s t (exp( r V, 1 (X v)n(X dv)ds) - l)dT G Q s s t + o • j r V, 1 (X v)n(X dv)ds) -1) -^0 - T (exp( r V, l r (X v)n(X dv)ds) -1) , u 0 and hence that T = exp(-0 r V. 1 (X v)n(X dv)ds) G Q s s R as r e q u i r e d . • We gather together the current hypotheses f o r the next r e s u l t : .x x P r o p o s i t i o n 10 Let (Q ) be strongly, r e g u l a t e d , and l e t ir s a t i s f y (8,6) and (8.9). Let (ft, F, P) be a complete p r o b a b i l i t y space, w i t h - 84 -a f i l t r a t i o n thereon, s a t i s f y i n g the 'usual c o n d i t i o n s ' . Let (X f c) be a r i g h t continuous process w i t h values i n F , which i s adapted to (^t) a n <* i s such that ( i t , F t , P, Q X) i s strong Markov. Let G be f i x e d , and l e t (Y ) be a p o i n t process, adapted to ' » s a t i s f y i n g (8.7) and (8.8). Let (K^) reg u l a t e (Q ) , and l e t J be given by Lemma 10. Then (a) For each k > 1 , there e x i s t s ( a f t e r e n l a r g i n g ft i f necessary) a sequence S^, ••• of e x p o n e n t i a l l y d i s t r i b u t e d random v a r i a b l e s A. w i t h u n i t means, which are independent of each other and of (Xj.) » and s a t i s f y the f u r t h e r c o n d i t i o n s : (b) I f we put T Q = 0 and T ± + 1 = i n f { t > T ±; Y t € V k and X t_ = , then i n f a c t , T. = i n f { t > 0 ; 1 i 1 (X ,v)n(X ,dv)ds > I s . } , P-a.s.; ) 0 s s i = l 2 ov, J k (c) For e > 0 , we put = 0 , and R ± + 1 = i n f { t >R i 5 K°(X.) > K°_(X.) + e} . Then f o r any p o s i t i v e f^..f£,g^..g^ e V , the random v a r i a b l e s g.(Y T_) , f j ( Y R ) l { Y ^ } , 1,3 = are c o n d i t i o n a l l y independent given (X f c) and the T^..T^, w i t h - 85 -c o n d i t i o n a l expectations r e s p e c t i v e l y 1 G , v ) g i ( v ) n ( X T ,dv) 0 i ' i lr (X ,v)n(\ ,dv) G0 T i ^ i {T.< <*>} , and l J ( X R _ , X R ) n ( X R _ , X R _ ; f ) 1 { R 3 3 3 3 3 proof; For C e F v a ( X ; t > 0) Tl-1 Z P(C S Tl-1 V k 1 (X ,v)n(X ,dv)ds > t ) 0 S S = l i m I p(.C., i-**>. j = l Tr.l 1*1 \ lr ( X , v ) n ( X ,dv)ds > t , G Q s s = l i m I e p(<J i-x" j = l 2 j k 1 (X v)n(X dv)ds > t , G Q s s T£ > j : * T£-l > r^) - t p(c, 1 (X s,v)n(X s,dv)ds) > t) , V i \ by Lemma 10 (d). Thus, c o n d i t i o n a l on F v o ( X . ; t > 0) - 8 6 -T£-l V k 1_ ( X . v ) n ( X ,dv)ds ^0 S has the law of the minimum of a mean 1 exponential random v a r i a b l e and the number 00 T f r v . ( X Q , v ) n ( X , d v ) d s . J J o s T£-l \ Thus we can f i n d an appropriate , P-independent of (X^.) and of F^ , . Repeating the argument gives $£_^ , and sin c e S£ i s independent of , may be chosen independent of S£ . By c o n t i n u i n g t h i s argument, we ob t a i n a proof of (a) and (b). The proof of (c) i s s i m i l a r , now usin g (b) and (c) of Lemma 1 0 ins t e a d of (d). C o r o l l a r y 4 Under the co n d i t i o n s of P r o p o s i t i o n 1 0 , the j o i n t law under P of ( X . , Y | ^ may be expressed i n terms of G , n , the Q , the i n i t i a l law of XQ under P , and the f u n c t i o n J obtained from an a r b i t r a r y r e g u l a t o r of (Q ) . proof: A l l r e l e v a n t p r o b a b i l i t i e s may be c a l c u l a t e d using P r o p o s i t i o n 1 0 . • Note that even without our r e g u l a t i o n hypotheses, there i s much that ( 8 . 7 ) t e l l s us about the d i s t r i b u t i o n of ( Y ) . Lemma 1 1 Let (ft, F, P) be a complete p r o b a b i l i t y space, w i t h a f i l t r a t i o n ( F t ) s a t i s f y i n g the usual c o n d i t i o n s . Let n(x, dv) - 87 -and Q X(du) be (F, E F ; V, f ) and (F, E F ; U F, U F) ke r n e l s r e s p e c t i v e l y , the Q being p r o b a b i l i t y measures. Let (X t) be a r i g h t continuous process w i t h values i n F , and adapted to (F ) , such that (X^, F f c , P, Q ) i s strong Markov. Let (Y t) be a poin t process adapted to (F ) and s a t i s f y i n g (8.7). Let A e f s a t i s f y sup n ( x , A ) < °° , x and l e t r > 0 . Put . T = i n f { t > r ; Y -e A> . Then: (a) There e x i s t s (upon e n l a r g i n g ft perhaps) an e x p o n e n t i a l l y d i s t r i b u t e d random v a r i a b l e S , of u n i t mean, which i s independent of F (but not n e c e s s a r i l y of a(X ; s ^ 0)) and s a t i s f i e s IT S t » T = i n f { t ; n ( X g , A)ds > S} . r (b) For f e V p o s i t i v e , f C v W X ^ ,dv) J E [ f ( Y _ ) | F_ ] = P-a.s. on {T«=°}. T " h ( X T _ , A ) (where again, 0/0 = 0) proof: (a) This i s proved e x a c t l y as i n Lemma 10 (d), except that s i n c e we no longer r e q u i r e S to be independent of a ( x s 5 s > 0) , we do not need to use the hypothesis of strong r e g u l a t i o n . - 88 -(b) By (8.7), {s; Y g e A} i s P-a.s. d i s c r e t e , so that Y^ , e A on {T < - o o } . Let B e F . Then E[B,t<T<°°, f ( Y _ ) ] = E[B, I f(Y )1 (Y ) ] S 6 ( r v t , T ] Y ^6 s .T E[B, r v t A f ( v ) n ( X g , d v ) d s ] / f ( v ) n ( X . d v ) = ECB, I A : W ] s e ( r v t , T ] n(X A) Y # .- s s (by Lemma 9) = E[B, t« T < 0 0 , 1 n(X T_, A) f ( v ) n ( X T _ , d v ) ] By monotone c l a s s arguments, t h i s s u f f i c e s . • (c) The Strong Markov Property: To f a c i l i t a t e f u t u r e a p p l i c a t i o n s , we w i l l consider the f o l l o w i n g f o r m u l a t i o n . Let E be a separable metric space, w i t h E i t s a - f i e l d of u n i v e r s a l l y measurable subsets. Let F be both a U-space and a u n i v e r s a l l y measurable subset of E . F i x a s i n g l e measurable space (ft , F ) ', a r i g h t continuous f i l t r a t i o n (F ) thereon, and an (E, E ; Q, F) k e r n e l P X(dco) such that each P x has unit mass,' and . (F,' F X isaugmented with'respect to the P^ . As above, l e t (V, V^). be a.separable measurable space. Write 1/ f o r the uni-ver.saL completion'of l / ° , and,let n(x,dv) be an •p (F, E ; V, I/) k e r n e l s a t i s f y i n g (8.6). - 89 -Let (X^.) be a r i g h t continuous F-valued process, adapted to x A x F F (F )^ , and w r i t e Q f o r the image law of P on (U , U ) under X. . I f now (Y ) i s any (V, l/)-valued p o i n t process adapted to (F ) , we w i l l say that ( X t , YT, F t , P x) i s strong Markov i f ^ ( ( ^ y i ^ e A l F j ) = ^ ( ( x . , Y|(0jot)  e A) whenever y i s a p r o b a b i l i t y on E , T i s an (Fy) stopping time, and A e ( I F S P . C o r o l l a r y 5 Assume i n a d d i t i o n to the above hypotheses, that each ( X t , F y , P y, Q X) i s strong Markov, and that (Q X) i s strongly, regulated. F i x G. , and assume that. (Y^_) i s a (V, 10-valued point process,• adapted to (Ffc) and . s a t i s f y i n g ('8.7) and (8.8). Then (XT,YFC, F f c,P x) i s strong Markov. proof: C o n d i t i o n (8.9) fo l l o w s immediately from Lemma 9, by d e f i n i t i o n X of the Q . We apply P r o p o s i t i o n 10 w i t h the p r o b a b i l i t y space (Q, F y , P y) , the f i l t r a t i o n ( F t } = ( F T + t } > 6 ? ( Y ) ( t ) * and the processes \ = W Y t = e ? ( Y ) ( t ) • Our hypotheses guarantee that those of P r o p o s i t i o n 10 are met, hence, - 90 -as i n C o r o l l a r y 4, the c o n d i t i o n a l law of (X., Y.) given F^ may X X be expressed using only n, G, the Q , a r e g u l a t o r f o r (Q ) , and the i n i t i a l law of X^ , showing the r e s u l t . (D) Existence We now consider the problem of c o n s t r u c t i n g p o i n t processes (Y ) s a t i s f y i n g (8.7). With f u t u r e a p p l i c a t i o n s i n mind, we w i l l do so i n a context s i m i l a r to that of (C). In a d d i t i o n to the general hypotheses of (C), assume that each ( X t , F 1^, , Q X) x 0 i s strong Markov, and that (Q ) i s regulated by a PAF (K f c) . F i x G , and assume th a t n s a t i s f i e s (8.9). We w i l l f u r t h e r assume F F that an ( F x F , E 8 E ; V, I / ) k e r n e l n(x,y;dv) may be found, w i t h n(x,y; V') < 1 and n(x,x; V) = 0 , such that 1 (x,v)g(G(x,v))f(v)n(x,dv) G 0 = g(G(x,v)) f (v')n(x,G(x,v) ;dv')n(x,dv) f o r every p o s i t i v e g e t " and f e V . (As remarked before, such a k e r n e l w i l l e x i s t , provided (V, l/^) = (U, iP) , where E i s a U-space.) We may then perform the f o l l o w i n g c o n s t r u c t i o n ; o b t a i n a f u n c t i o n J(x,y) on F x F from Lemma 10(a). I t i s e a s i l y checked that the f u n c t i o n J ( x ' y ) 1 { ( x , y ) ; n ( x , y ; V ) = 1} s a t i s f i e s the same c o n d i t i o n as J , so that we may assume that J(x,y) = 0 whenever n(x,y; V) ^ 1 . - 91 -Enlarge ft by setting ( f t ' , F') = (n x n j , F a c N) , where ft^ is the space of a l l right continuous functions [0,°°) -> N , and C is i t s cylinder o-field. We retain the notation F for the preimage of F under the projection "of 'ft' on i t s f i r s t coordinate. Similarly, we write'; Xfc for the composition of Xfc and this projection. Let B (t) , t > 0 and k > 1 , be the coordinate maps on the second factor. Then we can extend the P by the appropriate product measures, so as to make the (B (t)) Poisson processes of unit intensity, and independent of each other and of F , under each law P . Set t k k r T. = inf{t; B*( 3 1_ (X v)n(X .dv)ds) > j} Ix s s ° W i Let RQ = 0 , and put R K + 1 = inf{t > R K ; K°(X.) - K°_(X.) e ^-)} Now enlarge (ft', F') again, to (ft", F") , and redefine the P so that there exist (,Vu{<5} > l^Va ({6}))-valued random variables V ; 1-1,2 and j,k >1; which are P -conditionally independent given F and the T , for every u , with conditional laws n(X Tk, dv) 1 (X T k,v')n(X T k,dv') 0 j 3 W l V n <F^ V.\V. 1) ( XTk' v )'» l f ± = 1 ; 0 k k-1 j and - 92 -J(X k ,X k ) n ( X k ,X k;dv) + (1-J(X k ,X k ) ) £ ( S ( d v ) , i f i = 2 R. R. R. — R. R".~ R. J 3 3 " 3 3 3 (and w h i l e we're at i t , enlarge the space enough, so that more independent random v a r i a b l e s may be found i n the f u t u r e ) . Set Y = t V , i f t 3 = r 2 yk 3 i f t = R 6 , otherwise , H° = F v Y V ) , H° = F t v Y V t ) , and augment (H^, H^ ) w i t h respect to the P y , to form (H, H^) Then P r o p o s i t i o n 11; On the enlarged space °,",(Xt, Y , H^, P X) i s strong Markov, and s a t i s f i e s (8.7) and (8.8). proof; Let and w r i t e T (r) = i n f { t > r ; Yfc e V ^ V ^ , Xfc = Xfc_} , n k(A) = W l l G ( j ( X 8 , v ) l A ( v ) n ( X 8 ( d v ) Then f o r t > q > r , A e U , and every p r o b a b i l i t y p , we have by d e f i n i t i o n , that - 93 -P J J ( T k ( r ) e ( q , t ] , Y 6 A, T k ( T k ( r ) ) > t | FvH-.vor(T*;i#0) T f c(r) r J O I P y ( T ^ r < T j , T ^ C q , t ] , Y fc e A , T^+± > t | Fvtf^cr(T*;i^k)) -£.= 1 T n 1=1 i T i - i - v rq k t ? k k ,e r S n k(V)e q S ; n k ( A ) -/ n k , ( V ) d 8 ' S " Js s , , - e ds = e - i : n s ( v ) d s f k n~(A)ds . s S i m i l a r l y , P P ( T k ( r ) > t | F v H v aCrS i/k)) = e * S r k -/ n K(V)ds Now f i x N , and l e t T C D = k A N T k ( r ) , j n k ( . ) k=l Then by independence, P y ( T ( r ) e ( q , t ] , Y T ( r ) e A, T ( T ( r ) ) > t | F v ^ N I P y ( T k ( r ) e ( q , t ] , Y e A, T k ( T k ( r ) ) > t , T X ( r ) > t f o r i#c k=l T K ( r ) F v H ) r n (A)ds s - 94 -Lemma 12 Let V„ = u{A.; i=l..N"}, the A. d i s j o i n t elements of N 1 l V , and l e t e > 0 , e ^ e N . There e x i s t s processes B ^ ( t ) , i = l . . N " , and random v a r i a b l e s C.. , j > 1 , such that under each (a) Each B^(t) i s a Poisson process of u n i t i n t e n s i t y ; (b) Each C_. i s uniformly d i s t r i b u t e d on [0,1] ; (c) The B_^  and are independent of each other and of F ; ( d ) I A / V V ^ - ' V = B i ( s e ( r , t ] x 0 r n (A.)ds) - B( s x n (A.)ds) S X Y H s f o r each i and each r < t , a.s.; (d) Let P Q = 0 , 3+1 i n f { t > p . ; K°(X.) > K°_(X.) + e} , 3 t v. a T = n(X , X ; A.) , ~] p . - p . x J 3'-= 3 b! = . f . a k' "e [0,1] , 3 k £ r 3 and w r i t e = 0 , b ^ " + 1 = 1 , A^ I I +^ = Vu{<5}\V N Then c. e (b"^ \ b"!"] whenever Y e A. 3 3 3 P j 1 proof: Let T_ = 0 , " X Y = n (A.)ds s x - 95 -For each i , p i c k a process B Q such that under each P^ , B Q i s a Poisson process of u n i t i n t e n s i t y , and the B^ are independent of each other and of H . Set B ^ t ) = \ k, i f t e [ n s ( A i ) d s , k+1 n (A ) d s ) , t <y S X B . C y 1 - ) + B J C t - y 1 ) , t > y 1 and that (d) holds by d e f i n i t i o n . i Choose 0^, j > 1 to be uniformly d i s t r i b u t e d on [0,1] , and independent of each other, the B 1 and H , under each P^ . Set i-1 i ' C. = b? + a.C. 3 1 3 3 i f Y e A. , i = 1 N" + 1 . Then (e) holds by d e f i n i t i o n . p i 1 Since the T. used to d e f i n e (Y ) are independent of the 3 c R. , given F , we see that the B. are independent of the C. , given F . Let t 0 0' < * r < *! < ... < r N , < t N , , i r . i N . e U,...,N"} , and H £ F . Set j Q = 0 , j k = H+ 1* where Z i s the l a s t number before k such that i ^ = i ^ (Z = 0 i f no such number e x i s t s ) , and de f i n e j ( i ) to be the l a r g e s t of the j such that i = i (and i f there i s no such k , we set i t equal to 0 ) . To show that the B^ are independent - 96 -of each other and of F , i t w i l l s u f f i c e to check that the p r o b a b i l i t y of the set H n { T ^ € ( r k , t k ] , k = l . . N ' , sup x j ( i ) + 1 > t N , } agrees w i t h the r e s u l t we would o b t a i n under (a) and (c) (as f i n i t e unions of such sets form a B o o l i a n algebra which generates a . a - f i e l d f o r which a l l the are measurable). By the computational preceding t h i s lemma (and s i m i l a r arguments), P y(H; ^ ( V t k ] , k - i . . N ' . ; sup T J ( ± ) + 1 > t f c) - ^CH; T C V l ) e C r ^ ] , Y e A T ( T ( V l ) ) > t f c , k-1 k k = 1..N') -t k N ' -L n _ ( V ) d s r 1 E y [H, JTe k _ 1 k=l r k n_(A. )ds] k - '* )ds N' N' - J n (A. = E y c H , TTdTi:a-fiM )e "k-1 s 1 + 1=1 k=l k - / k n ^ W k + 6 . . e k-1 X 1 k r k n (A.)ds])] s 1 which i s the r e s u l t we would have obtained under (a) and ( c ) . To show that the C^. are independent of each other and of F , w i t h uniform d i s t r i b u t i o n s , l e t O ^ r . < t . ^ 1 , j = 1..N ; 3 3 and H e F. I t w i l l again s u f f i c e to show that the p r o b a b i l i t y of - 97 -H n {C. € [ r . , t . ] , j = 1..N} 3 3 3 agrees w i t h that c a l c u l a t e d u s i n g (b) and ( c ) . P y(H; Cr-:e [ r . , t j 3 , j = 1..N) I P^(H;C e [ r . ,t 3, Y e A , j = l.,N) i1..lNe£l..,N"+l} 2 3 3 Pj i -1 i .-1 N .r.-b. J t . - b . J = E [ H, I (TT H-2-^ , 2 . 2 ] n [0,131) i 1 . . i N e { l . . , N " + l } j = l a 2 j a x j N i . : ( T T a . J ) ] 3=1 2 N i . - l i . - l i . = E y[H, I TTI I r t ] n Lb 3 , b3 + a 3 31 3 i . . . y { l . . , K " + l } j = l 3 3 J J J I N N i -1 i . = E V[H, JJ ( J | [ r . , t . ] n [ b . 3 , b . J ] | ) j -j = l i e{l..,N"+l} 3 3 3 3 N = J T ( t , - r )P y(H)", 3=1 2 3 as r e q u i r e d . • As an immediate consequence, we see that there i s a a - f i e l d F 1 which i s independent of F under each P y , and such that Thus, f o r every (n'^+) stopping time T , we can f i n d (as i n - 98 -Lemma 7) a stopping time T w i t h respect to the f i l t r a t i o n .. n (F a F1) s s>t on ft" x ft" , such that T(to) = T(CO,OJ) f o r every co . Then A T(*,co) i s an C.F ) stopping time f o r each w e f t " , so that f o r F f e U , f p o s i t i v e , we get that E y [ f ( X T + > ) ] = l'T.(o)1,a)2-)+ 0Ptl(da)1)PM-('dtt»o) , XC<oi); Q • • ( u 1 ^ 2 ) ( f ) p W ( d U i ) ^ ( d U 2 ) E U[Q T ( f ) ] .. We now e a s i l y o b t a i n that each (X f c, tfy, P P, Q X) i s strong Markov. S i m i l a r l y each B X ( t ) ..remains strong Markov when we a d j o i n to i t s minimal a - f i e l d , the in f o r m a t i o n i n F v 0 ( B k , 1 ^ J ^ i , j > l ) v o ( 2 Y j ; k,j>l) v a c M ; j ^ t ) ) , and hence, f o r any (Hy) stopping time T , B a ( - + n x ( V ) d s ) - B X ( s n (V)ds) 0 ii k. i s a Poisson p r o c e s s i n d e p e n d e n t of F v H v o"-(B--; k^ i ) , under - 9 9 -each P . Thus, 9^(Y) i s constructed from (^ t+f) ^ n e x a c t l y the same way that Y was constructed from (X f c) , so that by the strong Markov property of each ( X t , n , P , Q ) , we see that ( X t , Y t, Hfc, P X) i s strong Markov. Cond i t i o n ( 8 . 8 ) f o l l o w s by d e f i n i t i o n . To show ( 8 . 7 ) , i t s u f f i c e s to show the.conclusion of•Lemma 9 , w i t h (Z •) of the form Z. = Z . 1 , -,(t) , Z , e , t r+ ( r , s j ' r+ r ' and h(x,v) of the form 1 A n V ^ x ' v ^ ' A 6 V > o r N ° 0 f ( v ) l { ( y, z) ;y*z ,b (y, z) c ^ , - ~ ) } U ' G ( x ' V > } ' U V P o s l t i v e -In the f i r s t case, by Lemma 1 2 , c o n d i t i o n a l on frj v a ( X g ; s > 0 ) , I h ^ s _ , Y ) s e ( r , t ] S Y^6 i s a P o i s s o n random v a r i a b l e , of mean t n (A)ds , s Showing the corresponding expression of Lemma 9 . In the second case, - 100 -I Z s h ( X s - ' Y s ) ] s>0 Y ^6 s = E vCz „X i ( r t ] ( R k ) J ( x ,x ) r+. 1 U . t J 3 R*_ R * J 3 3 f ( v ) n ( X ,X ,;dv)] R - R 3 ' 3 X E UCZ , E / [ t - r r+ Q j J { (x,z>,x^z,b(x,z)£[k>k::j)} f(v) n ( W F .W^dvJdK 0]] s s s x t - r F 1{(x,z);x^z,b(x,z ) e c i- k^ I)} ( Ws' y ) J(wF,y) b(WF,y) f(v)n(W F,y;dv)N(W F,dy)dK s]] X t - r i l J [ Z r + E Q r C j £ J {(x,z);x^z,b(x,z)eC k, k3Y) i K W V ) ) f(v')n(W F,G(W F,v);dv*)n(W F,dv)ds]] AU r h ( X s , v ) n ( X s , d v ) d s ] . • 9. Excursions away from a set The techniques of the foregoing s e c t i o n s can be used to examine the more general problem of c o n s t r u c t i n g a strong Markov process - 101 -from a 'process on the boundary', together w i t h the excursions away from that 'boundary'. In t h i s s e c t i o n we w i l l be content w i t h modifying a r e s u l t due to B.. Maisonneuve [38] , proving a p a r t i a l converse, along the l i n e s of Theorem 2 and P r o p o s i t i o n 5, and then showing some a u x i l l i a r y r e s u l t s . A p p l i c a t i o n s f o l l o w i n S e c t i o n 10. Let E be a U-space. F i x a m e t r i c d on E , compatable w i t h i t s topology. We w i l l assume that some i s o l a t e d p o i n t A e E has been s i n g l e d out, to f u n c t i o n as a 'cemetary', and we w i l l l e t 9(u> = i n f { t > 0 ; u ( t ) = A} , f o r u e U . Let P X(du) be an (E, E ; U, U) k e r n e l such that P K(U) = 1 f o r every x , and P^(W' = A f o r every t ) = 1 . Augment ( U , U f c ) w i t h respect to the P y to form (U,'0 ) . Assume that (w t, D, D t , P X) i s a r i g h t process s a t i s f y i n g the f u r t h e r c o n d i t i o n (6.5) of near-B o r e l m e a s u r a b i l i t y of the excessive f u n c t i o n s . Let F e E be ne a r l y B o r e l , w i t h A e F . We set P F = {t i 0; Wfc e F }., M = p p D t = i n f { s > t ; s- e M } , Kx) = E X [ e °], M Q={t;D t>D t_} . Since F i s n e a r l y B o r e l , each D g i s a (Q^) stopping time, and i), i s 1-excessive. Assume that (9.1) F = {x £ E; I|J(X) = 1 } - 102 -Then M has no i s o l a t e d p o i n t s , a.s.. Assume a l s o that (9 .2) i s r e g u l a r (see Blumenthal and Getoor [5] ) . Then there e x i s t s a continuous PAF 1 , s a t i s f y i n g oo • (9 .3) E X [ e - t d l t ] = i p(x) f o r every x e E , 0 whose set of i n c r e a s e i s a.s. e x a c t l y M . A word of c a u t i o n ; at t h i s p o i n t we are t r e a t i n g A j u s t as any other p o i n t of F, and hence DO NOT TAKE THE CONVENTION THAT (A) = 0 . In f a c t , (A) = 1 , so t h a t i n s t e a d of having l f c = l t A g , we have that ^ Q ^ t = t ^ o r e v e r y fc) = ^ • Of course, we could have i n s t e a d stopped 1 at 3 ; that i s , we could have used the f u n c t i o n IJJ(X) - E [e ] i n s t e a d of iKx) • A l s o , r e c a l l f o r l a t e r use, that a f u n c t i o n D : U -v [ 0,oo] i s c a l l e d a p e r f e c t t e r m i n a l time i f D = t + D de on {D > t} , f o r every t > 0 . I t i s c a l l e d exact i f D ° e •> D as t 4- 0 . Now, l e t (ft,G) be a measurable space, on which we are given a r i g h t continuous E-valued process ( X f c ) , and a r i g h t continuous f i l t r a t i o n (Gt) t o which (X ) i s adapted. Suppose that a x b P (da) i s an (E, E; P., G) k e r n e l such t h a t P i s the image -b law of P under X. . Suppose a l s o that (G, Gfc) i s augmented w i t h respect to the P y , and that ( X t , Gy, P P, P X) i s strong Markov f o r each u (so that a l s o ( X t , G» Gt, P ) i s a r i g h t p r o c e s s ) . We set L t = l t ( X . ) , S + ( s ) = i n f { t ; L t > s} , S~(s) = i n f { t ; L > s} Then S + and S are. the r i g h t and l e f t continuous inverses - 103 -of ( L t ) , r e s p e c t i v e l y , and s i n c e Clfc) i s continuous, they are s t r i c t l y i n c r e a s i n g whenever f i n i t e . A l s o , S +(s)' i s a (Gt) stopping time. Put ? = 3(X.) ; I = L ; Dfc = Dt(X.) ; (9.4) X„ = V ( s ) > l f S ( S ) « A , otherwise ; F s = GS+(s) ' F 0 - = G 0 - ; (9.5) Y -X(S-(s)+.)AS+(s) ' i f S _ ( S ) < S + ( S ) , otherwise ; b "b and l e t P Q be the image law of P on ( U , U ) , under X.