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Embedding in Brownian motion Falkner, Neil F. 1978

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EMBEDDING IN BROWNIAN MOTION by NEIL F. FALKNER B . S c , U n i v e r s i t y of Manitoba, 1973 M.Sc, U n i v e r s i t y of Manitoba, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES i n the Department of Mathematics We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September , 1978 (c) N e i l F. Falkner, 1978 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requ i rement s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I ag ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f M a t h e m a t i c s The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date September 18, 1978 Supervisor: Dr. R. V. Chacon Ab s t r a c t and H i s t o r i c a l Review: Let n be a p o s i t i v e i n t e g e r , l e t y be a p r o b a b i l i t y measure on ]R n , and l e t (B ) be Brownian motion w i t h i n i t i a l d i s t r i b u t i o n y . (For those u n f a m i l i a r w i t h Brownian motion we i n s e r t a b r i e f h e u r i s t i c e x p l a nation. Consider a drunk on a pub crawl i n ]R n . Imagine that h i s i n i t i a l p o s i t i o n i s u n c e r t a i n , and i s described by the p r o b a b i l i t y law y . Imagine that there i s a pub at each po i n t i n ]R n , and that the drunk wanders from pub to pub i n a t o t a l l y random f a s h i o n , as i s to be expected from h i s i n e b r i a t e d s t a t e . Imagine that the drunk takes only an i n f i n i t e s i m a l d r i n k at each pub, so that he keeps moving c o n s t a n t l y . We can describe the drunk's random progress through ]R n by c o n s i d e r i n g the set of a l l p o s s i b l e paths he can f o l l o w , and a s s i g n i n g each some i n f i n i t e s i m a l p r o b a b i l i t y . In more p r e c i s e mathematical terms, we d e s c r i b e the motion of the drunk by means of a c e r t a i n p r o b a b i l i t y measure P on the space C of continuous maps from [0, °°) i n t o 3Rn . Then B t (to) = co(t) f o r co e C , t e [0, °°) . The p r o b a b i l i t y space (C,P) , together w i t h the f a m i l y (B f c) of random v a r i a b l e s defined on i t , i s c a l l e d Brownian motion, a f t e r the b o t a n i s t Brown who, i n 1827, observed the random motion of microscopic p a r t i c l e s suspended i n water.) For each random time T l e t y^ be the d i s t r i b u t i o n of the random v a r i a b l e B^ . (This random v a r i a b l e i s the one defined by B (to) = B - .(to) .) I t i s n a t u r a l to ask which measures v on ]R n are 1 i ( to) of the form y T where T i s a stopping time. (A stopping time i s a - 11 -random time which "does not depend on the f u t u r e " ; f o r example, the f i r s t time (B^) h i t s some set i n H n i s a stopping time.) Skorohod [1] f i r s t considered t h i s s o r t of question. He showed that i f n = 1 and u i s the point mass at 0 , and i f v i s a p r o b a b i l i t y easure on E. w i t h centre of mass at 0 and w i t h f i n i t e v a r i a n c e m 2 a = x dv(x) , then there i s 'a "randomized" stopping time T such that v = and the expectation of T i s equal to the v a r i a n c e of v . (We remark that i f u i s the point mass at 0 and T i s a stopping time, p o s s i b l y randomized, w i t h f i n i t e e x p e c t a t i o n , and i f we define v to be , then v has mean 0 and v a r i a n c e equal to the expectation of T . Thus Skorohod's r e s u l t can be s t a t e d i n an " i f and only i f " form.) Dubins [1] and Root [ 1 ] , by d i f f e r e n t methods, showed that Skorohod's r e s u l t can be improved i n that a T which i s not randomized can be obtained. The method of Dubins y i e l d s a " n a t u r a l " stopping time T whenever v i s a p r o b a b i l i t y measure on ]R whose mean i s defined and equal to 0 , even when v does not have f i n i t e v a r i a n c e . The meaning of " n a t u r a l " here i s enlarged upon i n the paper of Chacon [ 1 ] , where standard stopping times are defined. We give a d i f f e r e n t but equivalent d e f i n i t i o n of standard stopping times i n 8.2, and a l s o another c h a r a c t e r i z a t i o n of them i n 8.13. Doob (see Meyer [3]) has pointed out that i f n = 1 and u and v are any two p r o b a b i l i t y measures on TR then there i s a stopping time T such that u = y . This i s the reason f o r the s p e c i a l i n t e r e s t i n standard stopping times. But we d i g r e s s . To get back to our s t o r y , Chacon and Walsh [ 1 ] , using p o t e n t i a l theory, gave a very transparent proof of the r e s u l t of Dubins and Root. A l s o using p o t e n t i a l theory, Rost [1] - i i i -ge n e r a l i z e d Skorohod's r e s u l t to Markov processes w i t h proper p o t e n t i a l k e rnels (eg., Brownian motion i n dimension greater than or equal to 3). Rost's method, however, produces a randomized stopping time even when a non-randomized one e x i s t s . Now a p h y s i c i s t has s a i d , "Give me f i v e parameters, and I w i l l f i t an elephant; give me s i x parameters, and w i l l make him wiggle h i s trunk!" In other words, i t i s n a t u r a l to ask when one can get a non-randomized stopping time, f o r n >_ 2 . Baxter and Chacon [1] have given s u f f i c i e n t c o n d i t i o n s f o r t h i s to be so, but t h e i r hypotheses are rat h e r strong — see the d i s c u s s i o n i n 8.21. In 7.11 and 8.20, we have succeeded i n proving r e s u l t s along these l i n e s which appear much c l o s e r to being best p o s s i b l e . They are not a c t u a l l y best p o s s i b l e , however, as the example 7.13 shows. I t i s much e a s i e r to get best p o s s i b l e r e s u l t s i f one allows oneself to work w i t h randomized stopping times. In 11.12, the 2 p r o b a b i l i t y measures v on R such that v = v , where T i s a standard randomized stopping time and u i s a p r o b a b i l i t y measure on ]R^ such that i s a p o t e n t i a l (see 1.4), are c h a r a c t e r i z e d . This r e s u l t appears to be new. A l s o , i n 10.5, we prove a p a r t i c u l a r case of the r e s u l t of Rost [ 1 ] , by a d i f f e r e n t method. Now Skorohod [1] al s o showed that i f (X^) i s a square-integrable martingale w i t h time set I = {0, 1, 2,...} , s a t i s f y i n g E(X^) = 0 , and i f (^t) i - s Brownian motion, i n one dimension, s t a r t i n g from 0 , then there i s an i n c r e a s i n g sequence (T ) of randomized stopping times w i t h f i n i t e e x p e c t a t i o n s , such that the processes (B^ ) and i (X.) have the same j o i n t d i s t r i b u t i o n . (Note that i f X. = X,. f o r l J l 0 a l l i , t h i s reduces to the r e s u l t f o r measures which we described - i v -f i r s t . ) Dubins [1] remarks that h i s method can be a p p l i e d here to y i e l d non-randomized stopping times, and that the c o n d i t i o n that 2 E(X_^) < «> can be dropped, though of course the T^'s need not have f i n i t e e xpectation then (though one can show that they w i l l be standard i f c onstructed as Dubins d e s c r i b e s ) . In 12.7 we prove a g e n e r a l i z a t i o n of t h i s to Brownian motion i n n dimensions. (For n >_ 2 , we use randomized stopping times.) We a l s o give a d e t a i l e d proof of Dubins' a s s e r t i o n concerning the one dimensional case. Monroe [1] has shown that a r i g h t - c o n t i n u o u s martingale ^t^0<t<°° C a n ^ e e m k e d d e d -*-n a n "enlargement" of one dimensional Brownian motion by means of a r i g h t -continuous i n c r e a s i n g f a m i l y (T f c) of minimal (= standard) stopping times. In 12.16, we prove a g e n e r a l i z a t i o n of t h i s r e s u l t to n dimensions. F i n a l l y , i n 12.18 we describe an example, discovered by R. V. Chacon, which shows that enlargement r e a l l y i s necessary i n the theorem of Monroe. Now f o r a few words about the s e c t i o n s of t h i s t h e s i s from which r e s u l t s have not yet been mentioned. Sections 1, 5, and 6 are mainly devoted to e s t a b l i s h i n g n o t a t i o n and terminology, and to s t a t i n g c e r t a i n known r e s u l t s , f o r ease of