^ D . Pa r t s of the f o l l o w i n g r e s u l t are taken from E l Karoui [20] . The reader i s s t r o n g l y encouraged to omit i t s proof, and pass d i r e c t l y to Theorem 3. Lemma 13 F i x a p r o b a b i l i t y u on E . (a) I f T i s a (Gy) stopping time, then L T i s an (F y) stopping time. I f T i s any random time, we have that Gy S-(L T)" Gl (b) I f T i s an (F y) stopping time, then S+(T) i s a (G^) s t stopping time, FT" " ^ - C T ) - ' FT = GS+(T) - 104 -(c) I f (Z ) i s (Gy) p r e d i c t a b l e , then (Z , i s (F y) t t o \S/ S p r e d i c t a b l e . I f (Z ) i s ( f y ) p r e d i c t a b l e , then (Z ) S S J_i^ i s (G^) p r e d i c t a b l e . A (d) ( x s) ^ s adapted to (.Fg) » and takes values i n F , a.s. . x "x F F.. For x e F , l e t Q be the image law of P on (U , U ) A under X. . Then (X t , r\, P p, Q X) i s strong Markov, and Q X ( W Q = X ) = 1 f o r every x e F . (e) (Y ) i s adapted to (F t) (f) The f o l l o w i n g h o l d a.s. : M Q(X.) = { S - ( s ) ; se CO,.")' , S-(s) < S+Cs)} ; 5 , . = S + ( s ) f o r every s > 0 ; and a \s) t, < oo . . i f and only i f S~(£) < °o . proof: (a) We already know that S + ( s ) i s a (Gy) stopping time, f o r s f i x e d , and that so i s D^ , , whenever T i s a (Gy) stopping time. Since C l t ) increases e x a c t l y on M , and S~ i s s t r i c t l y i n c r e a s i n g , i t f o l l o w s that S+(L^,) = D^ , a.s., and hence { L T < s} = {S+(L T) < S+(s)} e G y + ( s ) = F y . For T now a random time, G^_(1 i s generated by the - 105 -/s l i P - n u l l s e t s , and by sets A n { t < S~(L T)} = An {D t<T} , f o r A e G y c GJ? . Since i>t i s a (G t) stopping time, these sets l i e i n GjjL . (b) FjjL i s generated by the P y - n u l l s e t s , and by sets A n {s < T} = A n {S +(.s) < S"(T)> , f o r A g F y = G g + ^ . Because S + ( s ) i s a (G y) stopping t i V-S-(T) xme, they l i e i n G^/rpx- • Thus Before proving the reverse i n c l u s i o n , we w i l l show the remainder of Cb) . S +(T) i s a (G y ) stopping time, s i n c e {S+(T) < t} = {T < L. } e , F T M (as T i s an (F y ) stopping t i m e ) , and y by the above, and part ( a ) . S i m i l a r l y , i f A e Frp , then An {S+(T) < t} = An {T < L.} e F T y c G y , t L f c- t so that - 106 -\ C GS+(T) ' Conversely, i f A e G g + ^ » then A n {T < s} = A n {:S+(T) < S+(s)} e G ^ . ( ^ = F y . F i n a l l y , G g _ ^ _ i s generated by the P y - n u l l s e t s , and by sets of the form A n {t < S~(T)} = A n {L < T} , where A • ^ <= ^a t> These sets l i e i n FjjL s i n c e L.fc i s an ( F g ) stopping time. (c) Let (Z t) be l e f t continuous and adapted to ( G y ) . Then (z ) i s a l s o l e f t continuous, and f o r each s , b \ S ) V ( s ) e G S - ( s ) - = F s -by (b). Thus ( z s - ( s ) ) l s ( F g ) p r e d i c t a b l e f o r such (Z f c) , and hence by monotone c l a s s arguments, f o r every ( G y ) p r e d i c t a b l e (Z f c) . The 'converse' i s proved s i m i l a r l y . (d) By (6.2), Pj, i s a.s. closed on the l e f t , so that the argument of Meyer [40] , P r o p o s i t i o n 1, shows that M\M. c n a.s. U r Since S + i s r i g h t continuous, s t r i c t l y i n c r e a s i n g , and ( l t ) increases only on M, a.s., we get that S+(s). e (M\M ) (X. ) u {«>} f o r every s, a.s. , showing that X g e F f o r every s , a.s, . Since S i s r i g h t - 107 -continuous, so i s (X g) . I t i s c l e a r l y adapted to CF ) . I f T i s an (F y) stopping time, B e E*" c E , and h > 0 , then = P (X^eB) = Q ( W h £ B ) , P y-a.s. by the strong Markov property of (X f c) , showing (d). (e) S~(.s) and S +Cs) are adapted to CFg) . Thus, f o r E > 0 , T = i n f { s > 0 ; S+Cs) > S~Cs) + e} i s an CF ) stopping time. Because {s < t ; S+Cs) > S~(s) + E} i s f i n i t e f o r every t , a.s., i t t h e r e f o r e s u f f i c e s to show that Y T i s measurable, from (ft, F T) to (U, U) . Let 0 < t , < t 0 < ... < t. , and B. , .. , B, e E^ . Then 1 2 k 1 k { Y T C t ± ) £ B 1 , i = l . . k } - { X ( s _ ( T ) + t i ) A ^ . B . , i - l . . k } By Cb), S +(T) i s a (G ) stopping time, so that (t.io) ^ X t A S + ( T ) ( a ) ) i s a measurable f u n c t i o n from B » Gg+^^ to E^ . Since S-(T) 6 G S + ( T ) by ( a ) , i t f o l l o w s that the above set l i e s i n Gg+(T) = F^ , . By completeness, we can extend t h i s to {Y e A} , A e U. Cf) We f i r s t prove the t h i r d statement. For x e E we have that - 108 -e t > e V [ e " S d l s ] > F^C^] , so that 1 < 0 0 f o r every t , a.s., and hence that a.s., £ < whenever S ~ ( < 0 0 . Conversely, l e t T = 1 = i n f { t > 1; Wfc e F and l- t>l } and T k + 1 " \ + T ° 6T ' k For x e F we have that E X C l T ] > EXC e " S d l ] s 0 = Kx) - E X[e"%(W T)] > 1 - e 1 . Thus, P X ( 1 T > ( 1 - e ^  > \ f o r every x e F . Because WT e F a.s. on {T < °°} , we see by the strong Markov property of (Wt) , that P y(T < °° and 1 - 1 < 1 - e _ 1 f o r every k > K) = 0 , f o r every K and y . Since T (X.) <£=°° a.s. on {S _(£) = °°} i t f o l l o w s that rv % = l i m L = o o a.s. on that set. t->oo Now l e t M^ be the set on the r i g h t of the f i r s t statement - 109 -of (f) . For co such that L > (co) increases exactly on M(X#)(co) and S (*)(co) is s t r i c t l y increasing, we obtain that D s_ ( g )(co) = S+(s)(co) , and hence M (w) c MQ(X<)(co) . Conversely, for teMQ(X>)(co) , we w i l l have that teM^(co) provided t=S (s)(co) for some s>0 . It follows by choice of co , that Lt(co)>0 , and that t=S (L t) (co) . We need therefore only show that Lt(co)<°° . This follows from the third part of ( f ) . • Theorem 3 (Q ) i s regulated. There exists an exact, perfect F terminal W + ) stopping time D , and an (F, E ; U, U) kernel n(x, du) such that the following conditions are satisfied n(x,{D =0 or W ± WD for some t > D }) = 0 for every x e F . II (a) n(x, {supd(x,W ) > e } ) < °° , for every x e F and t c e > 0 h (b) lim h+0 n(X g, {sup d(x,Wt) > e})ds < » P2 -a.s, for every x e F and e > 0 III (1-e D ^ ) n ( x , d u ) < 1 for every x e F IV n(x,{u;D(u) > t, u e A , 0 t(u) e M}) A n P^ ( t )(M)n(x,du) , D>t} for t > 0 , A e ti , M e U, x e F - 110 -n(x,{u; u(0) e B, u e M » = J Pg ( 0 )(M)n(x,du) {u;u(0)eB} f o r x e F and B e E such that B n F = 0 . VI (a) Let m(x) = 1 - (1 - e _ D ( u ) ) n ( x , d u ) , x e F \ { A } ; m(A) = 0 ; and I = {x'e F; P X ( m(X g)ds > 0 f o r every t > 0) = 0} . 0 Then f o r each x e I and h > 0 , h n(X ,U)ds = » P X-a.s. . J 0 Cb) I f n'(du) i s a p o s i t i v e measure on (U,U) s a t i s f y i n g IV, such that n' (du) < n(x,du) f o r some x , then n' (WQ e F) = 0 or Further: ( 9 . 6 ) D = D(X ) f o r every s , a.s.; S S T * ( 8 . 7 ) , ( 8 . 8 ) and ( 8 . 9 ) h o l d , w i t h G(x,u) = • u(P(u)) , D(u) < co A , otherwise ; (X. , Yfc, Ft» P X) i s strong Markov; and ( 9 . 7 ) -7 m(X )ds + r^ nD(Y ) S +Ct) = 0 s seCO.t] s Y ^6 s - I l l -Note; 1. In the s i t u a t i o n of Thereom 2, VI (b) s t a t e s that i f n(U) = °° then ( v i ' c ) h o l d s , and i f n(U)'<•;<» then n(U a) = 0 . VI (a) becomes the remainder of ( v i ' b ) , that i f n(U) < 0 3 , then m > 0 . 2. As a consequence we conclude that c o n d i t i o n s (9.1) and (9.2) r u l e out the d i s c r e t e v i s i t i n g behaviour of ( v i ' a ) . 3. C o n d i t i o n VI (b) r u l e s out two d i s t i n c t types of behaviour, one i d e n t i c a l to that precluded by ( v i ' ) , and another not found i n the s i t u a t i o n of Thereom 2. In f a c t , using 11(a), c o n d i t i o n VI(b) i s seen to be equivalent to the c o n d i t i o n that f o r every x e F , both: n(x,{W 0 g F\{x}}) = 0 ; and n'(WQ = x) = 0 or «>, whenever n'(du) i s a p o s i t i v e measure on (U,U) s a t i s f y i n g IV, and such that n'(du) < n(x,du) . X 4. The proof that (Q ) ; i s regulated uses (6.5) . As remarked i n ,the i n t r o d u c t i o n to section. 8, - G z y l , [28 ] .shows ?'a. stronger r e s u l t , namely . that ( X p -is a - r i g h t , process. proof: We can apply the p e r f e c t i o n arguments i n Meyer [40] to the r i g h t process (Wt, U, U f c , P .) , to o b t a i n an exact p e r f e c t t e r m i n a l CU ,) stopping time D such that P x(D=D n) = 1 f o r - 112 -each x e F . Thus D = D(X , ) a.s. , q q+. f o r each q e Q , so that by exactness, we o b t a i n c o n d i t i o n (9.6). Note the f o l l o w i n g , from Meyer [40], For h > 0 , (h A D ° 0 ) i s a cadlag process adapted to ^t+2\^ ' s o t n a t i t : i s ^t+2h^ pr o g r e s s i v e . Thus so i s M' = { ( t ,u); l i m D 0 6 (u) = 0}- . s+t S s<t and hence M' i s (^t+) prog r e s s i v e by D e l l a c h e r i e et Meyer [IS] , IV. 14. Thus M i s i n d i s t i n g u i s h a b l e from a pro g r e s s i v e set r e l a t i v e to the f i l t r a t i o n (IT , ) (rather than (IT) . t+ t We can now apply P r o p o s i t i o n 9.2 of Maisonneuve [38] to (Wt, Q, 0 t , P x) and M' . Maisonneuve works w i t h (Wfc) k i l l e d at D , r a t h e r than stopped t h e r e , but h i s k e r n e l c l e a r l y extends to one which, by Lemma 13(c) and the above d i s c u s s i o n , s a t i s f i e s (8.7). I t f o l l o w s from h i s proof, and that of h i s Theorem 4.1, that t h i s k e r n e l n s a t i s f i e s I I I . We w i l l now show that (Q ) i s regu l a t e d . Let E denote the Ray Knight c o m p a c t i f i c a t i o n of E , and l e t g belong to the Ray cone of E . Thus g i s continuous on E , and g|^ i s a-excessive w i t h respect to the P , f o r some a > 0 . By (6.5), g i s thus n e a r l y B o r e l . F i x a p r o b a b i l i t y u on F . Then there e x i s t g , go e E , wxth - 113 -g ; L < g | E < g 2 , P y(g 1(W t) *g 2(W t) f o r some t ) = 0 . Now 8 i | F e E ' J? F 0 so that o Wt i s a CU _^  ) o p t i o n a l process. Thus A = t § 1 ( W F ) * g 2(W F) , or l i m g^wj) ^  , F or l i m g.(W ) does not e x i s t } e U . ^ °1 s s-*t s<t Since Q i s the image law of P under X. , and g o X, i s a.s. cadlag ( g | v being a - e x c e s s i v e ) , we see that a l s o QyCAg) = 0 . I f we now l e t g range through a countable dense subset of the Ray cone, we get that F ~ F u {W. i s cadlag i n the topology of E} e U , and has Q y-measure 1 , whenever u i s a p r o b a b i l i t y on F . " Augment ( U F , U F) w i t h respect to the Qy , to form ( 0 F , U F) , and w r i t e V > V > V F * f o r the l e f t l i m i t s of W. , X. , and X. at t , i n the topolo of E . We conclude from the above that f o r any p o s i t i v e - 114 -h e E F a E F , Kn = I h(W F_,W F)l C s e ( 0 , t ] S S I f £ F } s- s s-—F defines a PAF adapted to (Ufc) ( p o s s i b l y one which i s i d e n t i c a l l y i n f i n i t e ) . Since E i s a U-space, we conclude from the d i s c u s s i o n p r i o r F F to Lemma 9, that there i s an (Fx F , E & E ; U, U) k e r n e l n(x,y; du) , such that l A ( u ( D ( u ) ) ) f (u)n(x,du) = l A ( u ( D ( u ) ) ) f (y)n(x,u(D(u));dv)n(x,du) f o r every p o s i t i v e f e t/ , and A e E such that x (f A, Set h n ( x \ y ' ) ( 1 - e D ( u ) ) n ( x ' , y , ; d u ) v 1 {(x,y);x^y,n(x,y;U)=0} By the argument of Getoor [23] (13.4), we see once again that s- X f o r every s such that Y ^ 6 , a.s., so that by Lemma 9 (which d i d not r e q u i r e (Q ) to be regulated) and I I I , t ;F s e ( 0 , t ] ° S S h Q ( X g , u (D (u) ) ) n (Xg, du) ds ] Y ±& s 0 = E^C ( 1 - e D ( u ) ) n ( X g , d u ) d s ] < t A l s o , A. Benveniste and J . Jacod show i n [2] , that there i s a f u n c t i o n h^ e E such that the f u n c t i o n a l se I ^ ( X ) 1 ~ ~ (0 , t ] 1 S i X s - ^ s ' s - e E > - 115 -has a bounded 1 - p o t e n t i a l . But i f Y g = S and s < <;, then by (f ) of Lemma 13, S"(s) = S + ( s ) and so X s ~ " V ( s ) - ' Thus, i f we put h(x,y) = h (x>- A h Q ( x , y ) , f o r x,y e F , we have that h(x,y) > 0 whenever x £ y , and K < 0 0 f o r every t , a.s. . I t f o l l o w s once more by the argument of (13.4) of Getoor [23] , that WF = W*1 Q y-a.s. on {T < *>} , T— T -F u f o r any (JLT^) t o t a l l y i n a c c e s s i b l e stopping time. Thus, t a k i n g b = h , c o n d i t i o n (8.3) w i l l h o l d , provided Q y(K^_ + , x > 0) = 0 F u f o r any (U 1 ) p r e d i c t a b l e stopping time T . We may, without l o s s of g e n e r a l i t y , assume that T > 0 , and that x = °° on { T > . Write T = T ( X . ) . Then T i s (F y ) p r e d i c t a b l e , so that Y T = 6 P y-a.s., and hence S~(T) = S+(T) > 0 P y-a.s. (we use here that P y (T = •§«») = 0) . Let T f T , T < T P P-a.s. . n n Then - 116 -S +(T ) + S-(T) , S+( T ) < S-(T) P y-a.s. n n a l s o , so th a t S~(T) i s a (G y) p r e d i c t a b l e stopping time. Thus ~ F ~ XT- " V ( T ) -e i t h e r equals Xg+(T) = XT ' ° r d o e s n o t ^ e ^ n E ' ^ r o m which we conclude that Q y(K h * K h) = 0. T— T F u Conversely, l e t x be (U £ ) t o t a l l y i n a c c e s s i b l e , w i t h F F 11 W ^ W Q -a.s. on {x < °°} . x- X Then as above, = WF Q^-a.s. , X- T— so that by d e f i n i t i o n , K ^ K Q^-a.s. on {x < °°} . Thus x- x (K^) r e g u l a t e s (Q x) . (8.9) now f o l l o w s as a s p e c i a l case of Lemma 9. The strong Markov property of (X t,Y t) i s immediate from that of (X ) . (8.8) f o l l o w s from (9.6), and the d e f i n i t i o n of (Y ) . To show (9.7), put A(t) m(W ) d l + s s Since m e E , A ( t ) i s adapted to (Q ) , hence i s a continuous p e r f e c t a d d i t i v e f u n c t i o n a l of (W" ) . I t w i l l s u f f i c e to show - 117 -that A ( t ) = t f o r every t , a.s., and f o r t h i s i t s u f f i c e s i n t u r n that the corresponding 1 - p o t e n t i a l s agree. E X [ e tdA f c] e t d L t ] - E X [ e t ( l - m ( X 4 . ) ) d L ] + E X [ t t e _ t l E \ F ( X t ) d t ] 0 0 ~ x r -D-, c;xr = E [e ]-E [ - -v -r _ D o 9 t -e b t s ; ( l - m ( X ))ds]+E X[ £ e t ( l - e ~ Z ) ] o 3 t e M o u { 0 } = E X [ e " D ] - EXC I s>0 Y *S s + EXC I e" S ( S ) ( 1 - e ' S' ) ] s>0 Y ^6 s -D(Y J (by (8.7)) = E X [ e D ] + E X [ 1-e D ] =1 , as r e q u i r e d . Turning to I I , l e t x F, and R = i n f { t ; d ( x , X J >£> , T = L,. t K. - 118 -By r i g h t c o n t i n u i t y of (X )[•, R and hence T , are P -a.s, s t r i c t l y p o s i t i v e . Thus by (8.7) ,, n(X ,{supd(x,WJ > e})ds] 0 = E X [ I 1, se(0,T] L t Y^S )>e} v s ( Y J ] < 1 showing 11(b). We o b t a i n 11(a) by modi f y i n g n on a set of X - p o t e n t i a l zero, as f o l l o w s . Let B = {xeF; n(x,{supd(x,W ) > e}) = »} e E t c Then f o r each t > 0 , (S,OJ) l B ( X s ( o j ) ) i s measurable w i t h respect to the ds $ P y completion of ^ l [ g t ] S ' as a f u n c t i o n C0,t] x a •+ ]R . Thus t l B ( X s ) d s 0 i s F y measurable f o r each u , and hence T = i n f { t ; > 0} i s an (^t) stopping time. Let R = i n f { r > T; n(X ,{sup d(X W )t>f})ds > 1 , S ( ; i t z or d ( X r , X T) > |> Then R n(X ,{sup d(X W ) > e})ds < 1 , - 119 -so that by d e f i n i t i o n of T , we must have R = T a.s. . But by 11(b) and the strong Markov property of X at T , we have that R > T a.s. on {T < °°} , and hence T = <» ' a.s. . Thus we may replace n(x,«) by 0 for x e B , and leave the other properties unchanged. Properties IV and V follow from Theorem 5.1 of Maisonneuve [38] , i f necessary modifying n on a set of X-potential zero, as above. A s i m i l a r modification, together with (8.7), gives us I. F i n a l l y , consider VI. Let t » T = i n f { t ; mCX )ds > 0} , and J s 0 T, = i n f { t ; k n(X g,U)ds > k} , for k > 0 . Then TAT kAl E X C i '•; i : = E X C n(X s,U)ds] < k , 0<s<TAT, A l k u Y ^6 s so that S - increases at only f i n i t e l y many times i n (0,TAT^Al) Since S~ i s a.s. s t r i c t l y increasing, we must have that T^ = 0 "x P -a.s., whenever x e I . Since k was a r b i t r a r y , VI(a) holds. As i n Proposition 1, l e t C be the set of dyadic r a t i o n a l numbers, and f o r e , n > 0 , l e t B = {beE; P^(D>n) > E ) e E , and e,n ' 0 - 120 -H = u_{D>0, W e B f or .t e C n (0 ,A) } e,r) A>0 ' t e ,n v ' We o b t a i n as before , that f o r every x e F , n(x,H ) <^n(x,D > fi) < ^ — £ ' n £ eCl-e-n) (l-e'D)dn(x,.) e(l-e-n) F i x p . Because H e Un, , we can f i n d e li!? such that e ,n 0+ e ,n 0+ H => H , and E y [ n(X , H \H )ds] = 0 . s' e,n e,n We argue as before , that (as F i s n e a r l y Borel) T = i n f { t > 0; X„ e F, e H }} t t+. e,n i s a (G/) stopping time. Since c {D > 0} , and t e,n £"[ I 1 \ Y s ) ] = E y [ Y ^6 E > T 1 0 s n ( X , H u )ds] s e,n < t/e(l-e n ) < co , f o r t > 0 , we have that Y e H° and 3 L e F , P y-a.s. But by the strong Markov property of (X^.) , no excursion can s t a r t at a stopping time at which X l i e s i n F . Thus T = P -a.s., so that - 121 -Y s and hence I 1 A CYJ = 0 , F-a.s.., s>0 H n {W_ e F> £ ,n 0 n(X , H n {Wn e F})ds] = 0 , o f o r each e,...n and ji . We can th e r e f o r e modify n as befor e , to make (9.8) n(x, H n {W e F } ) = 0 S , T) U f o r every x, e and n. . Now observe t h a t i f VI(b) f a i l s at x , then there i s a f i n i t e , p o s i t i v e , nonzero measure n' concentrated on {WQ e F } , f o r which IV h o l d s , and a l s o n(x,du) > n'(du). The argument of P r o p o s i t i o n 1 shows that n' (H. rr. {Wn e F } ) > 0 f o r some E , n > 0 , c o n t r a d i c t i n g (9.8). Thus VI(b) holds. • We now tu r n to the converse. Let E be a seperable m e t r i c space, w i t h an i s o l a t e d point A s i n g l e d out. Let 9 be as before. The main r e s u l t of t h i s s