reference. Sections 2, 3, and 4 are devoted to e s t a b l i s h i n g those aspects of the f i n e theory of balayage which are needed throughout the r e s t of the t h e s i s . With the probable exception of the strong form of the domination p r i n c i p l e 2 f o r the l o g a r i t h m i c p o t e n t i a l i n ]R , the r e s u l t s of these three s e c t i o n s are not new. I b e l i e v e , however, that some of the proofs are novel. For example, the reader might f i n d i t amusing to compare our proof of 2.1 w i t h the proof of 8.43 of Helms [ 1 ] , or to compare our - v -proof of 3.2 w i t h the d i s c u s s i o n i n s e c t i o n s 9 and 10 of the paper of B r e l o t [ 1 ] . Secti o n 9 of t h i s t h e s i s i s concerned w i t h the development of the m a t e r i a l on randomized random v a r i a b l e s , and enlargements of p r o b a b i l i t y spaces, needed i n s e c t i o n s 10, 11, and 12, and i s e s s e n t i a l l y a review of, and enlargement on, p a r t s of the papers of Baxter and Chacon [2 and 3 ] . The theorem 9.13 however, though a simple r e s u l t , i s new and sheds l i g h t on the meaning of the " d i s t r i b u t i o n a l enlargements" defined i n Baxter and Chacon [ 3 ] . ( D i s t r i b u t i o n a l enlargements are e s s e n t i a l l y what we c a l l o p t i o n a l enlargements.) One l a s t t h i n g : towards the end of s e c t i o n 10, we d i s c u s s a couple of p o t e n t i a l t h e o r e t i c a p p l i c a t i o n s of the embedding theorem 10.5. - v i -Table of Contents Page 1. P o t e n t i a l Theoretic P r e l i m i n a r i e s 1 2. The I n t e g r a l Representation of Balayage 8 3. Thinness 23 4. The Strong Form of the Domination P r i n c i p l e 31 5. Brownian Motion P r e l i m i n a r i e s 46 6. P r e l i m i n a r i e s on Brownian Motion and P o t e n t i a l Theory 57 7. Embedding Measures i n Brownian Motion i n a Green Region, Using Non-Randomized Stopping Times 61 1 2 8. Embedding Measures i n Brownian Motion i n H or 1 , Using Non-Randomized Stopping Times 76 9. Randomized Stopping Times, and Enlargements of P r o b a b i l i t y Spaces 98 10. Embedding Measures i n Brownian Motion i n a Green Region, Using Randomized Stopping Times 124 2 11. Embedding Measures i n Brownian Motion i n H , Using Randomized Stopping Times 136 12. Embedding Processes i n Brownian Motion 147 13. Appendix of Miscellaneous N o t a t i o n and Terminology 184 References 189 Index of Selected N o t a t i o n and Terminology 192 - v i i -Ac knowled g ement s I wish to thank my s u p e r v i s o r , R a f a e l Chacon, whose guidance and encouragement have b e e n i n v a l u a b l e ' t o me. He has taught me much about how to do mathematics. I would a l s o l i k e to thank Maurice Sion, whose l u c i d teaching made i t i n f i n i t e l y e a s i e r f o r me to l e a r n the "heavy duty" measure theory needed i n the study of processes, and John Walsh, who was always ready to answer my questions about Brownian motion and P o t e n t i a l Theory. I wish to express my g r a t i t u d e to the N a t i o n a l Research C o u n c i l of Canada, and the Izaak Walton K i l l a m Memorial Fund of the U n i v e r s i t y of B r i t i s h Columbia, f o r p r o v i d i n g me w i t h f i n a n c i a l support. F i n a l l y , I would l i k e to thank Cathy Agnew and C a r o l Samson f o r t h e i r f i n e t y p i n g . - 1 -1. POTENTIAL THEORETIC PRELIMINARIES This s e c t i o n i s mainly devoted to e s t a b l i s h i n g the n o t a t i o n and terminology of p o t e n t i a l theory that we s h a l l use. 1.1. For each x e H , 6 denotes the Dirac measure at x: x 6 (A) = l . ( x ) f o r A £ B o r e l H X A. I f x = 0 we s h a l l j u s t w r i t e 6 f o r 6 1.2. A denotes the L a p l a c i a n on E. ; n .2 i = l 9x: We s h a l l f r e q u e n t l y use the L a p l a c i a n i n the sense of d i s t r i b u t i o n theory. In p a r t i c u l a r , i f u i s a superharmonic f u n c t i o n i n an open set D i n H n then -Au i s a ( p o s i t i v e ) measure i n D which i s c a l l e d the Riesz measure of u . On the other hand, i f T i s a d i s t r i b u t i o n i n D such that -AT i s a ( p o s i t i v e ) measure i n D then T a r i s e s from a unique superharmonic f u n c t i o n i n D . (We s h a l l have no need of the l a t t e r f a c t , though.) 