e c t i o n i s : Theorem 4 (a) Let F c E } A e F , F e E , and a ssume that ' F i s a U-space. Let (PQ) s a t i s f y (2.4), and assume that F i s n e a r l y o p t i o n a l f o r (Wt) and the P^ . Thus, f o r each p , the f i r s t h i t t i n g time DQ of F ; - 122 -D Q = i n f {t > 0; W't e F l ; i s a stopping time w i t h respect to the P y .completion of (U f c +) i n U . Assume that P y(W ± W n f o r some t > D ) = 0 f o r each y . Then D = inf{te(0,°°) n 1} ; W e F and f o r some e > 0 , W = W f o r every se[t,t+e) n Iff s t i s an exact p e r f e c t t e r m i n a l ( f t + ) stopping time which i s P ^ - i n d i s t i n g u i s h a b l e from D n v f o r each y . Let n(x,du) be an (F, E ; U, U) k e r n e l . Let (fi,F) be a measurable space, w i t h a r i g h t continuous f i l t r a t i o n (F f c) . Let P (du) be an (E, E; fi, F) k e r n e l c o n s i s t i n g of p r o b a b i l i t i e s , and assume that (F, F f c) i s augmented w i t h respect to the P y . Let (X f c) be a r i g h t continuous process with, values i n F , which i s A A adapted to (F ) , and w r i t e C = 9(X.) . For x e F , we denote ^ X > s X the image law of P under X., by Q . Assume that f o r each p r o b a b i l i t y y on E , the process ( X t , F*, P>\ Q X) i s strong Markov. Suppose that c o n d i t i o n s I-VI h o l d f o r these o b j e c t s . Let Y : 9, -> n be adapted to (F t ) , and s a t i s f y - 123 -(8.7) f o r t h i s n and (fc ) . Suppose a l s o t h a t : (X f c, Y , F , P X) i s strong Markov; / Y t ( D ( Y t ) ) , i f D(Y f c) < °o (9.9) I f Y . , then X ' = ) . L A , i f D(Y f c) = -(9.10) P y ( Y 0 e d u ) = P{j(du) f o r every u . Define m and S + by VI(a) and (9.7), and l e t L = sup{s; S +(.s) < t} . Then there i s a r i g h t continuous f i l t r a t i o n (J3^) of (ft,F) , a f a m i l y ( P X) of p r o b a b i l i t i e s on (U,(J) , and a xeE • r i g h t continuous process (X f c) adapted to (Gt) , such t h a t : Each (X t, G^ , P y, P X) i s strong Markov; (G, Gfc) i s augmented w i t h respect to the P ; (9.11) P P(X. _ kdu) = Py(.du) f o r each u ; A D Q 0 (9.12) Each S + ( s ) i s a (Gfc) stopping time, Gs+^g^ = F g , and (9.4) and (9.5) h o l d f o r every s , a.s.; (9.13) M = { t ; X -e I ' K i s ( G t ) - p r o g r e s s i v e , s a t i s f i e s (9.1) and (9.2) and M = { S + ( s ) ; s>0 , S+(s)«=° }. (9.14) For every (Gy) stopping time T, ( X > + T > L - L T ) i s c o n d i t i o n a l l y independent of G^  under P^ , given X^ , , w i t h c o n d i t i o n a l law (9.15) P ((X.,L.) e (•)) ; and (L^,) s a t i s f i e s (9.3). - 124 -(b) Moreover, i f the (PQ) a r e t n e t r a n s i t i o n laws of a r i g h t process s a t i s f y i n g (.6.5), and i f from other c o n s i d e r a t i o n s , we X X know that the excessive f u n c t i o n s f o r the (P ) are (Q )-nearly B o r e l , when r e s t r i c t e d to F , then i n f a c t ( X t , F, G t , P*) i s a r i g h t process s a t i s f y i n g (.6.5), In t h i s case, there i s a continuous PAF ( l t ) adapted to CU + ) , such that L t = 1 (X.) f o r every t , a.s. . Note: 1. We do not assume that (Q2*) i s reg u l a t e d . We can get away w i t h t h i s because (Y ) i s assumed to be given to us, and to be strong Markov. I f we were given only n , we would of course not know how to produce such a C v t) , unless (Q ) was regulated (see C o r o l l a r y 6 ). Al s o note that Theorem 4 uses only the strong Markov property of ( X t , Y ) , and not the knowledge of t h e i r exact j o i n t d i s t r i b u t i o n , which we could o b t a i n from C o r o l l a r y 4 provided (Q ) was known to be s t r o n g l y r e g u l a t e d . In t h i s case, the assumption that (X , Y f c) be strong Markov could be dropped, using C o r o l l a r y 5. Note that (.9.9) gives (8.8). 2. The c o n d i t i o n s of Theorem 4(a) are met i n the (v i ' b ) and ( v i ' c ) cases of Theorem 2, and i n t h i s s i t u a t i o n , Theorem 4(b) contains P r o p o s i t i o n 5. The proof of Theorem 4 i s e s s e n t i a l l y the same as that of Theorem 2, the c h i e f c o m p l i c a t i o n coming i n the proof of r i g h t c o n t i n u i t y . - 125 -3. The c o n d i t i o n that F be a U-space i s inc l u d e d i n order that Lemma 11 h o l d . This r e s u l t i s needed f o r Lemma 15. We w i l l use t h i s c o n d i t i o n again i n C o r o l l a r y 6, In that r e s u l t , c o n d i t i o n (9.9) again plays the r o l e of (8.8), but f o r now we use i t only to give (9.5). proof: R e c a l l that m and S + are defined as In VI and (9.7) Put t /m(u(s))ds + Z D(pCs)) e+/- 0 0<s<t s (t,P,u) = p(s)^6 . so that S + ( t ) = S + ( t , Y, X) . Let S~ ( t , p, u) = S + ( t - , p, u) , t t t oo ' and l e t V be the u n i v e r s a l completion of \P . As i n the proof of Theorem 2, we ob t a i n Cg+) stopping times £ t(p,u) , and fun c t i o n s x (p,u) measurable from 1/9 + to E^  . The set M of Lemma 2 becomes 0 = {(p,u); u(s) = A and p(s) = 6 f o r every s>9(u), S~(s,p,u)<«> f o r every se[0,3 (u)]n[0,») and i s s t r i c t l y i n c r e a s i n g at a l l such s , S~(s ,p ,u) •-><» as s ->- °° , and f o r every e >0 and te[0,3(u)) there i s an h > 0 such that sup d ( p ( s ) ( r ) , u ( t ) ) < e r whenever p(s) ^ 6 and se. ( t , t+h).} . - 126 -Because x t(p,u) = u(£t(p,u)) whenever t = S +(t,p,u) , the proof of Lemma 2 a p p l i e s , showing that t -> -x i s r i g h t continuous on 0 . Lemma 14 0 e 1/ , and (Y. , X,) e 0 a.s.. proof: Let 0' be the set of Cp,u) s a t i s f y i n g a l l but the l a s t c o n d i t i o n i n the d e f i n i t i o n of 0 . We see e a s i l y that 0' e \P . Thus consider 6 = { ( t , p , u ) ; (p,u) e 0' , and f o r every e > 0 there i s an h > 0 such that sup dCp(s) (r) , u (t)) < e r whenever p(s) 4 <5 and se - C t , t+h).} . Let T q kCp) = inf{s>q; p(.s) 4 8 and D(p(s)) > }^ The T k a r e CPfc+) stopping times, and {s; p(s) 4 8}= { T .(p) ; q e XJ , k > 1} , q , K. whenever (p,u) e 0' f o r some u . Thus 0 = Q u n { ( t , p , u ) ; ;(p,u) e 0' and 1 .3 K 1 , ( t ) l 1 ( x n , ( p ) ) S u p d ( p ( T n Av))(r),u(t)) (q-J,q)- Cq,q + j> q ' k ^ q» k - 127 -Since (t,u) -*u(t) i s measurable from B a 0 to E^ , and (v,x) -»- sup d ( v ( r ) , x ) r i s measurable, from IJ a E " to B , we see that 6 e B 8 I/" , and hence that 0 e f . Since 8(X.) = the f i r s t c o n d i t i o n of 0 f o l l o w s from (8. (take Z = 1," N ( s ) ) . I t f o l l o w s as i n the proof of Lemma 3, s (S,°°)v v that S~(s) < oo f o r s < ? , s < «>• . Let T = i n f { s ; S"(s) = S~(s+h) f o r some h > 0} , and R = i n f { s > T; Y s ^ 6}. They are both (F g) stopping times, Xj, e I a.s. on {!<?;}, and R 1 > E y [ I 1 ].=- E y [ se(.T,R] n(X ,U)ds] Y f$ s f o r every y . Thus, by V I ( a ) , R = T a.s. on {T < l) , so that T = £ a.s., and hence S~ i s a.s. s t r i c t l y i n c r e a s i n g on (0,£] n (0,») . Assume that VV (£ = 0 0 > S~(.s) A ° o as s oo) > 0 , and l e t - 128 -T = i n f { t ; I D(Y ) > k} k s e ( 0 , t ] S Y ±S s Then f o r some k , m(X g)ds < °°) > 0 , and hence k > E p [ D(u)n(X g,du)ds] > E [ ( l - m ( X s ) ) d s ] = - , 0 which i s impossible. F i n a l l y , we t u r n to the l a s t c o n d i t i o n . We argue as on p.94 of Blumenthal and Getoor [5] . F i x e > 0 , l e t T Q = 0 , and def i n e T f o r a a countable o r d i n a l number, by a T , = i n f { s > T ; d(X_, , X ) > e , or Y ^ 6 and a+1 a T. s s sup d ( X T , Y g ( t ) ) > e} , and t m T = sup T. , i f a i s a l i m i t o r d i n a l . Then T i s an (F ) stopping time f o r each .a . Let ot s R = i n f { s > 0; n(X. ,{sup d(X_,W. ) > e})ds > 1} M t By 11(b), R >.0 a.s., and by (8.7), CY„)] < 1 < 0 0 , se(0,R]{sup d(XQ,W )>e} Y ±6 t s - 129 -f o r every u . The integrand i s thus a.s. f i n i t e , and so T^ > 0 a.s. by r i g h t c o n t i n u i t y of X. . By the strong Markov property of ( x t ) > w e t h e r e f o r e o b t a i n that f o r each a < B , T < T„ a.s. on {T n <•-=} . a B B But f o r each u , -T h(a) = E y [ l - e a ] i s i n c r e a s i n g i n a , so that as u s u a l , there e x i s t s a countable o r d i n a l such that h(a) = h ( a Q ) f o r every a > . This i s only p o s s i b l e i f P PCT < ») = 0 . But i f T (OJ) = <» , then there i s an a < a_ such that a Q 0 t e CT (to), T .-(to)) , and by d e f i n i t i o n of T , a a+1 a sup d ( Y g . ( t t ) ( r ) , X t(to)) < 2e whenever s e ( t , T ..(to)') . " From" t h i s , we conclude that a+i Py((Y.,£.) e 0) = 1 , and as u was arbitrary.,, the lemma i s proven. • Now, f o r each u , l e t pQ_be generated by the P y - n u l l s e t s of F y , and de f i n e „„,,.-. 130 -HR,y G y r t and t h e i r i l k from the F y , as i n Theorem 2. We def i n e G = nG y . t y t The proof of Lemma 4 a p p l i e s to the p a i r of f i l t r a t i o n s ( F y ) , (Gy) , and s i n c e (9.5) f o l l o w s from (9.9), (9.12) i s now immediate. Conditions (9.11) f o l l o w s immediately from (9.10). Turning to (9.13), l e t M' = { S + ( s ) ; s>0 , S+(s)<°° ) Since X s + ( s ) £ F when S+(s)<°°,it f o l l o w s that M' c M . Conversely, to show that M c M' a.s., i t s u f f i c e s to show that Y (s) i F whenever t < t, and s < D(Y )• ; a.s.. F i x y , and l e t y'(A) = E^C e - S l A ( X s ) d s ] . By IV, V and the i n d i s t i n g u i s h a b i l i t y of D and D Q f o r each v , we see that there i s an A e U such that { DQ < D}' c A , and n(x,A)y'(dx) = 0 . F Thus by (8.7), P (Y e A f o r some t ) = 0 , as r e q u i r e d . S (Lp i s a (G t) p r e d i c t a b l e process, hence { t ; S-(L t) = t} i s (Gfc) pro g r e s s i v e . This set i s i d e n t i c a l to M', which i n tu r n - 131 -equals M a.s., so that M i s CG^.) progressive. Since S + i s rig h t continuous and s t r i c t l y increasing, (9.i) i s now immediate. Let MQ be the set of points of M, i s o l a t e d on the r i g h t . By (8.7), we cannot have P P(Y_ ^  <$) > 0 f o r any ( F Y ) predictable K S stopping time R > 0 . Let CT ) be a sequence of (G^) stopping times such that T + T and T < T on {T > 0} . Then n n S +(T ) + S _(T) , and S+(T ) < S"(T) & y-a.s. on {T = S _(L r p) > 0} . n n i Thus Y T = 6 P y-a.s. on t h i s set, and hence P y ( T e M 0 ) = 0 . Condition (9. 2) now follows, from which we conclude (9.13). Thus, a l l that remains of part (a) i s to show the strong Markov property of each (X t, G y, P y, P x) , together with (9.14) and (9.15) The analogue of Lemma 6 i s proved as i n that r e s u l t . Thus, for each x e F, the coordinate process (W t» U^, n(x,«), R^) Is strong Markov at every (U t +) stopping time R s a t i s f y i n g n(x,{WQ e F, R = 0}) = 0 . Because F i s a U-space, the argument of (13.4) of Getoor [23] applies once more, to show that X R_ e x i s t s P y-a.s. on {R < «>} f o r any ( F Y ) stopping time R such that Y D ^ 6 P y-a.s. t K on {R < °°} . Lemma 15 Let T be a (Gj.) stopping time such that L^ > 0 and T < S + ( L T ) on { T < o o } . Let p be a a - f i n i t e p o s i t i v e measure 132 -on E , and l e t H be the a - f i e l d F L - 8 1 U t ' T t on Q x U . Then there i s an (^t+) stopping time R such that (9.16) P y{oj; n(X. _(OJ), {u; R(u>,u) < «}) = °° , T < »} = 0 (9.17) R(a), Y T (a))) = ( T - S - ( L _ ) ) ( u ) , i f T(u>)' < «• LT 1 (9.18) R(OJ, u) = oo f o r every u e U , i f T(w) = <=° (9.19) P P ( Y e A, T - S " ( L T ) e B, T < ~ | F ) ( u ) T T = nQL ' {u;u'eA,ROa,u)eBn[0,«»)}) — , P - a . s . on {T<°°} n(X _(c o ) , (u;R(co,u) < °°>) T ' where A e U , B ;e B , and we make the convention that ° = 0 = -o » • proof: The c o n s t r u c t i o n of the stopping times R and R^ , and the proofs of (9.17) and (9.18) are as before, the only formal changes coming from the replacement of a by D , and the f a c t el that now the may take on the value oo . (9.16) w i l l again f o l l o w from (9.18), (9.19), and our convention that o o / o o = 0 . At t h i s p o i n t i n Theorem 2, we used a r e s u l t from I t o [32] . The corresponding r e s u l t here i s Lemma 11, and from i t we o b t a i n as before that - 133 -(9.20) P (C, Y eA, T - S " ( L T ) eB , T < »' ) LT 1 n ( X L _(.u),An{R(o),')eBnCO,»)}) T Cn{T<-} ^ ^ . ( c o ) , {R( U,.)<-}) P y(dw) whenever C e;FT _, A e t l , Be 8, and n(x,A) < °° f o r every x e F T But n(x, {D>l/k}) < oo f o r every x , so that the above a p p l i e s w i t h A replaced by A^ = An{D >1/k} . By our convention, the integrands f o r the A^ converge boundedly to the integrand f o r A, so that (9.20) and hence (9.19) h o l d , f o r every A e U. • With these r e s u l t s , the analogues of C o r o l l a r i e s 1 and 2 are proven as before (the former of these becoming that T = S+(L^,) P -a.s., whenever T i s a (G£) stopping time such t h a t X^, e F on {T < o o } ) . We now separate out the c o n t r i b u t i o n of Y^ to (X f c) , and w r i t e X. as a measurable f u n c t i o n x- = H ( V Yl(o,~)' *•> as before, and l e t , X / . \ f f-, / „ / „ l / s ? r / x s A u ( D ( u ) ) P"(A) = l A ( H ( u , Y| C Q J GO) , X, (co))P U K U W J J Cdo))PgCdu) The strong Markov property of (X t , GJ, P", P*) w i l l now f o l l o w as befor e , using now the strong Markov property of (x t , Y t , F t, P X ) . - 134 -Since L > + T - L T = £.(eJ(Y), x + T ) , t h i s strong Markov property i s used again, to argue as i n Theorem 2 that ( 9 . 1 4 ) holds. F i n a l l y , to show ( 9 . 1 5 ) , d e f i n e A ( t ) as i n the proof of ( 9 . 7 ) i n Theorem 3 . Because of ( 9 . 1 3 ) , we have by d e f i n i t i o n of S + , that A ( t ) = t f o r every t , a.s. . Thus we can reverse the argument i n Theorem 3 (using ( 9 . 1 0 ) ) , to o b t a i n ( 9 . 3 ) , showing part ( a ) . To show ( b ) , l e t f be a-excessive f o r (X f c) . Then f i s a l s o a-excessive f o r the C?Q) • As i n the proof of ( 9 . 1 3 ) above, we f i x ]i , and l e t y'(A) = E P [ e fclA(Xt)dt] . Because ( 6 . 5 ) holds f o r (W ) under the (P Q) , and f i s n e a r l y B o r e l f o r the (Q ) , by hypothesis, we conclude from IV and V that there are f ^ , e E^ such that f < f < f 2 , and n ( x , { f 1 ( W t ) ^ f 2 ( W f c ) f o r some t } ) y ' ( d x ) + P y ( f 1 ( W t ) *f 2 C W t ) f o r some t ) + Y N ( D ( Y ) ) + E y [ P U ( f 1 ( X t ) ^ f 2 ( X t ) f o r some t ) ] = 0 . - 135 -It follows immediately, that p y ( f 1 ( x t ) + f 2 ( x t ) f o r s o m e t ) = 0 , showing that (6.5) holds. The same w i l l hold f o r the canonical process (Wfc) on U , under the P , so that as i n the discussion preceding Lemma 13, applied to M = {t; Wfc e t }, there e x i s t s a continuous add i t i v e f u n c t i o n a l of (Wfc) with 1-potential x - D0 Kx) = E X [ e U ] By the perf e c t i o n arguments i n Meyer [40], we may choose a perfect version, adapted to CUfc+) • C a l l i t l t ( u ) . We need to show that 1 (X.) = L f c f o r every t , a.s. . Let (Jy be the P y completion of U , and adjoin the P y - n u l l sets of U y to Ufc , to obtain (Jy . Let M^ be the set of points of M that are i s o l a t e d on the r i g h t . Then as i n Maisonneuve [38] , p.409, the process ( l t ) i s the (U y)-dual predictable p r o j e c t i o n of t t -> -D oe 1 (W )ds + I ( 1 - e U S) ^ s seM n(0,t] under the measure P y . - 136 -Now l e t Then Z = Z 1 n ( s ) , Z e F y s ( t 0 , t ; L ] t ( -D(Y L ) Z g l F ( X s ) d s + I Z s ( l - e S ) ] s e M 0 t , - t = E y[Z E t 0 [ 1 0 lp(W s)ds + I -Doe (1-e S ) ] ] s e M ^ O , ^ - ^ ] & t - t X = E y[Z E t 0 [ 1 0 dl ]] s = E y [ z gd(i s(x.)) Thus, ( l t ( X . ) ) i s the ( F y ) - d u a l p r e d i c t a b l e p r o j e c t i o n of - D C * L Q ) t l F ( X g ) d s + I _ ^ ( l - e ) • seM Q n(0,t] But a d i r e c t computation shows that (L ) has the same property. Thus by uniqueness of dual p r e d i c t a b l e p r o j e c t i o n s , L = l t(X.) f o r every t , a.s., as r e q u i r e d . • For completeness, we i n c l u d e the f o l l o w i n g immediate c o r o l l a r y to P r o p o s i t i o n 11. C o r o l l a r y 6 Assume a l l the c o n d i t i o n s of Theorem 4(a), except those i n v o l v i n g -(Y ) . Assume that E i s a U-space, and that (Q ) - 137 -i s r e g u l a t e d . Then i n order that we may enlarge ft , F and the F , and f i n d a process (Y t) such that a l l of the co n d i t i o n s of Theorem 4 are met, i t i s necessary and s u f f i c i e n t that (8.9) hol d . proof: (8.6) holds by I I I , and as remarked i n the d i s c u s s i o n p r i o r to P r o p o s i t i o n 11, the c o n d i t i o n on the existence of kernels n(x,y; du) i s s a t i s f i e d when E i s a U-space. Thus P r o p o s i t i o n 11 a p p l i e s , g i v i n g (Y ) . Note that (9.9) i m p l i e s c o n d i t i o n (8.8), w i t h |u(D(u)) , D(u) < » A , otherwise . • G(x,u) = Note that i f we s t a r t w i t h (X t) and M as i n Theorem 3, v. and form ( X t ) , n , and (Y^) from them, then Theorem 4 w i l l apply, i and w i l l r e c o n s t r u c t ( X f c ) . S i m i l a r l y , a new process ( X f c ) may be constructed, using C o r o l l a r y 6 and then Theorem 4. As remarked i n s e c t i o n 8, the r e s u l t s of Gzy 1 [28]. show that (X^_) i s a r i g h t process, and so s t r o n g l y regulated. Applying C o r o l l a r y 4 , we see that ( X p has'the same law as ( X f c ) . I t f o l l o w s from a remark made f o l l o w i n g the proof of Lemma 2, that i n the s i t u a t i o n of Theorem 2, the constructed process . ( X . ) - 138 -w i l l be a.s. cadlag, provided that a.s., Y (•) has l e f t l i m i t s whenever Y ^ 6 . The f o l l o w i n g example shows that i n the present case, (X t) need not be cadlag, even when (X f c) and the Y^ are: Example 5: Let E be the u n i t square, w i t h the boundary p a r t i a l l y removed; E = ([0,1] x [0,1])\({1} x (0,1]) . Let F = [0,1] x {0} , and l e t r : [0,1) -»- ( 0 , c ° ) be continuous. Let P Q X ' ^ correspond to uniform downward motion on { x } x [ 0 , l ] at r a t e r ( x ) , s t a r t e d at (x,y) and stopped upon contact w i t h F . Let (X f c) be uniform motion to the r i g h t on F , stopped at the p o i n t (1,0) . I t i s a r i g h t process, so that (Q ) i s r e g u l a t e d . Set n(x,du) = -TTTZ ?~ - i—, \w p l X ' X ^ ( d u ) . 2 ( l - e x p ( - l / r ( x ) ) ) 0 We see that m(x) = 1/2 f o r x e [ 0 , l ) , so that VI(a) i s vacuous. Thus, c o n d i t i o n s I-VI are met, so that we can c o n s t r u c t (Y ) by C o r o l l a r y 6, and apply Theorem 4 to o b t a i n a process (X f c) . Let T = i n f { t ; X = (1,0)} . We have that T < °° a.s., and X = (1,0) f o r every t > T . There are only f i n i t e l y many excursions away from F i n any time i n t e r v a l [0,T-h] , h > 0 , but by l e t t i n g r ( x ) •> °° s u f f i c i e n t l y f a s t , as x -*• 1 , we can ensure that there are i n f i n i t e l y many excursions i n [0,T] , a.s., so that there i s no l e f t l i m i t at T , - 139 -We can impose c o n d i t i o n s to guarantee cadlag paths, as i n the next r e s u l t ; P r o p o s i t i o n 12 (a) Suppose that i n a d d i t i o n to the hypotheses of Theorem 3, we assume that (X f c) has l e f t l i m i t s on (0 , c ) , a.s. . Then h » (9.21) n(X g,{sup d ( X g , Wfc) > e})ds < » f o r every e > 0, h<£, a.s. 0 (b) Suppose that the hypotheses of Theorem 4(a) are v e r i f i e d , and that (9.21) hold s . Suppose a l s o t h a t : (X t) has l e f t l i m i t s on (0,5) a.s.; (X f c) has a l e f t , l i m i t at C a.s. on {£<°°, YA ^  6"}; and Y F C has l e f t l i m i t s on ( 0 , D ( Y T ) ] n ( 0 , 3 ( Y F C ) ) whenever Y 4 <5 , a.s. . Then we conclude that (X f c) has l e f t l i m i t s on (0,0 , a.s. . proof: Let T = T = i n f { t > 0 ; Y 4 6 and sup d(X ,Y (s)) > e} . ± t s t - t Since {(t,oo); Yt(.oj) 4 <5} i s i n d i s t i n g u i s h a b l e from a countable union of graphs of (F ) stopping times, as befor e , T must a l s o be an CFfc) stopping time. By the r i g h t c o n t i n u i t y of 0^) , T > 0 a.s. . Let T = T + T o 9 k+1 k BT. k Since CXfc) 1 S a.s. cadlag, i t has no o s c i l l a t o r y d i s c o n t i n u i t i e s a.s., showing that l i m T. > E a.s.. But - 140 -K ~ ^ A n. - ^ ( x . u h s u p d ( x , u ( t ) ) >:- £} ( Xs-' Ys ) ] s£(0,T k] Y ^8 s n(X ,{sup d(X' ,W.) > e})ds] S S u o by Lemma 9, showing (9.21) (b) We set T k = i n f { t ; n(X ,{sup d(X , W ) > E>)ds > k} s t s t 0 By Lemma 9, sup d(X , Y ( t ) ) > e and Y ^ 6 £ s s s f o r only f i n i t e l y many s e (0,T k] , and by hypothesis, l i m T k > t, . As i n Lemma 2, t h i s shows that Xfc_ e x i s t s at a l l times t = S~(s) A A (and equals x s - ) » s < C , and the ot h e r ; c o n d i t i o n s produce l e f t l i m i t s at a l l other r e l e v a n t times t . • Another d i f f e r e n c e between the general s i t u a t i o n and that of Theorem 2 l i e s i n that i n the l a t t e r we can have 'instantaneous' behaviour only when the measure n i s i n f i n i t e . The f o l l o w i n g i s an example i n which the c o n d i t i o n s of Theorem 4 are s a t i s f i e d , and n(x,u) < 0 0 f o r each x , but yet m E 0 . Example 6: Let E = [0,1] * [0,1] and F - C0,1] x {0}.. Let P Q X ' ^ corresponds to uniform downward motion on {x}x [0,1] , at ra t e 1 , s t a r t e d at (x,y) , and stopped upon reaching F . Let 141 (X f c) be uniform motion to the r i g h t on F , at r a t e 1, absorbed at the point (1,0) . Let r : [0,1]->- (0,°°) be B o r e l measurable, and set , , \ 1 ( x , r ( x ) ) r v n ( x ' d u ) " l - e x p ( - r ( x ) ) P 0 ( d u ) Then m = 0 . We may choose r such that a b dx r ( x ) f o r every a, b e [0,1) , a < b . In t h i s case, VI (a) i s s a t i s f i e d , so that we may apply C o r o l l a r y 6 and then Theorem 4 to o b t a i n the de s i r e d process (X t) . 10. A p p l i c a t i o n s (A). Skew Brownian Motion Let n be t h e . c h a r a c t e r i s t i c measure of the P P P of excursions of Brownian motion on H from 0 , and l e t PQ correspond to Brownian motion s t a r t e d at b , and absorbed at 0 . For a e [0,1] , l e t n 2 n^{W t > 0 f o r every t} n 2 n {W ^ 0 f o r every t} n = a n + + (1 - a)n . a Then f o r ft e Ufc, A e U we have that n ^ A , O Q > t , 9 ~ X A ) - 142 -= 2cm(A,a 0>t, e^A.W^O) + 2(l-a)n(A ,<y Q >t .O^A.W^O) W P 0 t ( A ) d n W t = 2a j P Q (A)dn+2(l-a) An{a0>t,Wt>0> Afi{a 0>t,W t<0} V ^ a > An{o Q>t} so that s a t i s f i e s ( i v ) f o r the PQ . Thus Theorem 2 produces a corresponding strong Markov process, which i s e a s i l y seen to be the "skew Brownian motion" examined i n Walsh [50] . This process was introduced by I t o and McKean [77] . They gave a c o n s t r u c t i o n i n terms of excursions, and i n t h i s context, Theorem 2 becomes a proof that the process they c o n s t r u c t i s strong Markov. We w i l l now discuss s e v e r a l p a r t i c u l a r techniques ( ( B ) , (C), and (D) below-) f o r producing objects n, (X t) , and (Y f c) , from which Theorem 4 w i l l produce a strong Markov process. They w i l l i n general be techniques that transform one set of such objects i n t o new ones. Each technique w i l l apply to the f o l l o w i n g s i t u a t i o n : (a) S t a r t w i t h a r i g h t process (X ) and a set M as i n Theorem 3. (b) Obtain a process on the boundary, (ft ) , and a k e r n e l n as i n that r e s u l t . Cc) Apply the technique under c o n s i d e r a t i o n to produce a new process (X ) and a new k e r n e l n' . (d) The c o n d i t i o n s of C o r o l l a r y 6 and Thereom 4 w i l l now be met, so that (Y^) and hence the strong Markov process (X^) , may be - 143 -constructed. We w i l l however show that the techniques work under more At general c o n d i t i o n s on (X f c) and n , than that they a r i s e as i n (b). This i s not mere f r i v o l o u s g e n e r a l i t y , as i t w i l l t e l l us that the techniques may be i t e r a t e d , provided t h i s i s done i n the order i n which they are presented. That i s , i f we s t a r t A. w i t h (X t) and n as i n ( b ) , we may perform (B) to produce A i (X f c) and n' . These w i l l meet the c o n d i t i o n s of (C), so that new A it (X^) and n" may be found, which i n t u r n meet the c o n d i t i o n s of (D). We can introduce yet another v a r i a t i o n , i n that i n c e r t a i n circumstances, there i s a corresponding transformation that produces i a new excursion process (-Y ) d i r e c t l y from the o l d one, (Y t) (without using C o r o l l a r y 6). Using. these d i r e c t c o n s t r u c t i o n s of the excursion processes, the transformations given i n (B), (C), and (D) may s t i l l be i t e r a t e d (however, i f C o r o l l a r y 6 i s used at any stage of the i t e r a t i o n , the end r e s u l t w i l l be as i f i t had been used at each sta g e ) . A p a r t i c u l a r example of t h i s i s the procedure of F. Knight and 0. P i t t e n g e r , f o r e x c i s i o n of exc u r s i o n s , which we w i l l o b t a i n i n (E) , below. (B) h-transformation: We w i l l change C P Q ) a n d n v i a A, h-transforms, w h i l e l e a v i n g (X ) unchanged. I owe t h i s to a - 1 4 4 -suggestion of Mike Cranston. X ^ Let (PQ) , (X £) , n and F be as i n Theorem 4 . Assume i n a d d i t i o n that (8.9) h o l d s , (Q ) i s r e g u l a t e d , E i s a U-space, and that upon augmentation of ( U , (J F C) w i t h respect to the P Y , the coordinate process (W^ _) becomes a r i g h t process. ( t h i s l a t t e r c o n d i t i o n w i l l h o l d i n the s i t u a t i o n of Theorem 3; see Sharpe [ 4 6 ] , p. ). Let h : E [0,1] be 0-excessive f o r the ( P X ) (with h(A)=0 ), and l e t C be the set of dyadic r a t i o n a l numbers. Set E, = {xeE; h(x) + 0} , h U = {:ueU; l i m i n f h(u(q)) > 0} . q4-0,q€C We l e t :.P X(du) be the t r a n s i t i o n laws f o r the h-transformed h 0 semigroup, as i n Meyer [ 4 1 ] . That i s , the ^ P Q are p r o b a b i l i t y measures on (U, U) s a t i s f y i n g ^PQCW,. 4 A f o r some t > 3 ) = 0 , and such that f o r every ( U + ) stopping time T , we have that hPX(An{3>x>) = \ T r ^ E X [ A n { B > T } , h ( X T ) ] , x e E h 0 » otherwise whenever A e . By 1 . 2 4 of Meyer [ 4 1 ] , the ^ P Q are the t r a n s i t i o n laws of a r i g h t process w i t h values i n E^ . Suppose now that h^(du) i s any p o s i t i v e measure on (U, U) s a t i s f y i n g IV, and (10.1) n Q(D > t ) < °° f o r every t > 0 - 145 -We w i l l d e f i n e a measure n^ = n^ such that (10.1) h o l d s , n O = 0) = 0 , and (10.2) n^(A n {3 > t ? n 9 t 1A') = An' h(Vhpot(A')dno 3>t}nU, h f o r every t > 0 , A' e l i and A e l i . To do so, we use t h i s (k) formula to c o n s i s t e n t l y d e f i n e n^ (B) (the p u t a t i v e . n^(B n {D e [ I , ^L.)}) ) , f o r k > 1 and B :a c y l i n d e r set i n the space of a l l f u n c t i o n s [0,») E . By (14.6) of Getoor [23] , and II I . 5 2 of D e l l a c h e r i e (k) et Meyer [13] , n^ extends to a f i n i t e measure on the c y l i n d e r a - f i e l d . The coordinate process i s immediately seen to have a (k) r i g h t continuous m o d i f i c a t i o n , so that n^ p u l l s back to a f i n i t e measure on (U, li) . We l e t n i - Z n i • x k x (The reason f o r i n s i s t i n g that n^ be concentrated on w i l l emerge l a t e r , when we consider property VI(b).) Since h ° W i s P y-a.s. r i g h t continuous on [0,°o) f o r each y , i t f o l l o w s that h o W i s n^-a.s. r i g h t continuous on (0,°°) . Thus we may use approximation by stopping times t a k i n g values i n , to show as u s u a l , that n^(A n {T<9}) = An h(W T)dn 0 x<3>nlL h - 146 -f o r every s t r i c t l y p o s i t i v e stopping time T , and A e (J T+ Since h < 1 we have that n^ < n^ Thus, i f n^(D= 0) = 0 , we have i n p a r t i c u l a r that f o r every p o s i t i v e f e U , (10.3) {D<3} f(W.)dn 1 = f(w.)h ( w D ) d n 0 {D<3}nU1 h We wish to apply C o r o l l a r y 6 and Theorem 4 to (X f c) , the x h ^ P Q , and the n(x,•) . Condi t i o n IV holds by d e f i n i t i o n , and co n d i t i o n s I , I I , I I I , and (8.9) f o l l o w from the corresponding h p r o p e r t i e s of n , s i n c e h < 1 i m p l i e s that n(x,«) < n(.x,») f o r each x £ F . We must thus only show co n d i t i o n s V and V I . ,x Observe f i r s t that h o w i s Pg-a-s. r i g h t continuous at 0 , so that PX(U\U\ ) = 0 f o r X £ E , and O h h. ' P X ( U h ) = 0 f o r x £ E\E h Thus n(x,{W 0eE h\F}\U h) { w o e V F } P " ( 0 ) ( U \ I I ) n ( x , d u ) = 0, and U h n(x,Uhn{W0£E\(EhuF)}) = PQ ( 0 ) ( U h ) n ( x , d u ) = 0 {W Q£E\(E huF) Now l e t B n F = 0 , B e E . Then hn(x,{W £B}) = l i m hn(x,{W eB, D, >t>) t+0 - 147 -= l i m t + 0{W 0eB,D >.t}nUh h(u(t))n(x,du) = l i m t 4 , 0{w (,eBnE, } U h Eg ( 0 ).[h(W t),D- >t]n(x,du) h(u(0))n(x,du) , {W„eBnE, } U h by monotone convergence. Further, i f A e U , t > 0 and B n F = 0 , then we can use t h i s to obtain that n(x,An{W 0eB, D' >t>) = h(u(t))n(x,du) {W_eB,D' >t}nAnIL 0 h {W.eBnE, } An{D >t} 0 h h(v(t))pJJ ( 0 )(dv)n(x,du) h(u(0)) p" ( 0 )(An{D. >t})n(x,du) h U {W_eBnE, } 0 h hPjJ ( 0 )(An{D >t}) hn(x,du) , {WQ€B} showing V. Since n < n , - 148 -1 - m(x) = ^n(x,D > t ) e t d t , and 1 - m(x) = n(x,D > t ) e tdt 0 h h i t f o l l o w s that i f m(x) = 0 , then m(x) = 0 and n(x,D n(x, D > t ) f o r every t . Thus a l s o n(x,») = n(x,*) . h „x now x e I , then there i s P - a . s . a t > 0 w i t h 1 (X )ds = 0 {hm=0} S Thus a l s o x e I , and f o r every r > 0 , rAt r A t n(X g,U)ds = n(X g,U)ds = - , showing V I ( a ) . F i x x , and l e t n^ be a p o s i t i v e measure on (U, (J) y s a t i s f y i n g I V f o r the ^PQ , and such that n n(du) ^ n(x,du) . Assume that n Q(W 0 e F) > 0 . We must show that n Q(W 0 e F Since n 0(U\U h) = 0 , and { l i m i n f h(W ) > e} e q*0,q£C M f o r each e > 0 , we may f i x e > 0 , and assume that n. concentrated on t h i s s e t . Let - 1 4 9 -8 = Ip. • E h y Since h i s 0 - e x c e s s i v e f o r the PQ , so i s g . By 1 . 1 9 of Meyer [ 4 1 ] , g/h i s 0 - e x c e s s i v e f o r the ^ P Y , so that we may form g/h N ! = N 0 ' But G / H ( P Y ) = P Y y so that as before , n, s a t i s f i e s IV f o r the P „ . A l s o , 1 g 0 n^du) < g / h ( h n ( x , d u ) ) = gn(x,du) . i s n e a r l y o p t i o n a l , so that as i n Theorem 3, there i s an exact p e r f e c t t e r m i n a l W T + ) stopping time which i s P Y i n d i s t i n g u i s h a b l e from i n f { t > 0 ; Wt i E h> , f o r each u . We have that f o r A e U fc , g P y ( A n {3>t}) = P y (An {8AD h>t}) Using t h i s , the formulae n^CD^ = 0 ) = 0 , and nt(A n e^ B n {D, > t>) = 1 t h An pJ ( t )(B) n iCdu) D h>t} (A e U , B e U) can be used to de f i n e a f u n c t i o n n* on the f i e l d - 150 -of elements of U , cons i s t i n g of f i n i t e unions of sets of the form {VJ £ C., i = 1. .k; D, e ( t . , t . ]} , {VJ eC. , 1=1. .k, D = t. l h i i + l t, I h l i where t, < ... < t, ,, i ° , j ^ k . On t h i s f i e l d , we have that 1 k+1 ' J ' n^ ^ n* < n(x,») , so that because n(x,«) i s countably a d d i t i v e , and there e x i s t U. i n the f i e l d with n(x,U.)<°°, n(x,U\uU.) = 0 l l i i we see that n^ extends to a measure on (U, U) such that + + x n^ < n^ ".< n(x,«) . This n^ s a t i s f i e s IV for (PQ) since n^ did f o r ( P X ) , so that by VI(b), n^(WQ £ F) = 0 or » . But n^ > n 1 > ^ = n Q , so that n^(WQ e F) > n Q(W 0 £ F) > 0 , and hence = n^(WQ £ F) = l i m n (WQ e F, D ^ D > q) q4-0,q£C :.lim § ( W ^ o ^ ° ^ £ C { W 0 £ F , D . > t } h ( W t ) ^ n Q ( W 0 £ F) , Thus H Q C W Q £ F) = 0 0 , showing VI Cb). C o r o l l a r y 6 and Theorem 4 now apply, to produce a corresponding strong Markov process. Note In one p a r t i c u l a r case, we can construct an excursion process ) d i r e c t l y , without appealing to Corollary 6. - 151 -That i s , suppose i n a d d i t i o n to the hypotheses of (B) , that (Y ) i s given as i n Theorem 4 (we could a l s o drop the assumptions that (8.9) hold and that (Q X) be r e g u l a t e d ) . Let F c F, F^ e E , and l e t h be the O-excessive f u n c t i o n h(x) = PQ(D<3 , WD e F x ). Then h(x) = l p \{ Aj.(x) f o r x e F . I t f o l l o w s by d e f i n i t i o n of , P. , that h 0 h P X ( D O ) = 1 ( x ) ^ E XCDO , h(W D) ] h = P 0 [ D < 3 > WD e F l ] h = 1 F (x) E h Thus f o r each x e F, h n ( x , D=3 ) = l i m h n ( x , { D=9>t }) = l i m t-H) t-X) {D>t}nlL h ( u ( t ) ) hPQ ( t )(D=3) n(x,du) Let = 0 . Yfc , i f y U h n {DO , WD e F x } [ 6 , otherwise . - 1 5 2 -Then ( Y ) i n h e r i t s the strong Markov property from 0^ .) » and f o r each p o s i t i v e f e U, and p o s i t i v e (F y) p r e d i c t a b l e process (^t) , we have that EyC I Z f ( h Y ) ] = F ? [ . I Z f ( Y ) 1 { (Y s s s > Q s s u h m u<d,w D€J ? 1 j-"Y 45 Y <^5 s s E y [ f ( u ) n ( X g , d u ) d s ] 0 {D<3,WneF_ }nU, D 1 n = E y [ f Cu) h Cu CP Cu) ) ) n (X ,du)ds] 0 {D<3-}nU, - ^ E [ fCu) nCX ,du)ds] by ( 1 0 . 3 ) and the above. Therefore, CX F C) , h n , (^ ) , C ^ Q ) , and F s a t i s f y the c o n d i t i o n s of Theorem 4, so that a new strong Markov process may be constructed. (C) E x c i s i o n : We w i l l e x c i s e pieces of the path of CX^.) , w h i l e l e a v i n g n unchanged. Suppose that ( X ^ ) and M are as i n Theorem 3 , and o b t a i n ( X T ) and F from them. Suppose that ( P Q ^ ' N ' A N D ' together w i t h (X ) and F , s a t i s f y the c o n d i t i o n s of C o r o l l a r y X 6 and Theorem 4 [note: We do not assume that (PQ) and n are a l s o constructed from C x t) and M as i n Theorem 3 . Since the - 153 -procedure of (B) l e f t (X t) unchanged, these hypotheses w i l l x h thus be s a t i s f i e d by C^PQ) A N D N ^ ' Assume now that we can w r i t e F = FQ U F ^ , where F^ and F ^ have d i s j o i n t c l osures i n the (X^)-Ray Knight c o m p a c t i f i c a t i o n E of E , and A e F : Suppose f u r t h e r , that (10.4) P X ( D •<•. o ° , and WF e F Q ) = 0 f o r x e E \ F Q . Let t A ( t ) = 0 1 (X )ds , A'(s) = i n f { t ; A ( t ) > s} *1 S Y' = y r i = r A t A ' ( t ) ' V ( t ) ' Since (X ) i s E - r i g h t continuous w i t h values i n F , i t f o l l o w s t h a t ; f o r every t there i s an e > 0 such that (10.5) *t+s = X A ' ( t ) + s f o r s e C0,e); a.s. . P r o p e r t i e s 11(b) and VICa) thus f o l l o w by the strong Markov property of (X f c) at A'(0) , and (8.9) f o l l o w s from (10.4). The remainder of p r o p e r t i e s I-VI are of course s t i l l s a t i s f i e d . Let 'x A x F F n ' Q be the image law of P on' (U , U ) under X. , so that as b e f o r e , each CV F y, P y, Q ' x ) F F ' i i i s strong Markov. Augment (U , (Jfc) w i t h respect to the Q _ T _ t to o b t a i n ( 0 , U^) . As i n Theorem 3, - 154 -{(W F) i s not cadlag i n E } e 0' and has Q y measure zero f o r each u , so that the r e g u l a t o r O x -' K F C f o r (Q ) , defined i n Theorem 3, w i l l be adapted to (Ufc) For every t , K ^ ( X . ) and K ^ ( T ) ( X . ) A d i f f e r by only a f i n i t e random v a r i a b l e , s i n c e ( X ) has no OS c i l l a t o r y d i s c o n t i n u i t i e s i n E , a.s., so that (X^) makes only f i n i t e l y many jumps from F^ to F^ i n any compact time i n t e r v a l . Thus (8.3) ho l d s . Write f o r the l e f t l i m i t s at t i n E of X. , X. , and W. . Then -'u as b e f o r e , i f T i s a s t r i c t l y p o s i t i v e (U ) p r e d i c t a b l e stopping time, we w r i t e T = T(X.) ,. and have that A'(T-) i s an (F y) p r e d i c t a b l e stopping time. Thus e i t h e r t -F A.F QU X A ' ( T - ) - = XA'(T-) ' E a - s ' o n { T < o o } But x F A'(T-)-l i e s i n the E-closure of F^ , so that i n the l a t t e r case, by (10.5), we have that A 1(T-) = A'(T) , and hence - 155 -~'F «' Thus G>(K° t K°) = 0 , T- T showing (8.4). Conversely, i f T i s ( 0 W ) t o t a l l y i n a c c e s s i b l e , we have as before, that vJ = W Q -a.s. on {x < oo} . T- T~ Thus i f WF t WF Q^a.s. on {T < o o } , T- i f -then a l s o 4 by d e f i n i t i o n of (K^) . (K^) i s t h e r e f o r e T- T t t 'x a r e g u l a t o r f o r (Q ) , so that C o r o l l a r y 6 and Theorem 4 apply, to produce a strong Markov process (X ) . Note: 1. There i s a d i r e c t c o n s t r u c t i o n of the excursion process a v a i l a b l e . That i s , suppose that i n a d d i t i o n , we are given '(Y ) as i n Theorem 4 ( c o n d i t i o n (8.9) need no longer be assumed), and put i Y = Y t V ( t - ) • As i n Lemma 13( c ) , ( Z A ( t ) ^ ^ S ^ t ^ p r e d i c t a b l e whenever (Z f c) i s (F..?y) p r e d i c t a b l e , so that i n t h i s case, - 1 5 6 -J Z., (X ) A(s) F v s j :f(u)n(X ,du)ds] = E [ fCu)n(X g, du)ds] ,. showing ( 8 . 7 ) . The strong Markov property of ? i t ( x t , Y T , F t > P X ) i s immediate from that of (x f c , Y T , F t , P X ) , so that Theorem 4 a p p l i e s . 2. C o n d i t i o n ( 1 0 . 4 ) i s only included f o r s i m p l i c i t y . We can remove x ' x i t by t a k i n g A e F^ , and changing ( P N ) to ( P Q ) , where the ' -X X P Q are obtained from the P Q by making p o i n t s x e F Q i n t o branch p o i n t s , w i t h p'^(X n e B ) = Q X ( T = i n f { t ^ 0 ; w f e F , } < » , W F e B ) The n(x,*) are a l t e r e d s i m i l a r l y , and we once again o b t a i n ( 8 . 9 ) . 3. I f , i n s t e a d of assuming that F n and F^ have d i s j o i n t c l o s u r e s i n E , we assume that t h i s holds w i t h the r e g u l a r topology of E , and that (X ) i s cadlag i n t h i s topology, then we would s t i l l get ( 1 0 . 5 ) . I f i n a d d i t i o n we had assumed that (X f c) was a Hunt process, then by ( 1 3 . 8 ) of Getoor [ 2 3 ] , (X ) - 157 -and ( x t_) a r e i n d i s t i n g u i s h a b l e . In t h i s case, the proof of r e g u l a t i o n s t i l l works, so that we could apply the arguments of (C) to c o n s t r u c t a strong Markov process. 4. S i m i l a r l y , i f we wish to use the d i r e c t c o n s t r u c t i o n of note 1, i t i s enough to assume that we are given (X t) , n , (Y ) , (PQ) , F and ( F F C ) as i n Theorem 4, together w i t h FQ and F ^ which have d i s j o i n t c l osures i n the r e g u l a r topology on E , such that F = FQ U F ^ , (%t) i s cadlag i n t h i s topology, and (10.4) ho l d s . The l a t t e r c o n d i t i o n could be removed as i n note 2. In t h i s case, we need no longer show (8.9); the p a r t i c u l a r form of the branching behaviour on FQ i s now needed to guarantee (9,9). (D) S c a l i n g and Time Change: We s c a l e n and time change (X ) . Let (X f c) , (F ) , ( P X ) and n s a t i s f y the c o n d i t i o n s of C o r o l l a r y 6 and Theorem 4. Let f : F -> C0,°°) and g : F -* 10,<?) be E -measurable, and assume that f i s bounded, and bounded away from 0 . Assume a l s o that f o r each x, f ( x ) g ( x ) < 1 , and that (10.5) g(x) ( 1 - e D ) d n ( x , 0 < 1 , w i t h e q u a l i t y only i f m(x) = 0 . Put t » A ( t ) = fCX )ds , A'(s) = i n f { t ; A ( t ) > s} 0 * t = V ( t ) ' F t " F A ' ( t ) ' a n d n' (x,du) = g(x)n(x,du) . - 158 -Then f o r Q the image law of P under X. , we have as befor e , A ' ' I J A i l that (X^, F £ , P , Q ) i s strong Markov. Since A i s continuous and s t r i c t l y i n c r e a s i n g , the t o t a l l y i n a c c e s s i b l e stopping times f o r the completions of (U^ .) w i t h respect to the QV and Q ^ ,correspond e x a c t l y (under a time change), so that 'x '0 (Q ) i s reg u l a t e d by the appropriate t r a n s f o r m a t i o n (K ) of (Y?) . C o n d i t i o n ( I I I ) holds by (10.5), and except f o r 11(b), V I ( a ) , and (8.9), the remaining c o n d i t i o n s of C o r o l l a r y 6 and Theorem 4 are now immediate. F F Let h e E ® E be p o s i t i v e . Then EQ' :J 0 {D<°°} hr \ / , (WF,WT,)h(WF,W1,)dn,(WF,-)ds] i(x,y);x#y} s D s ' D s' E X [ A» ( t ) 1 { ( x , y ) ; x ^ } ( X s ' W D ) h ( X s ' V g ( X s ) d n ( X s ' - ) J 0 {D<°°} f ( X )ds] s < i x [ I h(x x ) i (x x )] seCO.A'Ct)] U x , y ) , x ? y l s s Y 48 s s e ( 0 , t ] (K } s- s s i n c e f g < 1 , showing (8.9). C o n d i t i o n 11(b) f o l l o w s s i m i l a r l y . To show V I ( a ) , observe that s i n c e e q u a l i t y holds i n (10.5) only - 159 -i f m(x) = 0 , we have that m'(x) = 0 only i f m(x) = 0 and g(x) = 1 . Since f i s never zero, we see that i f x e I 1 , then a l s o x e I , so that t * n(X ,U)ds = oo f o r every t > 0 , P -a.s. . j 0 Because f i s bounded from zero, and g(x) = 1 whenever m'(x) = 0 , i t f o l l o w s that a l s o t » " A X n'(X g,U)ds = °° f o r every t > 0 , P -a.s., 0 showing V I ( a ) . Thus C o r o l l a r y 6 and Theorem 4 apply. Note: 1. Because f (x)g(x) . may be s t r i c t l y l e s s than 1, the f u n c t i o n J'(x,y) given by Lemma 10(a) may have values•decreased from those of J ( x , y ) . Thus fewer of the jumps of (X g) w i l l correspond to A excursions of (xp than was the case f o r (X g) and (X f c) . This w i l l be the case when f = 1 , g < 1 , so that even i f we s t a r t w i t h a continuous process (X f c) , the process (X^) w i l l i n general have many jumps, between p o i n t s of F . 2. I f on the other hand, f and g balance, so that f g H i , and f > 1-m (with e q u a l i t y only i f m = 0 ) then we can o b t a i n (YV) d i r e c t l y . That i s , suppose CY ) s a t i s f i e s the c o n d i t i o n s of Theorem 4, and l e t Y ' = Y y t V ( t ) • We see as i n Note 1 to (C), that (Y') s a t i s f i e s the c o n d i t i o n s - 160 -of Theorem 4, f o r (X^) , n' , e t c . . . , so that a strong Markov process (X^.) can be constructed. Let (X t) be the strong Markov process a s s o c i a t e d to (X f c) , (Y f c) and n , and set t t B(t ) = W C X s ) d s + ( f + T n - l ) ( X s ) d L s 0 B' (s) = i n f { t ; B ( t ) > s} . We w i l l see that X t -X B ' ( t ) , B'(t) < , B'(t) = <» f o r every t , a.s, Since the excursions are l e f t unchanged, i t w i l l s u f f i c e to show t h i s f o r t of the form S + ( s ) . Since » A | A s'+(s) = X s = X A ' ( s ) = XS+(A*(s)) ' i t w i l l s u f f i c e to show that B ' ( s' +(s)) = S + ( A ' ( s ) ) , o r , by r i g h t c o n t i n u i t y , that s' +(s) = B ( S + ( A ' ( S ) ) ) . S+(A'(s)) S+(A'(s)) B ( S + ( A ' ( S ) ) ) = L ^(X )dr + E\F r y (f+m-1)(X r)dL r A'(s) re[0,A'(s) ] Y ±6 r D(Y ) + r (f+m-1)(X r)dr I D ( Y J + refO,s] Y ^6 c f ^ z i ) C x ; ) d r = S' + ( S) , 0 - 161 -s i n c e ( f + m- l ) / f = 1 + (m' - 1) = m' . Thus (X^) i s a time change of (X ) on F , and no new jumps as i n note 1 are introduced. We summarize i n : P r o p o s i t i o n 13 Let (X ) and M be as i n Theorem 3, and o b t a i n X F , (PQ) and n from them. Let h : E -* [0,1] be 0-excessive f o r the P Q . Let F = F Q u F^ , where F Q and F ^ have d i s j o i n t c l o s u r e s i n the (X f c)-Ray Knight c o m p a c t i f i c a t i o n of E , A e , and , P X ( D < co and W e F N ) = • 0 f o r x £ E \ F . . h 0 u v U p Let. f : F-> C0,«>) 9 g : F [0,°°) be E -measurable and s a t i s f y : f i s bounded and bounded below on F^; f = 0 on FQ ; f g < 1 on F ^ ; and g(1 - m) < 1 on F^ , w i t h e q u a l i t y only where m = 0 . Let (X ) be the time change of (X t) by f , as above, and l e t n'(x,du) = A g(x) l ln(x,du) , x. e F , x £ F Q X * ' Then the c o n d i t i o n s of C o r o l l a r y 6 are met, f o r (j^O^ ' a n d n' , so that we may construct a strong Markov process (Xj.) from them by Theorem 4. • ( E ) the K n i g h t - P i t t e n g e r procedure S t a r t w i t h (Xj.) and M as i n Theorem 3, and o b t a i n F , (X t) , ( P Q ) J • C ^ T ) > n a n d m as there. Assume that (X f c) i s - 162 -cadlag i n E , and that F = F^ u F^ where F^ and F^ have d i s j o i n t c l o s u r e s i n - E, and AeF^. Let h be as i n the note to (B), and apply the d i r e c t c o n s t r u c t i o n i n that note to o b t a i n ) and m . Apply the d i r e c t c o n s t r u c t i o n i n Notes 1 and 4 of (C), to o b t a i n (X ) , (Y ) and n' (observe that (10.4) holds i n t h i s case by (10.3)). Apply the d i r e c t c o n s t r u c t i o n i n Note 1 of (D) to (\) , (Y^) , (Yj.) and n' , w i t h f = 1 - m + m > 1 - m (note that i f f = 1 - , then m = 0 so that \i = 0) , to o b t a i n (X ) , (Y^) and n" , and then use Theorem 4 to r e c o n s t r u c t a strong Markov process (X ) . Let t A ( t ) = 0 f C X j l - , (X )ds , A'(s) = i n f { t ; A ( t ) > s} . S r ^ S By our choice of f , we have that m'f = m , so that t t m"(X )ds = s A' ( t ) m " ( X A , ( s ) ) d s A'(t) m " ( X s ) d A s - m ( X s ) l ^ ( X s ) d s 11 As i n Note 1 of (D), i t f o l l o w s immediately that (X f c) can be described as f o l l o w s : K i l l (X f c) at the l a s t e x i t time from F^\{A} (that i s , a f t e r t h i s time, i t jumps to A ) , and then e x c i s e from the path of (X t) a l l those excursions away from F^ , that e i t h e r meet F^ or l i e i n U\U^ . By Lemma 16 below, there - 163 -are almost s u r e l y no excursions l y i n g i n U\U, that come back to h F other than at po i n t s of FQ U {A} . Since a l l the excised excursions occur before the l a s t e x i t time from F ^ \ { A } , they must a l l come back to F , and hence a l l meet FQ . Thus we have shown that the process (X f c) i s obtained from (X f c) by e x a c t l y the procedure given by Knight and P i t t e n g e r i n [35] . Our r e s u l t s t h e r e f o r e give another proof of t h e i r theorem, that the process constructed i n t h i s manner i s strong Markov. Note that i n order f o r our techniques to work, we have had to impose c o n d i t i o n s (.(X , F, F , P ) i s a r i g h t process, { t ; X t e FQ U F 1 > s a t i s f i e s (9.1) and (9.2)) guaranteeing the exi s t e n c e of a l o c a l time. They do not need to do so. We c l o s e t h i s s e c t i o n w i t h the lemma used above. Lemma 16 n(x, {D< 3,{WD e F 1>\U h) = 0 f o r every x e F^ . proof: F i x x e F^ , and e , s > 0 . Then n(x,{D< 9, WD e -P± , h(W j / 2k) < e f o r some j / 2 k e (0,s)}) [ 2 k s ] = n(x,{D.e(j/2 K, » ) , ^ e F ; L \ { A } , h(W j / 2k) <e , h(W ± / 2k) > e f o r i<j}) - 164 -C2 ks] f W . k ? Q 2 , Z (D<», = I 3=1 J {h(W / 2 k ) < e , h(W 1 / 2k)>e f o r i < j ) WD e F1\{A})dnCx,«) k < e n(x, h(W y 2k) <~e f o r some j/2 e (0,s)}) Thus, ( l - e ) n ( x , {D < 3 , WD e F x , hCW / 2 k ) ' < e f o r some j / 2 k e (0,s)}) < e n(x, {D = 3, or D < 3 and WD e F Q . }) . L e t t i n g k-» °° and then s -> 0 , we see that ( l - e ) n ( x , {D<3 , W ^ F ^ }\Uh> < e n(x, {D=3', or D <3 and WD e F Q • }) f o r every e > 0 . The r i g h t hand s i d e i s f i n i t e by 11(a) and I I I , so that we o b t a i n the d e s i r e d c o n c l u s i o n upon l e t t i n g e 0 . • - 165 -PART 2. MARTIN BOUNDARIES 11. Introduction Recall that using Theorem 4 and Corollary 6 of Part 1, we constructed a strong Markov process from a 'process on the boundary' ( x t) » a n c ^ a kernel n(x,du) . Our applications i n section 10 consisted of constructing these objects from other s i m i l a r ones, thereby obtaining a transformation that produced new strong Markov processes from old ones. A more far reaching program would hope to b u i l d the process on the boundary, and the kernel n , from scratch, knowing only the laws (P x) of the process stopped at the boundary. The problem of constructing a process on the boundary seems hard i n general; Condition (8.9) gives information about what i t s Levy system must be, and in a general state space, i t seems d i f f i c u l t to produce a process with a given Levy system (how does one 'compensate'?). For r e s u l t s i n Euclidean space, see Bass [ 1 ] , and Stroock" [47] . As to the kernels n(x, du), one might f i r s t try to construct the kernels n(x, y; du) of Section 8D. The natural approach i s v i a h-transforms, the hope being that n(x, y; •) = ^ fo r an appropriate (Pg) - excessive function h , depending on y . The f i r s t step i n t h i s approach would therefore be to determine those excessive functions h for which x i s an entrance point for the h-transformed process. In the second part of t h i s t h e s i s , we w i l l approach t h i s problem from a d i f f e r e n t perspective, i n the p a r t i c u l a r case of Brownian motion. The - 166 -arguments w i l l be completely independent of those given i n P a r t 1, and except i n t h i s i n t r o d u c t i o n , we w i l l f e e l f r e e to use completely  d i f f e r e n t n o t a t i o n from that used I n P a r t 1, i n an attempt to conform to the n o t a t i o n used i n the Martin boundary l i t e r a t u r e . The numbering of the r e s u l t s w i l l s t a r t over again a l s o . The major new r e s u l t s are Theorem 6 and C o r o l l a r i e s 1 and 2. 12. h-transforms and Martin boundaries In t h i s s e c t i o n , we w i l l o u t l i n e those p a r t s of the theory of h-transforms and Martin boundaries that w i l l be needed l a t e r . The r e s u l t s given below are taken from a set of notes by J . B. Walsh, but were o r i g i n a l l y proven almost e n t i r e l y by J . L. Doob ( [15 ], [17]), and by R. S. Martin ( [ 3 7 ] ) . The arguments using t i m e - r e v e r s a l seem to be l e s s w e l l known than they should be, so proofs w i l l a t l e a s t be sketched. In t h e i r present form, these arguments are due to J . Walsh, but many of them d e r i v e from the paper [30] of G. A. Hunt. Any mistakes are my c o n t r i b u t i o n . Many of the other r e s u l t s may be found i n the i n t r o d u c t i o n to Walsh [49]. In a d d i t i o n to the o r i g i n a l a r t i c l e s , proofs may a l s o be found i n Meyer [41]. The d e f i n i t i v e treatment w i l l be (once i t appears) J . L. Doob's book [19] on t h i s whole su b j e c t . Let E be a domain (an open connected subset of R^) , and l e t E be the a - f i e l d of i t s B o r e l subsets. Let 6 be some a d d i t i o n a l p o i n t which w i l l a c t as a cemetary. Let ft be the c a n o n i c a l space of f u n c t i o n s co : [0, ») E u'.{&} such that there e x i s t s £ > 0 w i t h co continuous on [0, £) , and {t ; co(t) = •£'} = [£', °°) : Let X " ; " t _> 0 be the coordinate maps on ft , and l e t - 167 -F = a(X ; s < t) , F =.F t s — « We w i l l sometimes need to work w i t h the space °, of f u n c t i o n s a) :(0, °°) E u { 6 } , which are continuous up to t h e i r l i f e t i m e , as above. By the appropriate abuse of n o t a t i o n , we w i l l use the same symbols V' , F, X t, e t c . . . f o r the corresponding o b j e c t s onn ft. Of course, i n t h i s context, X i s only defined f o r t > 0 . S i m i l a r l y , 8 w i l l denote the B o r e l a - f i e l d on both [0, °°) and (0, °°) . We f o l l o w the convention of making a l l f u n c t i o n s vanish at 6 unless otherwise s p e c i f i e d . Let P f c(x, dy) be the t r a n s i t i o n f u n c t i o n of Brownian motion on , k i l l e d upon l e a v i n g E , and as u s u a l , set P (x, .6) = 1 - P t ( x , E) , P ( 6 , { 6 } ) = 1 . I f now h i s excessive f o r (^t) > w e ^-et E, = {x e E ; h(x) < «>} h h P t ( x , dy) = \ 1 P f x , dy) h(y) , x £ E, h(x) t v " ' x y / ' h 0 , otherwise (note that unless-^ h 5 0 , {h = 0} = 0) . Theorem 1 (a) I f h > 0 i s excessive, then ( j ^ t ) forms a sub-Markov semigroup on E . Extend i t as above to form a Markov semigroup. Then: (b) For x eE , there e x i s t s a p r o b a b i l i t y measure ^ P x on (ft, F) - 168 -under which (X t) i s strong Markov f o r the semigroup (^ p t) a n ^ the f i l t r a t i o n > and f o r which h p X ( X n = x ) = ^ » (c) I f T i s any ( F t + ) stopping time, and A e F f c + , then h P X ( A , C > T) (x) E X[A, h(W T)] ; h (d) v i s h-excessive i f f v = 0 on EXE^ , and there i s an excessive f u n c t i o n u w i t h u = vh on E, ; h (e) Every excessive or h - e x c e s s i v e • f u n c t i o n i s - a.s. continuous along the path of (X t) , up to i t s l i f e t i m e . In general, i f we are given a measurable space,a p r o b a b i l i t y ..P-thereon, ;and aarandom v a r i a b l e Y-with values , i n ) (ft, wF) , wet say that >Y- i s an h-transf orm under P , i f i t i s strong Markov w i t h respect to P and the f i l t r a t i o n Y _ 1 ( F t + . ) w i t h t r a n s i t i o n semigroup hP t • We use the same n o t a t i o n i f Y takes values i n ft , but now the strong Markov property i s at s t r i c t l y p o s i t i v e stopping times. Now l e t (A, A) be a measurable space, and u a measurable f u n c t i o n : (A x E , A 8 E) + ( [ 0 , » ) , 8) such that each u(a,•) i s excessive. Let v be a measure on (A, A) , and set h(x) = u(a, x)v(dx) . Then i f h i s not i d e n t i c a l l y i n f i n i t e , i t i s excessive, and (f ) J> X(A) = 1 li v'1' h(x) J u(a, x) s P X ( A ) v ( d a ) u(a,•) on E, , f o r A e F n - 169 -R e c a l l that an excessive f u n c t i o n h i s c a l l e d minimal i f whenever h = u + v , where u and v are both excessive, then both u and v are m u l t i p l e s of h . (g) I f E has a Green f u n c t i o n G(x,y) , then G(•, y) i s a minimal excessive f u n c t i o n . Let the i n v a r i a n t a - f i e l d be I = {A e F ; e"3^ A n > -0}) =f A n { C > t}} . X (h) I f h i s minimal, then I i s t r i v i a l under each P . I f h > 0 h i s excessive and A e I , then ,P'(A, £ > 0) i s h-excessive. n ( i ) Let u and v be excessive, and l e t ( Y F C ) and P be a process and a p r o b a b i l i t y such that ( Y ) i s both a u and a v-transform under P . I f P ( Y Q ¥* <S) > 0 , then u and v are m u l t i p l e s of each other. ( j ) Let U be a subdomain of E , and l e t h be excessive on E . Let (Y ) be an h-transform under P . Then k i l l i n g (Y t) upon f i r s t l e a v i n g U y produces a process which i s a transform by the r e s t r i c t i o n of h to U , under P . I f A c E i s B o r e l , w r i t e T and L f o r the f i r s t h i t t i n g time and l a s t e x i t time of A , r e s p e c t i v e l y ; T = i n f { t > 0 ; X e A} L A = sup{t > 0 ; X e A} (sup(0) = 0) . - 170 -A l s o , f o r h _> 0 , l e t R Ah(x) = E X [ h ( X T )] . (k) I f h i s excessive, then R Ah(x) = h P X ( T A < ») h(x) . R.h i s a l s o the excessive r e g u l a r i z a t i o n of the infimum of a l l A excessive fu n c t i o n s m a j o r i z i n g h on A . (1) I f (X t) i s an h-transform under P , then k i l l i n g (X t) at L^ produces a process which i s an R Ah-transform under P . (m) I f (X f c) i s an h-transform under P , and EXA i s a subdomain of E , then the process A i s a v-transform on E\A , under P , where v(x) = h ( x ) h P X ( T A = ») [ f o r t h i s , see 5.1 of Meyer, Smythe and Walsh [43]] . Note that the new processes of (1.) and (m) may be i d e n t i c a l l y equal 6 , w i t h p o s i t i v e p r o b a b i l i t y . • Now suppose that E has a Green f u n c t i o n G(x,y) . Let t ^ > 0 and l e t <j> be bounded and i n c r e a s i n g , w i t h <j>"(t) = t f o r t e [0, tg) Let y^j e E be f i x e d , and suppose that <£ i s concave, so that p(x) = 4>(G(x, y 0 ) ) i s excessive. F o l l o w i n g Helms [29], l e t - 171 -I = {x e E ; G(x, y Q) > tQ} . I t i s an open neighbourhood of y^ , w i t h compact c l o s u r e i n E . Define the Martin f u n c t i o n to be K(x, y) = G(x, y)/p(x) , x, y e E . Let the Martin metric d on E , be Let E be the completion of E under d , and c a l l A = E \ E the Martin boundary of E . Theorem 2 (a) E i s compact, and E i s a dense open subset. The topology induced on E by d i s the usual Euclidean one. (b) K extends to a continuous f u n c t i o n from E x E to (0, °°] . I t takes on the value °° only on {(x, y) ; x = y e E} , and never vanishes. (c) For x e A , K(x, •) i s harmonic, hence excessive. We say that x e A i s minimal i f K(x, •) i s . Let A^ be the set of minimal x e A . (d) For x e A , x i s minimal i f f f o r every open neighbourhood A of x , d(x, y) = |K(x, Z) - K(y, z ) | ( l + { K(x, z) - K(y, z)|)dz I R 'AnE K(x, •) = K(x, •) (e) i s B o r e l . For every excessive h > 0 , there i s a unique measure v on A^ u E such that - 172 -r h = J K(x, -)v(dx) h Is harmonic i f f v(E) = 0 ( f ) Let h be harmonic, and continuous on an open subset U of E Let v represent h , and l e t y^ represent the f u n c t i o n i d e n t i c a l l y equal to 1 . Then v(dy) = h(y)p n(dy) on U . • I f h i s exc e s s i v e , w r i t e (, U X ) , n f o r the r e s o l v a n t of n A>U ( hP t) • Let X = t , x , t e (o , O be the reverse of (X ) from i t s l i f e t i m e . Observe that X e f t , t and that t h i s d e f i n i t i o n a p p l i e s regardless of whether X # i s defined at 0 . Theorem 3 (a) Let h be exc e s s i v e , and assume that (t; < °°) = 1 f o r every x £ E (as u s u a l , i t s u f f i c e s that t h i s ' holds f o r some xxt£"e,E, ) . . L-Let un y be a p r o b a b i l i t y on E . Then under the p r o b a b i l i t y ^P^ , the process (X ) „ i s a v-transfbrm, where F t t>0 v(x) = G(z, x) sj \ :,\ Vi (dz) h(z) - 173 -(b) I f P is.any p r o b a b i l i t y on (ft, F) under which ( x t ^ t > o i s an h-transform, then there i s some excessive f u n c t i o n v such that (X ) . i s a v-transform under P . t t>0 proof: (a) F i r s t , we show that v < 0 0 a.e. . Let A = {x e E ; P X(C > n) < 1/2} , n h B = t J n A J 0 n 1 A (X )ds , T(j.) = i n f { t ; Bfc > jn} Since X t U P U > n) i s ,P - a.s. continuous i n t < £ , i t f o l l o w s that f o r each x £ E , h ^ P X ( T ( j + l ) < - |T(j) < » < hP X(£ o 8 > n | T(j) < -) = h E X [ hP T ( j ) ( £ > n) | T(j) < ~] < i . Thus G(x, y)h(y)dy = ,P (x, A )dt h t n = , E X [ B ] h 0 0 < n I P X(B > jn) <_ n £ 2, J = 2n j=0 h " j=0 I n t e g r a t i n g w i t h respect to u , we see th a t - 174 -r j v(x)h(x)dx <_ 2n f o r each n . n Since (P ) i s strong F e l l e r , and h i s l . s . c , we see that A = {E X[h(X )] < h(x) / 2} n n i s open. By hypothesis, t E , so that not only i s v < 0 0 a.e. on E , but v(x)h(x)dx i s a Radon measure thereon (compact subsets of E have f i n i t e measure). The occupation time measure of (X f c) i s v(dx) = y(dz) U ( z, dx) = v(x)h(x)dx , J h • so that by the r e s u l t s of C a r t i e r , Meyer, Weil [ 6 ] , (a) w i l l be shown, once we show that ( IT'S and ( U^) are i n d u a l i t y w i t h respect to h A>0 V A>1) Let G^(x,y) be the X-order Green f u n c t i o n . Then i f f and g are p o s i t i v e and B o r e l , we have that < f, U g > ' v 5 v f ( x ) v ( x ) h ( x ) I G (x,y)g(y)v(y)dydx v(.x) J g(y)h(y)v(y) 1 h(y) J f ( x ) h ( x ) GX ( x , y ) d x d y by symmetry of G^  , as required. (b) Let (E(n)) be a sequence of subdomains of E., such that E(n) has compact c l o s u r e i n E(n+1) , and uE(n) = E . J \ - 175 -Let T(n) = i n f { t ; X e E(n)} . Then ( X T ( n ) + t ) t > 0 (defined by r i g h t c o n t i n u i t y a t 0 " i f T(n)"v==) 0) i s s t i l l ' a n h-trans'form £-'~ under P , and i t does have an i n i t i a l d i s t r i b u t i o n , so that (a) a p p l i e s . Le t (X^) be the reverse of ( X x(n)+t^ ' S O t' i a t t n e r e e x i s t s an excessive v. such that (X n) i s a v -transform under P . Let n t n S = i n f { t > 0 ; X n t E(n)} n t L = sup{t > 0 ; X n -k E(n)} . n t Then X R = Xn+^~ i f t < L , and i n p a r t i c u l a r , i f t < S . From ( i ) t t n c n and ( i ) of Theorem 1, i t f o l l o w s t h a t there i s a constant c > 0 such J ' n that v , = c v on E(n) . We may m u l t i p l y each v by a constant, n+1 n n J c J n j to make each c = 1 , so that an excessive f u n c t i o n v on E may be n defined to make v = v on E(n) , f o r each n . Since each (X n) i s n t a v -transform, i t f o l l o w s that (X ) i s a v-transform under P . • n t Lemma 1. Let h > 0 be harmonic i n E , and l e t A be compact i n E . Then , P X ( L . < = 1 f o r each x e E . n A proof: By ( f ) of Theorem 1, and (e) of Theorem 2, i t s u f f i c e s to show t h i s f o r h minimal. The event {L = c,} i s i n v a r i a n t , so that i n t h i s case , P X ( L A = £•} = 0 or 1 f o r each x h A - 176 -(by (h) of Theorem 1). Since = E , t h i s same r e s u l t shows that t h i s f u n c t i o n i s h-excessive. Thus by (d) of Theorem 1, i t i s e i t h e r i d e n t i c a l l y zero or i d e n t i c a l l y one. Assume the l a t t e r . Then f o r each x e E , i t f o l l o w s from (k) of Theorem 1, that h(x) = h(x) h P X ( L A = 5) £ h(x) h P X ( T A < ») = R Ah(x) . Since A i s compact, R h i s a p o t e n t i a l . This i s a c o n t r a d i c t i o n , since the only harmonic f u n c t i o n dominated by a p o t e n t i a l i s i d e n t i c a l l y zero. • We w i l l use the n o t a t i o n P x = ... X P X . y K ( y , 0 whenever y e E . Theorem 4 Let h(x) = EuA K(z, x)y(dz) 0 2£ be ex c e s s i v e . Then f o r each x e E, , X e x i s t s i n E , P -a.s., and n c,- n has d i s t r i b u t i o n h P x ( X ? _ e dy) = ^ " K ( y > x)y(dy) proof: By ( f ) of Theorem 1, i t s u f f i c e s to show that P X ( X = y) = 1 f o r y e E u A y £- 0 - 177 -Case 1: Suppose y e E . By (1) of Theorem 1, and Lemma 1, there e x i s t s ' an excessive*:, f u n c t i o n u u c suchUthat y e E „iand •-uHtransformsahave " -i u f i n i t e l i f e t i m e s . By Theorem 3, (X ) i s a G(y, •) transform under P^ , so that there e x i s t s a p r o b a b i l i t y v such that u y P V U e (0, ~) , X ?_ = y) = 1 . This holds f o r every \3 by m i n i m a l i t y , as i n Lemma 1. Case 2: Suppose y € Ag . In t h i s case, there i s no reason why £ should be f i n i t e . Let \\ be an open neighbourhood of x e E , w i t h compact c l o s u r e i n E . Let (E(n)) be a sequence of subdomains of E', w i t h each E(n) having compact c l o s u r e contained i n E(n+1) , and such that E ( l ) contains the cl o s u r e s of £ .and ^l'ifi andv.uE(n)E=,E By Lemma 1, y E(n) fo r each n , so that we may form the reverse X11 of X from h, . E(n) and o b t a i n from Theorem 3(a) (and (1) of Theorem 1 ) , that X n i s a G(x, •) transform under P x . Let y k(w, z) = G(w, z)/G(w, x) , f o r z e I and w e E*" = EX(£ u ^ ') . Let T(n) = i n f { f > 0; X ^ E'} . Since k('» z) i s bounded on E' , f o r z e £ , i t f o l l o w s as u s u a l , that k ( X n _ ,, z) tv\T(n) ' - 178 -i s a bounded martingale under P X , f o r each z £ Y . I f U denotes ° y u n the number of upcrossings i t makes, of a f i x e d i n t e r v a l [a, b] , we th e r e f o r e o b t a i n that |a| + sup k(w, z) E X [ U ] < W £ E ' y n - b — a But U i s a l s o the number of upcrossings of [a, b] by k(X , z) , n t f o r t £ (Lr v» > L„. v] 2,UA E ( n ) J Thus U t U , and si n c e the above bound d i d not depend on n , we see n oo that U < oo p x -a.s. . Thus by the usual argument, i t f o l l o w s that oo y l i m k(X , z) t+£ e x i s t s y P X -a.s., f o r each z e \ . But k ( • , z) i s bounded away from zero on E' , and K(w, z) = k(w, z)/k(w, y Q) f o r w £ \ , so that i n f a c t l i m K(X z) t+£ e x i s t s P X -a.s., f o r each z £ ) . By F u b i n i ' s theorem, t h i s holds y ^ P x -a.s., simultaneously f o r a.e. z £ £ , so that i n f a c t d(X , X ) - > 0 as s , t + S , P x -a.s. . Thus, the E - l i m i t X e x i s t s . v s' t y ?-- 179 -We must now i d e n t i f y i t as y . Let (A(n)) be a decreasing sequence of neighbourhoods of y , w i t h nA(n) = {y} . We must show only that T. , N < 0 0 P X -a.s. A(n) y But K ^ Z> = R A ( n ) n E K ^ > 0 ( X ) " x ) y P X ( T A ( n ) < - ) by (k) of Theorem 1, and (d) of Theorem 2, so that t h i s holds. • This proof traces i t s ancestry to Hunt [30], The technique of k.,.1 "i killdhg?at J-the - l a s t J - e x i t f t>imesE,of; compact',subsets, of" E" , e l e t us H _ . • eljjaiSa-te'hisTu^ chains.' We say that a p o i n t y e A_ i s a c c e s s i b l e i f P X ( ? < 0 0) = 1 0 y for some (and hence every) x e E . Let Acc denote the set of a l l a c c e s s i b l e p o i n t s . Theorem 5 Let u(x) = G(x, z)>»(dz) be an a.e. f i n i t e p o t e n t i a l , and l e t y e Acc . Suppose that K(y, z)v(dz) < °° Then there e x i s t s a unique p r o b a b i l i t y P^ on Q , such that (X^) i s u-transform under P^ , and u P y(X_ -> y i n E as t + 0) = 1 u t J proof: Let - 180 -a = I K(y, z ) v(dz) , y(dz) = - K(y, z ) v(dz) Because y i s accessible, we can reverse from £ to obtain that under , ( 5 0 i s a transform by the function y ' t — — T G(z, x)y(dz) = i -K(y,z) a G(z,x)v(dx) = — u(x) Since li m X = lim X = y , P1^ t . J.T- t y ' y -a.s. t+0 by Theorem 4, we have shown existence. To show uniqueness, we w i l l show that i f P i s a p r o b a b i l i t y on , as i n the theorem, then ( x t) ^ s a K(y, •) transform, with i n i t i a l measure y , under P . Thus the law of (X t) , and hence o f f (Xfc) i s determined. I t follows from Theorem 3(b), that since (X^) i s a u-transform under P , ( x p i s an h-transform for some h . Write h(z) = K(x, z)n(dx) By Theorem 4, X t P(X ?_ e dx , £ > t) = E[ hP -..OX e d x > l = E[ ; K(x, X )n(dx) , £ > t] , 1_ / IT \ L h(X t) for every t > 0 . But X^_ = y P-a.s., so that n must be a point mass at y , and hence h(z) = cK(y, z) for some c . - 181 -By Theorem 4, we use a s i m i l a r argument to see that X^_ £ E P - a . s . . Let u' be the d i s t r i b u t i o n of . I t i s then the i n i t i a l d i s t r i b u t i o n of (X^) , so that by Theorem 3 ( a ) , (X f c) i s a transform by ( -i G(z, x)y'(dz) = v(x) . K(y, z) I t f o l l o w s from ( i ) of Theorem 1, that v = cu f o r some c > 0 Since a p o t e n t i a l determines i t s measure, we conclude that 1 y'(dz) = c v(dz) , K(y, z) or i n other words, y' = acy , so that c = 1/a and y' = y . • We wish to examine the set D = {(x, y) e (Acc u E) x (Acg u E) ; there e x i s t s a p r o b a b i l i t y ^ P X on ft ' under which (X f c) i s a K(y, •) transform, and X Q + = x a.s.} Theorems 4 and 5 show that D •=> (Acc * E) u (E x Acc) • The arguments given above a l s o s u f f i c e to show the f o l l o w i n g statements: I f (x, y) e D then there i s only one law ^P s a t i s f y i n g the r e q u i r e d c o n d i t i o n s ; and (12.1) D i s symmetric (that i s , (x, y) e D i f f (y, x) e D) , and i f (x, y) e D and A e(ft, F) , then x P y ( A ) = y P X ( ( X t ) e A). - 182 -R e c a l l that a f u n c t i o n f on E i s s a i d to have f i n e l i m i t a , at a poin t y e Ag , i f P X ( l i m f ( X J = a ) = 1 y tH t f o r some (and hence by m i n i m a l i t y , as i n Lemma 1, f o r every) x e E By the c o n s t r u c t i o n of.Theorem 5, t h i s i s equ i v a l e n t to P y ( l i m f ( X ) = a) = 1 X t+0 t f o r some (and hence every) x e E Lemma 2 Let A be open i n E , w i t h y e A n Ag . Then h = R E \ A K ( ? ' '> i s a p o t e n t i a l ; that i s , f o r some measure v , we have h = [ G(z, -)v(dz) = U°v . J • E proof: Write r K(z, -)y(dz) . h = EuA J0 Since K(z, •) i s p r o p o r t i o n a l to G(z, •) , i t w i l l s u f f i c e , by Theorem 4, to show that h P x ( X ? _ e Ag) = 0 f o r x e E But P X ( X r = y) = 1 , y 5-- 183 -and so /< L E\A < 0 * 1 as w e l l . Thus by (1) of Theorem 1, . P X ( X e A ) = P X ( X T e A ) = 0 . • h £- 0 y L E U 0 Theorem 6 Let x, y e Ace . Let A be an open subset of E such that y e A and x does not l i e i n the c l o s u r e of A . Let z e A n E , and set f(w) = K(y, w)/G(z, w) . Then the f o l l o w i n g c o n d i t i o n s are equivalent (a) (x, y) e D (b) Let Then K(x, z)v(dz) < 0 0 (c) f has a f i n i t e f i n e l i m i t at x (d) For some w e E , l i m i n f f ( X ) < «, p w - a . s . proof: We w i l l show that (a) <^ > (b) , and (a) => (c) => (d) => (a) . ((a) => (b)) : Let ( Y ^ be (X f c) k i l l e d at L ^ , so that (Y f c) i s an Rgv^ K(y> ') transform under ^ P x . Let y be the d i s t r i b u t i o n of LE\A under P x . Let (Y ) be the reverse of (Y ) ..from-its l i f e t i m e , y t t By (12.1), (y, x) a l s o belongs to D , and - 184 -y P X ( ( Y t ) t > ( ) e A) = x P y ( ( X ) t > Q e A) = x p y ( ( X t ) t > 0 e A) , f o r A e(ft, F) Thus, by Theorem 3(a), (Y^.) i s also a transform by ' G(z, •) K(x, z) y(dz) I t f o l l o w s from ( i ) of Theorem 1, that there i s a constant c w i t h 7 ^ 4 y(dz) K(x, z) G(z, -)v(dz) , and s i n c e a p o t e n t i a l determines i t s measure, i t f o l l o w s that K(x, z)v(dz) = c < 0 0 . ((b) => ( a ) ) : Let By Theorem 5, there i s , a p r o b a b i l i t y , P X on Q under which (X ) h t i s an h-transform s t a r t e d at x • Define a p r o b a b i l i t y P' on ft x ft by. P'(A) = X T (u) l A ( u , w') P A (du') ,P X(dco) , A y h and l e t P be the image law of P' on ft under the random v a r i a b l e Y t(u), co') Xt(u>) X t - T A ( U ) < U " , t < T A ( o 3 ) t >_ TA(oa) Then f o r f p o s i t i v e and bounded, we have that - 185 -E [ f ( X t + s ) F 8 ] E[f(W' T A ^ S I F s ] + *K<W' T A > S + t ' F s J + E [ f ( X t + s ) , T A e(s, s + t ] | F g ] X X E [ f ( V ^ V ^ s ^ ^ [ f ( V ' T A > t ] V ^ s } A • + 1 {T A>s} X Ez[f(X„ )] .P S ( T A e dr, X e dz) y L v t - r h A TA X X = E y ^ V 1 1 { T A < s } + y E [ f ( V > T A > t ] ^ s ) + 1 {T A>s} X E^[f(X. .)] P & ( T , 6 dr, X £ dz) y t - r y A T A X y E S [ f ( X t ) ] Thus (X f c) i s a K(y, *) transform under P , as r e q u i r e d , ((a) => ( c ) ) : The Green f u n c t i o n f o r a K(y, •) transform i s We have that i f i s a compact neighbourhood of z , contained i n E , then f o r w i \ , G(w, z) = G(w', z) y P W (T£« £ dw') sup G(w', z) < we 9 r y - 186 -Then as u s u a l , Z = G(X , z) t y t A T £ i s a bounded ^ P W - martingale, f o r w e E\| ?•. Since (x, y) e D , and (Zj.) i - s bounded, we o b t a i n from the Markov property, that ( z t ) t > n i s a bounded P x - martingale as w e l l . Thus, i t convergex P -a.s. to a random v a r i a b l e Z „ , as t+0 . Since Z. > 0 f o r y 0+ t t > 0 , ZQ_|_ cannot be i d e n t i c a l l y zero. Let (E(n)) be a sequence of open subsets of E such t h a t E(n) has compact c l o s u r e i n E(n+1) , and uE(n) = E . Let ~n , t > L. E(n) Then by Theorem 3, (X t> i s a K(x, •) transform under P , and / ^ O ' 6 ) " y P X ( L E ( n ) = °> +° By m i n i m a l i t y of x , Z„, i s P x -a.s. constant on {X ^6} , so that J 0+ y 0 i n f a c t , there i s a constant a 4 0 , such that Z 0+ a P - a . s . Thus al s o ?PX(x£ t 6) = Px(xJ 4 6, l i m G(X t, z) = a) t'f £ P W ( l i m G(X„, z) = a) P X ( ) C e dw) x m y t y 0 E tT£ ' * - 187 -so that -,w, t + £ P ( l i m G(X_, z) = a) > 0 x y t f i r ^ f o r some w e E • By m i n i m a l i t y , t h i s p r o b a b i l i t y equals 1, showing ( c ) . The i m p l i c a t i o n (c) => (d) i s t r i v i a l . ((d) => ( a ) ) : By (d) of Theorem 1, f i s G(z, O-excessive, so that ( f ( X . ) ) i s a P W - supermartingale, f o r each w e E . Thus t z ( f ( X t ) « n) i s a bounded ^ - supermartingale f o r each w , so that by the Markov property, ( f ( X f c ) A n) ^ i s a bounded X X P -supermartingale. Thus, i t converges .at, 0 ,, P -a.s., and hence zh- - . . 8'. - •• .\ he-: z L •• ' i n f a c t l i m f ( X ) e x i s t s i n [0, °°] , P -a.s. t+0 z By m i n i m a l i t y and time r e v e r s a l , t h i s l i m i t i s a constant a (non zero, but a p r i o r i , p o s s i b l y i n f i n i t e ) . As i n Lemma 1, a l s o f ( X J -> a as t + 0 , P - a . s . t w f o r each w e E , so that by (d), we have that i n f a c t , a < Therefore E X [ f ( X )] = l i m E x [ f < X T ) A n] < a < » . A n-*» A Let R AK(x, •) = U^v . Since z e A , we have that L^ > 0 P Z -a.s., and so x ' - 188 -E X [ f ( X T )] = E ^ f C X , )] • U°v E Z [ f ( V ] 1 f(w) G(w, z)v(dw) (Theorem 4) U°v(z) J 1 K(x, w)v(dw) Thus K(x, w)v(dw) < oo and by.the previous i m p l i c a t i o n , that (b) => ( a ) , we get that (y, x) e D . Thus (x, y) e D by (12.1), as r e q u i r e d . • Note that the argument f o r ((d) => (a)) shows that i f h i s excessive and x e Acc , z e E , then h / G(•, z) has a f i n e l i m i t at x ( p o s s i b l y i n f i n i t e ) . This was f i r s t shown by Nairn [44]; she showed i t more g e n e r a l l y , f o r x e A^ . We could o b t a i n t h i s more general r e s u l t by combining the above argument w i t h the argument of Theorem 4, i n which upcrossings are estimated by k i l l i n g ( x t ) a t times -^E(n) » under the p r o b a b i l i t y x P z , and then r e v e r s i n g time. Theorem 7 Let r > 0 . Let E be a bounded domain such that f o r every po i n t x e 3E , there are open b a l l s B and C , of r a d i i greater than r , both c o n t a i n i n g x i n t h e i r c l o s u r e s , and such that B c E , C c R d \ E . Then the Martin c o m p a c t i f i c a t i o n i s homeomorphic to the Euclidean c l o s u r e of E , and moreover D = {(x, y) e 3E x 3E; x ^ y}. - 189 -proof: The f i r s t statement i s a consequence of the r e s u l t s of de l a V a l l e e Poussin [11], and these r e s u l t s , together w i t h Lemma 5.1 of Martin [37] s u f f i c e to show (d) of Theorem 6. • 13. A Martin Boundary i n the plane Let D' = {(x, y ) ; x, y e Acc u E and x ^ y} . I t i s not i n general true that D = D' . An example of a domain D f o r which t h i s f a i l s may be found i n Example 1 of Martin [37 ] . M a r t i n 3 constructed a c e r t a i n bounded domain E i n R , together w i t h a subset EQ of i t s Martin boundary, and showed that l i m sup K(x, z) = °° z y z e E f o r x, y e EQ • One can modify the parameters i n h i s c o n s t r u c t i o n s l i g h t l y , and o b t a i n . ( a s i n Theorem 9(b) below), that a l s o l i m K(x, w)/G(z, w) = °° w->y we-E f o r z e E and x, y e EQ . Thus c o n d i t i o n (d) of Theorem 6 f a i l s , so that (x, y) k D f o r any x, y e EQ . Modulo showing that EQ contains at l e a s t two d i s t i n c t a c c e s s i b l e p o i n t s , t h i s gives us our counterexample. This l a t t e r property i s i n f a c t not d i f f i c u l t to show. . 2 One embedds E i n a product E' = R x" E" , where E" c R i s a bounded domain. By a r e s u l t of M. Cranston and T. McConnell ([10]; Theorem 8 below), a l l minimal Martin boundary p o i n t s of E" are a c c e s s i b l e . - 190 -One can t h e r e f o r e construct a harmonic h on E' such that the h-transform X = ( Y t , Zfc) has f i n i t e l i f e t i m e £ ; (Y f c) i s a Brownian motion up t i l l time £ , independent of ( z t ) given C, ; and w i t h p o s i t i v e p r o b a b i l i t y , X e E f o r every t < £ ,- and Xfc converges i n E as t + t, , w i t h l i m i t i n EQ . From t h i s , one sees immediately that E ^ n Acc has many p o i n t s . Rather than d w e l l i n g on t h i s , we w i l l consider an analogous example 2 i n R , i n d e t a i l . The two dimensional s i t u a t i o n i s d i s t i n g u i s h e d from that of higher dimensions, by the a v a i l a b i l i t y of the Riemann 2 mapping theorem. That i s , i f we are given a domain E ^ c R whose complement ( i n the Riemann sphere) has n components (n >_ 1) , none of which i s a s i n g l e p o i n t , then there i s a conformal equivalence $ mapping E ^ onto a bounded domain E 2 w i t h smooth boundaries. I f (X^_) i s a Brownian motion s t a r t e d at x and k i l l e d upon l e a v i n g E ^ , then we o b t a i n a Brownian motion on E^ by time changing Y = $(X t) by the in v e r s e A'(t) to the a d d i t i v e f u n c t i o n a l rt A t | * ' ( * 1 ( Y ) ) | 2 d s 0 S Thus, i f and are the Green f u n c t i o n s on E ^ and E 2 r e s p e c t i v e l y , then f o r every B o r e l f >_ 0 , f ( * ( z ) ) G (*(x), * ( z ) ) | *'(z) | 2dz E l r f ( y ) G (*(x), y)dy E2 = E [ o f ( Y A , ( t ) ) d t ] - 191 -= E[ f ( * ( X )) *'(xt) dt] f ( f ( z ) ) G 1 ( x z) | $'(z) | 2 dz . That i s , G 2 ( $ ( x ) , $(z)) = G-^x, z) . Thus, $ w i l l a l s o preserve the Mart i n •'functionif.nanHehence,extends©to aohomeomorphismiof *>the',~Martin c o m p a c t i f i c a t i o n s of and E 2 , t a k i n g minimal p o i n t s to minimal p o i n t s . There i s of course no reason why $ should behave so n i c e l y on the Euclidean boundary of E^ . In our case, a l l the Mar t i n boundary p o i n t s of E 2 are a c c e s s i b l e . ( a n d minimal) by Theorem 7, so that by the c r i t e r i a of Theorem 6, i t f o l l o w s that D = D' . In dimension _> 3 , we could o b t a i n p a t h o l o g i c a l domains homeomorphic to the u n i t b a l l , but t h i s shows that to do the same i n dimension 2, we must consider domains of i n f i n i t e c o n n e c t i v i t y . U n l i k e the case of m i n i m a l i t y , the a c c e s s i b i l i t y or i n a c c e s s i b i l i t y of a Martin boundary po i n t w i l l not i n general be preserved under a conformal map. This problem i s resolved by the f o l l o w i n g r e s u l t of Cranston and McConnell [10] • Theorem 8 There e x i s t s a constant C such that f o r each domain E i n o R and each excessive f u n c t i o n h on E , we have that Thus, a l l minimal M a r t i n boundary p o i n t s of bounded domains i n R , are a c c e s s i b l e . 2 We w i l l now con s t r u c t a domain E c R f o r which D ^ D' . Let E X[£] <_ C • Area of E , f o r x e E . • ,2 - 192 -a(n) , b(n) e (0, 1) , n = 1, 2 Set s(0) = 0 and n s(n) = I b(n) , n >_ 1 . 1=1 Assume that a(n) and b(n) are decreasing i n n , and th a t s(°°) = 1 . Let E = (-1, 1) x (0, 1) \ u [a(n) - 1, l - a ( n ) ] x {s(n)} n=l We w i l l show that f o r every f i x e d sequence (b(n)) as above, the a(n) may be chosen to converge to zero s u f f i c i e n t l y f a s t that D ^ D' . Let c(n) = (0, (s(n) + s ( n - l ) ) / 2 ) , , n >_ 1 f (1 - a ( n ) , 1) x {s(n)} , n >_ 1 A(n) = [ (-1, a(n) - 1) x {s(n)} , n <_ -1 A+ u A(n) , n>l A- = u A(n) , n<-l A = A+ u A- , and B(n) = A(n) u A(-n) . SCO -. ..- -—— — < -1 Fip . 4. Let x Q be a l i m i t p o i n t i n E of the sequence (c(n)) . Define - 193 -the Martin fiinctionwielative^to'the,point- ,c'(l) , so that hh = K K ( X Q , : > ) i s symmetric on E (that i s , h(-z) = h(z) ; as the reader w i l l have gathered, we are allowing ourselves to pass f r e e l y between notations 2 x used for R and for C .) • We w i l l write P for the law of Brownian motion, started at x.e E and k i l l e d upon contact with 8E . c ( l ) For ease of notation, we w i l l write Q for the p r o b a b i l i t y ^P Martin's argument w i l l give us the f i r s t part of the following. Theorem 9 Suppose that l i m sup b(n) l o g ( l / a ( n - l ) ) > 41T . Then: n -»• °° (a) h(c(n)) •> °° as n -»- °° ; (b) Let z e E . Then h(w)/G(z,w) -»• °° as Im(w) + 1 . The only reason t h i s r e s u l t does not immediately give us that D 4 D' , i s that i t i s conceivable that the speed at which a(n) -> 0 forces that part [-1, 1] x {1} of the Euclidean boundary of E , to collapse to a sing l e point of the Martin boundary. I f we could show that h was not minimal, then there would have to be at l e a s t two minimal Martin boundary points x^ and x 2 , such that w -*• x^ implies that Im(w) -»- 1 , i = 1, 2 . In t h i s case, (b) of Theorem 9 together with c r i t e r i o n (d) of Theorem 6 would show that D ^ D' . Our main r e s u l t w i l l be: Theorem 10 Suppose that a(n-l)/b(n) 0 . Then Q(TA_ = oo) > o . - 194 -By symmetry, a l s o Q ( T A + = °°) > 0 , so that a l s o Q ( T A ( n ) < 0 0 f o r I n f i n i t e l y many n >_1) e (0, 1) , and hence we conclude from (h) of Theorem 1, that h cannot be minimal. Thus we have C o r o l l a r y 1 Suppose that l i m sup b(n) l o g ( l / a ( n - l ) ) > 411 . n -> °° Then D 4 D' . Now to work. A common fe a t u r e of many of the arguments i s that we o b t a i n estimates using the exact form of c e r t a i n S c h w a r z - C h r i s t o f f e l transformations. To do so, we w i l l e x t e n s i v e l y use e l l i p t i c i n t e g r a l s , f o r which we r e f e r to Gradshteyn and Ryzhik [26]. The n o t a t i o n (GR8.1133) w i l l r e f e r to the appropriate formula i n [26] . For ease of use, we w i l l use a nonstandard n o t a t i o n , w r i t i n g F(a, k) = r a dx , a, k e [0, 1] (1-x ) ( l - k x ) For now, we w i l l only r e c a l l that (GR8.1133) (13.1) F ( l , 0) = n/2 , F ( l , k) ^  l o g ( l / / U k ) as k + 1 . proof of Theorem 9: F i x n , and consider E' = E n (-1, 1) x (0, s(n)) For x e E 1 , l e t P , x be the law of Brownian motion s t a r t e d at x and stopped upon l e a v i n g E' . A l s o , l e t W(n) = (-1, 1) x ( s ( n - l ) , s(n)) c E' , C(n) = (-1, 1) x {(s(n) + s ( n - l ) ) / 2 } , and U(x, dy) = P' X(X e dy) . 8W(n) - 195 -Let $ be the S c h w a r z - C h r i s t o f f e l transformation mapping the u n i t d i s k to W(n) and t a k i n g the p o i n t s -1, 0, and 1, to (-1, (s(n) + s ( n - l ) / 2 ) , c(n) , and (1, (s(n) + s ( n - l ) ) / 2 ) . Since composing $ w i t h a Brownian motion on the d i s k produces a time changed Brownian motion on W(n) , i t f o l l o w s that $ preserves h i t t i n g d i s t r i b u t i o n s . Thus, by the corresponding r e s u l t f o r the d i s k , we see that f o r each x e W(n) , there e x i s t s a d e n s i t y u(x, y ) , continuous i n y , f o r y(x, dy) w i t h respect to y ( c ( n ) , dy) , and that f u r t h e r , there e x i s t s a constant k(n) w i t h U/X' ^ i k ( n ) f o r x e c ( n > a n d y» y' e x ) x {s( n>> • u v.xy ) Let m(n, w) = k(n) 2/y(w, (-1, 1) x { s(n)}) . Thus, f o r x, w e C(n) and y e (-1, 1) x (s(n)} , we have that y(x, dy) = U i X ' y\ y(w, dy) u(w, y) < uJx^Zl u ( w , d y ) - u(w, y) (-1,1) x { s(n)> u(x, y j ) , -; \\ y(w, dy') u(w,y') i n f { U ^ X > y | j ; y ' € ( - l , l ) x { s ( n ) } } y ( w , ( - 1 , 1 ) x { s ( n ) } ) < m(n, w) y(x, (-1, 1) x {s(n)})y(w, dy) . Let R(0) = 0 , and R ( i + D = R ( i ) + ( T ^ ^ ^ + T c ( n ) o e ) o e R , i > o atn-i; Then f o r x, w e C(n) and y e (-1, 1) x (s(n)} , - 196 -P' X(X e dy) i3E l = I E , x [ R ( i ) < », P' ^ V J - ; ( R ( 1 ) = - , X e dy)] i=0 9E' = I E ' x [ R ( i ) < », y ( X R ( i ) , dy)] i=0 99 <_ I m(n,w)y(w,dy) E ' x [ R ( i ) < », y ( X R m , (-1,1) * ( s ( n ) } ) ] 1=0 K } = m(n,w)y(w,dy) P , X ( X e (-1,1) x {s(n)}) i3E' <_ m(n,w)y(w,dy) . A l s o , one e a s i l y shows by the appropriate S c h w a r z - C h r i s t o f f e l t r a n s f o r m a t i o n , that there i s a constant M such that f o r every k , the f i r s t h i t t i n g d i s t r i b u t i o n of Brownian motion s t a r t e d a t c ( l ) , on the boundary of (-1, 1) x (0, s(k)) , has a den s i t y w i t h respect to the arc l e n g t h measure, which i s bounded by M . Now l e t h be the f u n c t i o n of the theorem. Then f o r w e C(n) , h ( c ( l ) ) h ( y ) P ' X ( T B ( n ) < " • XT , x £ d y ) B(n) p » C ( l ) / rp < O  Y * ( T C ( n ) ' T e dx) < P' A (T_ f < °°) m(n, w) C(n) h(y)y(w, dy) < 2 M a(n-l)m(n, w) h(y) P' w(X r r e dy) T3E' = 2 M a ( n - l ) m(n, w)h(w) Thus, provided a(n-l)m(n, c ( n ) ) -> 0 as n -* °° , i t f o l l o w s that - 197 -h(c(n)) -> °° . Now, consider G(z, •) . Let z belong to the c l o s u r e of W( j - l ) , and suppose that the n used above s a t i s f i e s n > j . G(z, x) i s bounded on B ( j ) (say by M') , and converges to zero as x approaches 3E . Thus, si n c e y(w, (-1, 1) x {s(n)}) = y(w, (-1, 1) x { s ( n - l ) } ) ^ P ' W ( T B ( j ) ' V f°r W £ C ( n ) ' a l s o G(z, w) <_M'y(w, (-1, 1) x ( s ( n ) } ) f o r w e C(n) . Combining our two estimates, we see that h(w) > hCfe'(l)) G(z,w) - 2 MM' a(n-l)m(n,w)y(w,(-1,1) x {s(n)}) h(c:(l)) 1 c n , s = ^ v ,' ' , f o r w e C(n) . 2MM' 2 Z M M a ( n - l ) k ( n ) Z Let i ( n ) be the expression on the r i g h t hand s i d e . Then i f a(n) 0 s u f f i c i e n t l y f a s t , a l s o i ( n ) -> 0 0 . Let E(n) = (-1, 1) ¥ ( ( s ( n - l ) + s ( n ) ) / 2 , (s(n) + s(n+l))/2) n E . Then E(n) i s reg u l a r f o r the D i r i c h l e t problem, so that i f v(x, dy) i s the h i t t i n g d i s t r i b u t i o n of 3E(n) by Brownian motion s t a r t e d a t x , then whenever g i s harmonic on E(n) , w i t h a continuous extension to i t s c l o s u r e , we have that g(x) = g(y)v(x, dy) 3E(n) - 198 -These c o n d i t i o n s hold f o r both h and G(z, •) . Both f u n c t i o n s v a n i s h on 3E(n) n 3E , so that we have h _> i ( n ) G(z, •) on 3E(n) . By the i n t e g r a l r e p r e s e n t a t i o n , t h i s i n e q u a l i t y holds throughout E(n) . We have t h e r e f o r e shown that (a) and (b) hold, provided that both m(n, c(n))a(n-.l) and • lc(n) 2 a ( n - ~ l ) converge to zero .• "'Since y ( c ( n ) , (-1, 1) x 4s(n) 1) + 1/2, the f i r s t of these lis i r r e l e v a n t , and hence-the exact"'? tatement 'of the theorem w i l l follow'from t h e " f o l l o w i n g estimate of k(n) . Theorem 9 w i l l then be proven. • Lemma 3 b(n) l o g ( k ( n ) ) 211 as n -»• °° . proof: Write b f o r b(n) . Let $ be the conformal map of the u n i t d i s k onto (-1, 1) x (0, b) , mapping the p o i n t s -1, 0, and 1, to (-1, b/2), (0, b/2), and (1, b/2) . Let e 1 9 = 4 _ 1 ( ( 1 , b)) ; 0 e (0, II/2) , and w r i t e dy f o r normalized arc l e n g t h , on the boundary of the u n i t d i s k . - The f i r s t h i t t i n g d i s t r i b u t i o n of t h i s boundary, by Brownian motion s t a r t e d at x i s then 1 - | x [ 2 . 1— a d y ' |x - y| so that s i n c e $ preserves h i t t i n g d i s t r i b u t i o n s , we have that 2 • 2 k(n) =.sup i1 " / ^ " i IM(x) = 0 , s, t e [8, n - 6]} i i t i i i s I |x - e I |x - e | - 199 -By simple c a l c u l u s , t h i s becomes k(n) = 1 + cos9 1 - cos6 We must ther e f o r e examine the dependence of 6 on b rii-e b 2 $ ( eX t ) |dt / dt dt He ) |dt , and $(w) = (j) -1/2 [(z - e ) ( z + e ) ( z - e ) (z + e )] dz + <J> 0 fo r some constants (j> and <j>' . Thus , . i t r / i t ±0. . i t , IB. i t - 1 0w i t , -19M = |<J> i e [(e - e ) (e +e )(e - e ) (e + e )] -1/2 = 2 U cos 2t - cos 20 1-1/2 -1/2 (cos 2t - cos 28) ' dt = 2 -1/2 (cos 2t - cos 20) dt dx COS0 / ( l - x2 ) ( x 2 - c o s 2 0 ) = y/I F ( l , s i n 2 0 ) (GR 3.1529) , and s i m i l a r l y rn-e -1/2 (cos 20 - cos 2t) dt = /2 ('COS© Jo dx / ( l - x 2 ) ( c o s 2 0 - x 2 ) = Jl F ( l , cos 0) (GR 3.1527) - 200 -Thus by (13.1), as 0 -> 0 we have n b ^ iog(i / e ) 2 Since k(n) ^  ( 2 / 6 ) , we are done. • The proof of Theorem 10 w i l l be given along the f o l l o w i n g l i n e s : Because we know l i t t l e about h (other than i t s 'symmetry'), we w i l l reduce the problem to that of e s t i m a t i n g c e r t a i n q u a n t i t i e s not depending on. h . (An e s s e n t i a l i n g r e d i e n t i n t h i s i s Theorem 8, which estimates the expected l i f e t i m e of an h-transform, but i n a manner independent of h .) These q u a n t i t i e s are l o c a l , i n that they i n v o l v e p o t e n t i a l t h e o r e t i c p r o p e r t i e s of c e r t a i n subsets of E . Sc h w a r z - C h r i s t o f f e l transformations are used to r e l a t e p o t e n t i a l t h e o r e t i c questions about these subsets, to questions about the u n i t d i s k , and questions about the parameters d e s c r i b i n g the transformation. The answers to questions of the f i r s t type may be e a s i l y obtained, whereas the answers to the second type form the main n o n - p r o b a b i l i s t i c i n g r e d i e n t of our proof. Before proceeding w i t h the proof of Theorem 10, we w i l l f i x some n o t a t i o n . Let S(0) = T(0) = T . For k >_ 0 , define N(k) = n , i f X T ( k ) e A(n) T(fcfi) = T<k) + T M B ( N ( k ) ) o e T ( k ) s ( k + i ) = T(k) + T M A ( N ( k ) ) o e T ( k ) . Let U(n) = E n (-1, 1) x ( s ( n - l ) , s(n+l)) , - 201 -U(n) F i g . 5 and w r i t e P x f o r the law of Brownian motion s t a r t e d at x and n stopped upon l e a v i n g U(n) . Define the Mar t i n f u n c t i o n K n ( x » y) on U(n) , r e l a t i v e to the base po i n t c ( n - l ) , so th a t K i s symmetric n (that i s , K (-x, -y) = K (x, y)) . n n As usual P X and P X w i l l denote the laws of the transformations ' h n y n by the f u n c t i o n s U(n) , K (y, 0 r e s p e c t i v e l y , but we w i l l a l s o introduce the n o t a t i o n ^P^ f o r the transformation by the f u n c t i o n K n ( y , ') + K n ( - y , •) . Note that t h i s object would be changed i f we had used a base p o i n t other than c(n ) . Set &(x, y) = F (Sgn N ( l ) - Sgn N(0) , S ( l ) = T ( l ) ) , i f y n x e B(n) and y e B ( n - l ) u B(n+1) f o r some n . 0 , otherwise . !(n, m) = i n f { a ( x , y ) ; x e B(n) , y e B(m)} ; n, m _> 1 - 202 -y(x, y) = 1 E* [ T ( l ) ] , i f x e B(n) and y e B ( n - l ) u B(n+1) y n 0 fo r some n otherwise X(n, m) = i n f { y ( x , y) ; x e B(n) , y e B(m)} ; n, m >_ 1 . Note that B(n, m) = X(n, m) = 0 unless |m - h| = 1 . The estimates of these q u a n t i t i e s that we w i l l need are contained i n the f o l l o w i n g r e s u l t , whose pro<5f we defer u n t i l the end of t h i s s e c t i o n . Lemma 4 Let a( n - l ) / b ( n ) -»- 0 . Then there e x i s t constants > 0 and > 0 such that f o r every n _> 1 , (a) (1 - B(n, n+1)) v (1 - g(n, n-1) < exp(-C /b(n)) (b) A(n, n+1) A A(n, n-1) >_ C 2B(n) 2 . Now, l e t I be the f u n c t i o n that replaces the x-coordinate by i t s absolute value. That i s , K z ) = ; z, Re(z) •> 0 - z , Re(z) < 0 R e c a l l that F = a(X ; s < t)' . We l e t t s' — G = a ( I ( X T ( k ) ) ; k > 0) Lemma 5 (a) Let H(x, y) be p o s i t i v e and j o i n t l y measurable i n x and y - 203 -Let T and. S be + ) stopping times, and set H'(x) = h E X [ H ( x , X T ) ] . Then E Q [ H ( X S > x s + Toe >l F s + ] = H ' ( V Q" A ' S- • (b) I ( X t ) i s strong Markov w i t h respect to Q and ) . (c) Let Z e F be p o s i t i v e , and set H(x, y) = E^fZ] , f o r x e B(n) , y e B ( n - l ) u B(n+1) . y n Then h E X [ z | l ( X T ( 1 ) ) ] = H(x, I ( X T ( 1 ) ) ) a.s. (d) Q(Sgn N(n) = Sgn N(n-l) , T(n) = S(n) | F T ( n _ 1 ) + v G) = a( x T( n_i)» I ^ X x ( n ) ^ ' ) a* S'' f ° r e v e r y n _> 1 . proof: ( a ) : I f H + H , then H (x, •) t H(x, •) , so that H 1 + H' . * n n n The c l a s s of f u n c t i o n s H f o r which (a) holds i s there f o r e a monotone c l a s s , and i t contains f u n c t i o n s H(x, y) = H^(x) H^Cy) by the strong Markov property. ( b ) : By symmetry of E and of Brownian motion, we have that P X ( I ( X . ) e A) = P I ( x ) ( I ( X . ) e A) f o r every x e E and A e F . We have chosen h so that h(x) = h ( I ( x ) ) f o r every x . Thus, i f f , . . . , f n e E are p o s i t i v e , T i s an ( F t + ) stopping time, and t ^ <..< t , then - 204 -X = J. T[nf.(i(x. ))] h 1 t . i A h lxT E [ h ( x x ^ ± ^ x t ) ) ] h ( I ( X T ) ) n E 1 [h(i(x ))nf ±(i(x t ))] , n i so that I(X^) i s strong Markov, w i t h semigroup h P t ( x , dy) + h P t ( x , -dy) . ( c ) : Let the i n t e g r a l r e p r e s e n t a t i o n of h on U(n) be h = K (y, -)y(dy) By uniqueness of the r e p r e s e n t a t i o n , and the f a c t that h and K n are preserved by the transformation x -* - x of U(n) , i t f o l l o w s that y(dy) = y(-dy) . Thus, f o r A c A(n+1) u A(n-l) and Z e F p o s i t i v e , E XCZ, K X T ( 1 ) ) e A] = . E J Z , X , . , , , e A V- A] h n h n l ' T ( l ) h(x) K (x, y) EA[Z]y(dy) n y n Au-A h(x) (K (x, y) EX[Z] + K (x, -y) - E X [ Z ] ) y ( d y ) n y n n -y n h(x) J (K (x, y) + K (x, -y)) ^ [ Z l y C d y ) n n y n h(x) J K (x, y) EA[Z]y(dy) n y n A u-A - 205 -(by symmetry of u) " h ^ W W ' I < X T ( l ) ) € A 1 ' as r e q u i r e d . ( d ) : F i r s t , observe that i f Z e F T ( n ) » t n e n E Q [ Z l F T ( n - D + V G ] = E Q [ Z | F T ( n - l ) + V ° ( I ( X T ( n ) » ] ' by part ( b ) . Now l e t f e E be p o s i t i v e , and set H(x, y) = f ( I ( y ) ) a(x, y) . Also put H*(x) = h E X [ H ( x , X ^ ) ] , and Z 1{Sgn N ( l ) = Sgn N(0), S(0) = T(0)} By ( c ) , we have that H'(x) = h E X [ f ( I ( X T ( 1 ) ) ) a(x, K X T ( 1 ) ) ] = h E x [ f ( I ( X T ( 1 ) ) ) Z] . Thus, i f a l s o A e F T ( n _ ] _ ) + > t n e n E [A, f ( I ( X T ( n ) ) ) , Sgn N(n) = Sgn N ( n - l ) , T(n) = S(n)] = E Q [ A , f ( K x T ( n ) ) ) , Z o e ^ j ] = [A, h E X T ( n - 1 ) [ f ( I ( X T ( 1 ) ) ) Z ] ] " E Q U ' H ' ( X T ( n - l ) ) ] = EQ [ A' H ( X T ( n - l ) ' X T ( n ) ) ] ( b y ( a ) ) = EQfA' ^ V^)" a ( X T ( n - l ) > I ( X T ( n ) ) ) ] - 206 -as r e q u i r e d . • proof of Theorem 10: For n _> 1 , and m = n + l or n - 1 , l e t N(n, m) be the number of k _> 0 such that |N(k) | = n and |N(k+l) | = m . Let H(x, y) = a(x, y) , and apply (a) of Lemma 5. We have that H»(x) = h E X [ Y ( x , X T ( 1 ) ) ] = h E X [ T ( l ) ] (by ( f ) of Theorem 1 ) , so that E Q[T(k+l) - T(k)] = E Q [ H ' ( X T ( k ) ) ] = E Q [ Y ( X T ( k ) , X T ( k + 1 ) ) ] > E [X(|N(k)|, |N(k+l)|)] . By Theorem 8, we there f o r e have that oo oo > E [?] = I E [T(k+1) - T(k)] + E [T(0) ] 4 k=0 ^ 4 CO >_ I [X(n, n+1) E [N(n, n+1)] + n=l ^ + A(n, n-1) E Q[N(n, n-1)]] . Since l o g ( l / ( l - x ) ) <^  2x f o r x._> 0 s m a l l , we have by Lemma 4, that there e x i s t s a constant C w i t h l o g ( l / 3 ( n , m) <_ C A(n, m) f o r n j > l ,m = n + l . Thus a l s o - 207 -oo > I [ l o g ( l / B ( n , n+l))E [N(n, n+1)] + n=l ^ + l o g ( l / B ( n , n - l ) ) E Q [ N ( n , n-1)]] = E n [ - l o g ( fl B(n, n + l ) N ( n > n + 1 ) B ( n , n ^ l ) N ( n ' " " ^ J y n=l The integrand i s t h e r e f o r e f i n i t e almost s u r e l y , so that a l s o E [N(0) = 1 , n B(|N(k)1, |N(k+l)|)] W k=0 = E n [ N ( Q ) = l , n g(n, n + l ) N ( n ' n + 1 ) B ( n , n - l ) N ( n ' ^ n=l > 0 . But by (d) of Lemma 5, j i i i E n[N(0) = 1 , n B(|N(k)|, |N(k+l)|)] ^ k=0 £E Q[N(0) = 1 , E Q[Sgn N ( l ) = Sgn N(0) , = SCI) = T(1).E [... ... E Q[Sgn N(j) = Sgn N ( j - l ) , S(0) = T ( 0 ) | F T ( j _ 1 ) + v G]... . . . 1 | F T ( 0 ) + v G]] = Q(N(0) = 1, Sgn N(k) = Sgn N(k-l) and ' . S(k): = T(k->, k = l , . . . j ) = Q(T A_ > T ( j ) ) L e t t i n g j -> oo f w e see that Q ( T A _ = °°) > 0 , as r e q u i r e d . • Before proving Lemma 4, we w i l l examine a s l i g h t l y d i f f e r e n t s i t u a t i o n ; Consider a S c h w a r t z - C h r i s t o f f e l transformation - 208 -$(w) = <f> rw ie 6 ie. -1/2 (z-e U ) n (z-e J ) j = l dz + <J>' , where <|> and cf>' are constants, and 6^ < 6^ < 9,. < 6^ < 6^ < 6^ < 9 1 < 60 = 97 + 2 1 1 * N o t e t h a t t h e e f f e c t o f c n o o s i n g d i f f e r e n t branches of the square root f u n c t i o n , i s to change the constant <|> . Then $ maps the u n i t d i s k 0 , to a region V w i t h s t r a i g h t s i d e s ; 0 \ 2 F i g - 6 For every such c o n f i g u r a t i o n of p o i n t s x ...x- , there w i l l 0 o e x i s t such a map $ , which i s unique up to a Mobius transformation of the u n i t d i s k . Since such transformations are uniquely s p e c i f i e d by the images of three d i s t i n c t boundary p o i n t s , we may always f i n d a $ w i t h 6Q, 0 and 0^ t a k i n g on some f i x e d v a l u e s . We w i l l take the image of $ to be V = [ ( - 1 , 1) x (0, b + b ' ) ] \ [ ( - l , 1-a) x { b ' } ] . F i g - 7 - 209 -and w i l l consider i\> and i j / such that ^ e 1 ^ ) = He±r) = ( a - 1 , b') , ij, e ( 6 ^ 0 Q) , V e ( 0 ? , Sg) . We w i l l a l s o l e t L be the Mart i n f u n c t i o n on V , defined w i t h base p o i n t $(0) . Thus, L ( * ( x ) , *(y)) = ( l - | x | 2 ) / | x - y| 2 . Lemma 6 Suppose th a t b' + 0 and a/b + 0 i n such a way that b <_b' . Then (a) i j i , 0 1, 6 3 0 2 and , Q^, &6 -> ©5 (b) l i m i n f b.log (l/(\\>-Q )) >_ 211 l i m i n f V . l o g ( 1 / ) ) >. 211 (c) sup{L(x; , $ ( e 1 1 : ) ) / ( L ( x , $ ( e 1 S ) ) A 1) ; x e (1-a, 1) x {b' } , and t , s e [0^, IJJ] u , 0^]} remains bounded. i 0 Q i 0 2 i 0 5 proof: ( a ) : Let the distances from e to e and e be at l e a s t 2e , and l e t 0 be the i n t e r s e c t i o n of the u n i t d i s k 0 w i t h " o a d i s k of radius e and centre at e . Then there i s a constant C > 0 (we w i l l use the n o t a t i o n C f o r any constant we do not wish to keep t r a c k of) such that no matter what a, b and b' are, we have that |$' (z) | _> C | <f> | f o r z e 0\0 £ . Now l e t g Q : [0, r ] -> 0 be.the u n i t speed curve whose image under $ i s [1-a, 1] x {b'} . Since - 210 -A r g ( — * ( g Q ( t ) ) ) i s constant, we obt a i n that r r a = | * ( g 0 ( r ) ) - *(g (0)) = dt • ( S o C t ) ) ^ ! Since r r d F * ( 8 0 ( t ) ) | d t Let Then U| f ( t ) = | ^ • ( e i t ) b = 2 b' = 2 f ( t ) d t / f ( t ) d t / r°2 r 2 f ( t ) d t f ( t ) d t f ( t ) d t = f ( t ) d t 2 - a = 2 f ( t ) d t / f ( t ) d t a_ b |*(g(r)) - *(g(0)) f ( t ) d t > e C / f ( t ) d t , we must have that - 211 -f ( t ) d t °° Thus, as b -> 0 and 2 - a -> 2 , a l s o r 2 r 0 f ( t ) d t and f ( t ) d t -»- «> Since e i f c - e 1 6 - 21 e 1 ( t + 6 ) / 2 s l n ( ( t - 6 ) / 2 ) , we have that f ( t ) = .. t-8 6 t-0. -1/2 i | s i n (-2-^) | | n sinC-yJ-) | 3=1 Thus, since s i n ( ( 0 Q - t ) / 2 ) <_ si n ( ( 0 & ^ t ) / 2 ) whenever t e [0.^ 0Q] , we have f o r each n > 0 , a constant C such that r 0 f ( t ) d t < C ( t - e i ) - 1 / 2 dt u l "1 whenever 0^  e + n» 0Q] • Since the r i g h t hand side i s bounded as a f u n c t i o n of 0^  , we conclude that 0^  -»- 82 . S i m i l a r l y , f o r each n > 0 there i s a constant C such t h a t r 1 f ( t ) d t < C [ ( t - e 2 ) ( e 1 - t ) ] 1 / 2 d t [ t ( l - t ) ] 1 / 2 d t < oo whenever 03 e [0^, © 2 - n ] . Thus . We have l i k e w i s e , that - 212 -V w4 "5 * I t f o l l o w s from the above estimates, that f o r n > 0 f i x e d , 90 f ( t ) d t e1+n remains bounded, so that Jl+n f ( t ) d t / f ( t ) d t + 1 Since J f ( t ) d t / 3 i js we see that ^ -> 0„ . S i m i l a r l y , ^' -* 6 f ( t ) d t = a/2 , (b) : Let r 1 • / 0 ^ • -3/2/2~"5s C = - s m ( — ) sxn (—- ) , k = ( e 2 - e 3 ) / ( e 1 - e 3 ) 1/2 By pa r t ( a ) , we have immediately that r 2 f ( t ) d t * C -1/2 [ ( e 1 - t ) ( e 2 - t ) ( t - e 3 ) ] dt = 2 c ( e 1 - e 3 ) 1 / 2 F ( I , k) (GR 3.1313) f ( t ) d t ^ C [ ( e 1 - t ) ( t - e 2 ) ( t - e 3 ) ] 1 / 2 dt - 213 -2 C ( 0 1 - 0 3 ) 1 / 2 F ( 1 , 1-k) (GR 3.1315) f ( t ) d t ^ C [ ( t - 0 1 ) ( t - 0 2 ) ( t - 0 3 ) ] 1 / 2 dt = 2 0 ( 6 ^ ) 1 / 2 F ( p , k) Thus b ^ 2F(1, l - k ) / F ( l , k ) , so that k •> 1 , and by (13.1), t h i s becomes that b ^ 2n/log ( ( 6 - O ^ / ^ - e ^ ) . S i m i l a r l y , F ( p , k ) / F ( l , 1-k) <\, r 1 f ( t ) d t / f ( t ) d t = a/b 0 , so that p 0 . Thus al s o (^-0-^ / ( 0 ' A N D H E N C E b l o g ( l / ( ^ 0 1 ) ) > Misgd/dii-ep) + iog((*-e 1)/(e 1-e 2)) + + l o g ( 0 1 - 0 3 ) ] •+ 2n . The second h a l f of (b) f o l l o w s s i m i l a r l y . (c) : R e c a l l , the path t —>. g (t) of par t ( a ) . We had th a t ^ * ( g o ( t ) ) d t | = *'(go(t)) dt , from which i t f o l l o w s that g Q minimizes - 214 -| * * ( g ( t ) ) [ d t 0 over a l l r ' > 0 , and a l l u n i t speed curves g : [0, r'] 0 such t h a t i 9 0 i t g(o) = e , and g(r') = e f o r some t e (8^,6^) . In p a r t i c u l a r , r |*'(g ( t ) ) | d t 0 remains bounded. Suppose now that i G 2 , (13.2) l i m i n f [ i n f { | g ( t) - e |; t e [0, r ] } ] = 0 . b' 0 a/b -> 0 There are constants C and n n such that whenever e e (0, n n ) , and b' and a/b are so small that 16. i 6 _ 18, 18. 1 - 1 - / I I -5 £• I |e - e |, |e - e | < e , then |$'(z) | > C(|z - e | + 2e) J / Z f o r z e 0 w i t h i 6 2 I i ^ z - e | e (e, n n ) I f 18 2 |g(t) - e | < e f o r some t , then s i n c e g i s a u n i t speed curve, i t f o l l o w s that - 215 -0 {z; Iz - e e (e, n) } ( g ( t ) ) d t >_ n - e f o r every n e (e, n^ ) In p a r t i c u l a r , under c o n d i t i o n (13.2), we have that ,-3/2 l i m sup b' •+ 0 a/b + 0 0 *'(g ( t ) ) |dt >_ C(n - e ) ( i r + 2e) f o r every e and n w i t h 0 < e < n < TIQ • Let e •» 0 and then n 0 to o b t a i n that the l e f t hand side i s i n f i n i t e ; a c o n t r a d i c t i o n . Thus the path of g^ stays away from exp(i02) as b' -> 0 and a/b'-* 0 . By a s i m i l a r argument, i t stays away from exp(iQ^) . The conc l u s i o n of p a r t (c) now f o l l o w s immediately from the d e f i n i t i o n s of L and g Q , and the r e s u l t that f o r every e and n w i t h 0 < e < n , sup 1- x x-e . i t . 2 / ( ,. i s i 2 (>x-e I 2 1 9 5 A 1) ; x-e I A Ix-e > n , r i l r 2 i i l r 5I , . and t , s e l r ; | e -e A e -e | £ £ J | i s f i n i t e . • proof of Lemma 4: (a) We must show that there i s a constant > 0 such t h a t P X(X C... e A-) < exp(-C./b(n)) y n S ( l ) — 1 f o r x e A(n) and y e B ( n - l ) u B(n+1) , n >_ 1 . We w i l l o b t a i n t h i s - 216 -i n two p a r t s ; f i r s t e x h i b i t i n g such a bound f o r I ^ x , y) = / X ( X S ( 1 ) e A(-n)) , and then f o r I 0 ( x , y) = P X ( X Q n , e A ( - ( n - l ) ) u A(-(n+l))) . 2 y n o(.l; Let V(n) = [ ( - 1 , 1) x ( s ( n - l ) , s ( n+1))]\[(-1, l - a ( n ) ) x (s(n)}] . Write A(n) f o r the Martin boundary of V(n) , and w r i t e P^ f o r the law of Brownian motion s t a r t e d at x and k i l l e d upon l e a v i n g V(n) Resume the n o t a t i o n of the l a s t lemma, so that we have a c a n o n i c a l conformal.equivalence $ from the u n i t d i s k to V(n) , and we define the Martin f u n c t i o n L on V(n) , w i t h respect to the base po i n t n $(0) . Also note that p o i n t s of A(-n) c 3V(n) s p l i t i n t o two p o i n t s of the Martin boundary A(n) of V(n) ; one being the l i m i t of p o i n t s above A(-n) , and the other from below. We l e t A' (-ri) be the c o l l e c t i o n of a l l p o i n t s . o f the Martin boundary of V(n) , that are a s s o c i a t e d to a p o i n t of A(-n) i n t h i s way. Let u(A) = P^°^(X^ e A) , f o r A c A(n) measurable, n £-A l s o , l e t C be the bound given by (c) of Lemma 6. F i x a po i n t y e B ( n - l ) u B(n+1) , and l e t e(z) = K (y, z) + K (-y, z) n n Let v be.the measure representing e on A(n) , so that L (w, ')v(dw) n A(n) - 217 -By ( f ) of Theorem 2, and Theorem 4, we have that v(dw) i s the sum of the measure e(w)u(dw n A'(-n)) , and of two point masses, one at y and the other at -y . Thus, by ( j ) of Theorem 1, V X ' y ) = ePn(XS(l) £ A ( " n ) ) - e p ; ( v € A ' ( - n ) ) e(x) L (w, x) PX ( X e A'(-n))v(dw) n w n t,-A' (-n) A'(nn) L (w, x) n  e(x) e(w)y(dw) Since e ( $ ( e l t : ) ) = 0 unless t e [ 9 3 > u [ip', 0^] , we have from (c) of Lemma 6, that e(x') _ L n ( z , x ' ) L (w, x 1 ) L n ( z , x') ^ e(w)y(dw) < C L n(w, x) L ( z , x) n e(w)y(dw) = C 2 — * ; ( x ) . , f o r x, x'e A(n) , z e A'(-n) L n ( z , x) Since e(x') = e(-x') , we can choose x' e A(n) to maximize e over A(n) , and s t i l l have e(w)/e(x') < 1 f o r w e A'(-n) . Thus - 218 -I ^ x , y) < C 2 e(x') A'(-n) L (w, x') n e(w)y(dw) 2 _< C u(A'(-n)) sup{L n(w, z) ; w e A'(-n), z e A(n)} < c 3 [ ( ^ e 1 ) + ( e 5 - ^ ' ) ] / 2 n We thus o b t a i n a bound on I ^ ( x , y) of the c o r r e c t form, by (b) of Lemma 6. Now consider ^ ( x , y) . Suppose f o r now, th a t y e B ( n - l ) , so that we may assume without logs of g e n e r a l i t y , that y e A(-i(n-l)) R e c a l l that W(n) = (-1, 1) * ( s ( n - l ) , s(n)) . Let P x be the law of Brownian motion s t a r t e d at x and k i l l e d upon l e a v i n g W(n) , and define the Martin f u n c t i o n L^ on W(n) usi n g the base p o i n t c(n) . Consider Then (Y t) i s a v-transform on W(n) , under £ P X , where v i s a harmonic f u n c t i o n on W(n) , given by (m) of Theorem 1. Because e i s symmetric ( t h a t i s , e(z) = e(-z) f o r z eW(n)) , the same w i l l be true of v . By Theorem 4 and our choice of base p o i n t , we th e r e f o r e have that v i s a m u l t i p l e of L n ( y , O + L n ( - y , •) Thus - 219 -I„(x, y) < P X(Y_ e A(n), Yr e A ( - ( n - l ) ) ) L — e n U £— < sup / ^ x r _ = y> z e A(n) v n ? L (y, z) T n sup ±= z e A(n) L (y, z) + L (-y, z) n n To bound t h i s , we b a s i c a l l y repeat the proof of Lemma 6, but f o r a simpler transformation. Let $ be the conformal map from the u n i t d i s k to W(n) , mapping -1 , 0, and 1 to (-1, (s(n) + s ( n - l ) ) / 2 ) , c(n) , and (1, (s(n) + s ( n - l ) ) / 2 ) . Then L n O ( x ) , *( z ) ) = (1 - | z | 2 ) / | x - z| 2 . Let 1^ e 1 9 = $ _ : L ( ( 1 , s(n))) , e 1 = $ _ 1 ( ( l - a ( n ) , s ( n ) ) ) , i * 2 - i ( ( l - a ( n - l ) , s ( n - l ) ) ) , where -n/2 < \\> < -6 < 6 < ip < II/2 . Thus I i s i t i 2 |e - e | (13.3) I 9 ( x , y) < sup r - — T — . r — „ z — , I I S I t IZ I i ( I I-s) l t | 2 se [^ 2 r . e . ] . le -e | + |e -e | t e i e , ^ ] We have t h a t $(w) = (J) [ ( z - e i 9 ) ( z + e i 8 ) ( z - e l 9 ) ( z + e ± 8 ) ] dz + <j>' 0 f o r some <j> and <)>' . Let - 220 -2|<f)|f(t) dt He ) | , so that b ( n - l ) = 2 f ( t ) d t / n - e f ( t ) d t a(n) = 2 f ( t ) d t / n - e f ( t ) d t a ( n - l ) = 2 f ( t ) d t / •n -e f ( t ) d t Let 2 2 cos 6 - cos \\i P j cos6 1 - cos \b. 3 1 , j = 1, 2 As i n Lemma 3, we have that f ( t ) d t = JT F ( l , s i n 2 6 ) n - e f ( t ) d t = JT F ( l , cos 8) , and s i m i l a r l y , rcose f ( t ) d t = JT dx 2 2 2 c o s i ^ f ( l - x ^ ) ( cos 6-x ) = JT F ( p 1 , cos 6) , and f ( t ) d t = JT F ( P 2 , cos 8) Since 15(n-1) ->- 0 , i t f o l l o w s that 6 -»• 0 and - 221 -b( n - i ) ~ n/iog(i / e ) . Since a ( n - l ) / b ( n - l ) and a(n)/b<n-l) + 0 , i t f o l l o w s i n tu r n that V P2 ~* ° ' S ° t h a t a l S ° *i» ^2 ° ' T h u S (1 - 6 2/2) - (1 - i>./2) 1/2 * [ - J ] = (1 - (6/i|>.) ) J * 2/2 J J 1/2 and hence b ( n - l ) l o g ( l / i f O * b ( n - l ) ( l o g ( l / i p ) + log(i|> /6)) * n J J J By (13.3), there i s a constant C w i t h I 2 ( x , y) < C ( ^ 2 ) 2 1 2 C ( ^ 2 ) , and we ob t a i n a bound of the d e s i r e d form. The case y e B(n+1) i s handled i n the same manner, showing ( a ) , (b) By symmetry, X(n, n+1) = i n f { E X [ T ( 1 ) ] ; x e B(n), y e A(n+1)} . y n Consider the square Z(n) = ( l - b ( n + l ) , 1) x ( s ( n ) , s(n+l)) . S ( M - | ) | Let F i g . 8 Y t " Xt+L U(n)\Z(n) - 222 -Under P x , (Y ) w i l l be a v-transform f o r some v on Z(n) y n t (by (m) of Theorem 1 ) , Y^ = y , and Y n l i e s on the lower or l e f t s i d e s of Z(n) . Let Z = (0,1) x (0,1) , and define $ : Z(n) -y-.Z by *(z) = [z - ( l - b ( n + l ) , s(n))]/b(n+l) . Let Y t = $ ( Y t b ( n + l ) 2 ) 6 Z • Because t h i s s c a l i n g preserves Brownian motion, (Yj_) w i l l be a transform by v o $ ^ , under P x . Let P2, be the law of Brownian y n Z motion s t a r t e d at z and k i l l e d upon l e a v i n g Z . Thus X(n, n+1) >_ i n f { y E x [ l i f e t i m e of (Y f c) ] ; y e A(n+1) , x e B(n)} > b ( n + l ) 2 i n f { E™ m ; z € ( l - a f f i , 1 } * { 1 } ' — z L b(,n+l; w e [0, 1] x {0) u {0} x [0, 1]} . Since a(n)/b(n+l) -> 0 , t h i s infimum remains bounded away from 0 as n -> <» . The corresponding estimate on X(n, n-1) f o l l o w s s i m i l a r l y , showing (b ) . • 14. Smallness of D'\D The r e s u l t s of the l a s t s e c t i o n show that even f o r planar Brownian motion, i n general D f D' . The d i s c u s s i o n of Section 11 suggests that i n order to co n s t r u c t extensions of the process c o n s i s t i n g - 223 -of Brownian motion on R d , k i l l e d upon l e a v i n g a domain E , we should suppose that D i s p r e t t y w e l l a l l of D' . We w i l l i n f a c t use the existence of such extensions ( f o r example, u n k i l l e d Brownian motion on R d) , to show that D'\D i s s m a l l . Let E be a bounded domain i n R d , and l e t U be a l a r g e open b a l l c o n t a i n i n g the c l o s u r e of E . We r e t a i n the n o t a t i o n ( 0 , F ) f o r paths w i t h values i n E u {&} , and w r i t e (V, I/) f o r the corresponding o b j e c t , where now paths take values i n U . The coordinate process on Q w i l l s t i l l be ( x t ) .'We l e t the-coordinate process on V be (Wfc) . Write G(x, y) f o r the Green f u n c t i o n of U , and w r i t e P x and Q x f o r the laws of Brownian motion, s t a r t e d at x and k i l l e d upon l e a v i n g U i n the f i r s t case, and E i n the second. Let y and v be p r o b a b i l i t i e s on U , such that a set i s y - n u l l i f and only i f i t i s v - n u l l . Assume that n e i t h e r one charges some neighbourhood of the c l o s u r e of E . Write h = G(z, -)v(dz) , and P = P^ . Let P be the law under P of the reverse of the h coordinate process ( x t) from i t s l i f e t i m e . Then we have that P(A) = P(A) = G ( x / ( x y ) y P X ( A ) y ( d x ) v ( d y ) f G ( X ( y f y P X ( A ) v ( d x ) y ( d y ) f o r A e (/ . Thus, since y and v share the same n u l l s e t s , the same w i l l be t r u e f o r P and P . Let - 224 -M = {t > 0; W k E u {6} , and l e t MQ be the set of p o i n t s of M which are i s o l a t e d on the r i g h t . Let Y t ( s ) = \ w , , s < T . o 0 t+s U\E t > 6 , s > T , o 0 - U\E t and l e t p(A) = E[ I l A ( Y t ) ] , f o r A e (ft, F) . t e M 0 Since every excessive f u n c t i o n f o r Brownian motion i s r e g u l a r (see Blumenthal and Getoor [ 5 ] ) , the hypotheses of Maisonneuve [38], P r o p o s i t i o n (9.2) are met. Thus, there e x i s t s a k e r n e l n(x, du), and an a d d i t i v e f u n c t i o n a l (^j.) such that r5 p(A) = E[ I n(w , A)dL ] j0 f o r every measurable subset A of ft , and ( x t ) i s a n h-transform under each n(x, •) . Thus ( x t) i s a l s o an h-transform under p . I t f o l l o w s from Theorem 5 and a time r e v e r s a l argument, that Z = l i m X , and Z' = l i m X t tit, t t+O e x i s t i n the topology of the Martin c o m p a c t i f i c a t i o n of E , p -a.s. Because v does not charge E , the l i m i t a c t u a l l y l i e s i n Acc c A . Let ri be the measure on A x A , defined by n(dx, dy) = p(Z e dx, Z' e dy) . - 225 -Theorem 11 n(D'\D) = 0 . Thus, w i t h p r o b a b i l i t y one, every excursion i n t o E s t a r t s and f i n i s h e s at p o i n t s z, z' of the Martin boundary, f o r which ( z , z') e D . Lemma 7 n i s a - f i n i t e . proof: By m i n i m a l i t y , we have that f o r every y e i ^ and x e E , e i t h e r there e x i s t s a p o i n t e(y) e 3E such that Q X(X -»- e(y) i n the Euclidean m e t r i c , as t + O = 1 , or y^ t Q X ( X t converges i n the Euclidean m e t r i c , as t t £) = 0 . In the l a t t e r case, we w r i t e e(y) = 6 . Since the Euclidean l i m i t s of X , as t + £ and t + 0 , both e x i s t p -a.s. , i t f o l l o w s that p(e(Z) = 6 or e(Z') = 6) = 0 . Since Brownian motion does not r e t u r n to p o i n t s , we have by the strong Markov property, that a l s o p(e(Z) = e(Z')) = 0 . Thus, i t w i l l s u f f i c e to show that p(|e(Z) - e ( Z ' ) | > 2e) < °° f o r every e > 0 . Let T(0) = 0 , T(k+1) = i n f { t > T ( k ) ; |xt - | > e} . Because h i s bounded and bounded away from zero, on a neighbourhood of the c l o s u r e of E , we o b t a i n from (f) of Theorem 1, a constant m > 0 226 -such that h E x [ T ( l ) ] > m whenever x l i e s i n E . Thus >. m E[ I l , i I , ] = m p( |e(Z) - e(Z') | > 2 e) , as r e q u i r e d . • proof of Theorem 11; Since x\ i s a - f i n i t e , we may use the ' c l a s s i c a l argument' of the l a s t s e c t i o n of Maisonneuve [38], to o b t a i n a k e r n e l p(x, y; du) such that P = p(x,y; - ) n ( d x , dy) Now, l e t K be the Martin f u n c t i o n on E , and suppose that A e F I t i s e a s i l y checked (by ( f ) of Theorem 1 ) , that h E X [ A | z ] = a(x, Z) , where a(x, y) = yE^[A] . Suppose i n a d d i t i o n that f ^ and f ^ are p o s i t i v e and measurable on A , and that B e F . Then r s E p [ f f J ( Z ' ) f o o ( Z ) , A o 8 s, B] = E p [ f Q ( Z ' ) h E ^ [ A , f J Z ) ] , B] = E p [ f Q ( Z ' ) ^ ( Z ) a ( X s , Z'), B] from which we see that (X f c) i s a K(y, •) -transform under p(x,y; •) - 227 -f o r n -a.e. (x, y) . Thus (x, y) e D f o r n -a.e. (x, y) , as r e q u i r e d . • Now f i x y^ e E , and define harmonic measure A. on A by yo X(dz) = Q (Z e dz) . By (h) of Theorem 1, and ( f ) of Theorem 2, we have that f o r every y e E , the measure h Q y ( Z e dz) shares the same n u l l sets as X . Lemma 8 X 8 X i s a b s o l u t e l y continuous w i t h respect to n . Thus we ob t a i n a more p o t e n t i a l t h e o r e t i c v e r s i o n of Theorem 1; C o r o l l a r y 2 X 0 X (D'\D) = 0 . proof of Lemma 8: Let A be a compact subset of E , w i t h nonempty i n t e r i o r , and set n k(dx, dy) = p(Z' e dx, Z £ dy , T A < °°) . We w i l l a c t u a l l y show that X 8 X i s a b s o l u t e l y continuous w i t h respect to n . k We can w r i t e X, nk(A) = {TA<°°} A T A 1 A(Z', y) hQ A ( Z £ dy)dp I f p(Z' £ B) = 0 , then P(Z e B) = P(Z' £ B) = 0 , - 228 -so that since P i s a b s o l u t e l y continuous w i t h respect to P , a l s o n(AxB) = P(Z e B) = 0 ( t h i s i s the reason f o r our choice of h) . Thus n k(AxB) = 0 , and hence \ hQ (Z e B) = 0 , p -a.s. . Because A has nonempty i n t e r i o r , we have P(T < 00) > 0 , so that by our above remark on the n u l l sets of X , a l s o X(B) = 0 . That i s , X(dz) i s a b s o l u t e l y continuous w i t h respect to p(Z e dz) . I f now nk(A) = 0 , then f T A 1 A(Z', y) hQ (Z e dy) = 0 p -a.s. . Thus 1 A ( Z \ y) X(dy) = 0 p -a.s. , J and so a l s o 1 (x, y)X(dy) = 0 f o r X -a.e. x . J ^ This gives immediately that X 8 X ( A ) = 0 . • - 229 -Bi b l i o g r a p h y [I] R. F. Bass. 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" F i e l d s , o p t i o n a l i t y and measur-a b i l i t y " , Amer. J . Math. 87 (1965), pp. 397-424. [8] K. L. Chung and J . B. Walsh. "To reverse a Markov process", Acta Math. 123 (1969);, pp. 225-251. [9] K. L. Chung and J . B. Walsh. "Meyer's theorem on p r e d i c t a b i l i t y " , Z. Wahr. 29 (1974), pp. 253-256. [10] M. Cranston and T. McConnell. A r t i c l e to appear. [ I I ] C. de l a V a l l e e Poussin. "Propri£t£s des f o n c t i o n s harmonique - 230 -dans un domaine ouvert l i m i t e par des surfaces a courbure bornee", Ann. R. Scuola Norm. Sup. d i P i s a ( 2 ) , 2 (1933), pp. 167-192. [12] C. D e l l a c h e r i e . CapaciteV et Processus Stochastiques, Springer V e r l a g , 1972. [13] C. D e l l a c h e r i e et P. A. Meyer. P r o b a b i l i t e s e t P o t e n t i e l , Chap. I-IV, 2nd e d i t i o n , Hermann 1975. [14]- C. D e l l a c h e r i e et P. A. Meyer. P r o b a b i l i t e s et P o t e n t i e l , Chap. V-VIII, 2nd e d i t i o n , Hermann. 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Notes on D u a l i t y , and on M a r t i n Boundaries, (unpublished). [53] J . B. Walsh and M. W e i l . "Representation de temps terminaux et a p p l i c a t i o n s aux f o n c t i o n n e l l e s a d d i t i v e s e t aux systemes de Levy", Ann. S c i . Ecole Norm. Sup. (4) 5 (1972), pp. 121-155. [54] S. Watanabe. " P o i n t processes and ma r t i n g a l e s " , S t o c h a s t i c A n a l y s i s , pp. 315-326, Academic Press 1978. [55] G. Weidenfeld. "Changements de temps de processus de Markov", Z. Wahr. 53 (1980), pp. 123-146. [56] M. W e i l . "Conditionnement par rapport au passe s t r i c t " , Seminaire de P r o b a b i l i t e s V, Lecture Notes i n Mathematics 191, pp. 362-372, Springer V e r l a g 1971. 

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