1.3. Let $ be the f u n c t i o n on ]Rn defined by $(x) = 2 1_ 2TT x i f n = 1 l o g x i f n = 2 i f n > 3 (n-2)a x n ,n-2 - 2 -where a i s the n - 1 dimensional Lebesgue measure of the surface n of the u n i t sphere i n ]R n . (we take $(0) = + °° i f n >^  2 .) Then <3> has the f o l l o w i n g three p r o p e r t i e s : a) $ i s superharmonic. b) A $ = -6 . c) $ i s i n v a r i a n t under r o t a t i o n s about the o r i g i n . In f a c t , these p r o p e r t i e s determine $ to w i t h i n an a d d i t i v e constant. Of course property b) has the simple form that i t does because of the way we normalized $ . We s h a l l a l s o use the l e t t e r $ to denote two other f u n c t i o n s : f i r s t , the f u n c t i o n on ]R n x ]R n defined by $(x,y) = $(x- y ) ; second, the f u n c t i o n on [0,°°) s a t i s f y i n g 4>(x) = $(| |x| |) . The context w i l l always make c l e a r which f u n c t i o n we mean. Note that $ depends on n but t h i s dependence i s not made e x p l i c i t . I t i s p a r t i c u l a r l y important to keep t h i s i n mind when $ i s regarded as a f u n c t i o n on [O,00) . 1.4. I f y i s a measure on the B o r e l subsets of H n (not assumed f i n i t e on compact sets) then we de f i n e U^, U^: H n —>• [0,°°] by t r > ) = u V ) = $ +(x,y)dy(y) $ (x,y)dy(y) and we d e f i n e U^, on the set where and are not both i n f i n i t e , - 3 -by We say i s a p o t e n t i a l i f f U M i s everywhere defined and super-harmonic on ]R n • This happens i f f i s f i n i t e at a l l p o i n t s of E.n and i s f i n i t e at at l e a s t one point of 3Rn . In t h i s case i s l o c a l l y Lebesgue i n t e g r a b l e and y i s f i n i t e on compact sets (so and y can be regarded as Schwartz d i s t r i b u t i o n s on ]R n) and AU^ = -y; i n p a r t i c u l a r , y can be recovered from . The f o l l o w i n g three r e s u l t s give more e x p l i c i t c h a r a c t e r i z a t i o n s of those measures y f o r which i s a p o t e n t i a l . We s t a t e these r e s u l t s without proof, but the statements are so d e t a i l e d that they almost prove themselves. 1.5. P r o p o s i t i o n : Suppose n = 1 . a) = 0 and U y = -U y b) I f U y i s f i n i t e at two d i s t i n c t p o i n t s then y i s f i n i t e , i s f i n i t e at a l l p o i n t s and U y i s continuous; indeed U y i s L i p s c h i t z and yOR) i s a L i p s c h i t z constant f o r x dy(x) < °° c) i s a p o t e n t i a l i f f yOR) < 0 0 and 1.6. P r o p o s i t i o n : Suppose n = 2 . a) I f U^(x) i s f i n i t e and y({y £ K. : | |y-x| | <_ r}) i s f i n i t e 2 f o r some x i n K. and some r i n ( l , 0 0 ) then y i s - 4 -f i n i t e , U.^  i s f i n i t e at a l l p o i n t s and i s continuous; 1 2 indeed i s L i p s c h i t z and — u0R ) i s a L i p s c h i t z constant f o r . b) I f u i s f i n i t e on compact sets then i s l o c a l l y Lebesgue i n t e g r a b l e . c) U P i s a p o t e n t i a l i f f u i s f i n i t e and log +||x||dp(x) < » . 1.7. P r o p o s i t i o n : Suppose n >_ 3 . a) = 0 and U P = . b) I f U P i s f i n i t e at one poin t then u i s f i n i t e on compact sets and U P i s l o c a l l y Lebesgue i n t e g r a b l e . c) i s a p o t e n t i a l i f f lA$du < 0 0 • One can combine 1.5 through 1.7 i n t o a s i n g l e r e s u l t . 1.8. P r o p o s i t i o n : U P i s a p o t e n t i a l i f f u i s f i n i t e on compact • sets and I $(x) I du (x) < 0 0 . J||x||>l 1.9. Consider an open set D i n ]R n . I f f o r some poin t X Q i n D the f u n c t i o n $(XQ,«) has a harmonic minorant i n D then f o r a l l x i n D, $(x,*) has a harmonic minorant i n D, and indeed has a greatest one which we s h a l l , f o r the moment, denote by h(x,•) . In t h i s case we say D i s a Green re g i o n and the f u n c t i o n G: D x D —*• [0,°°] defined by - 5 -G(x,y) = *(x,y) - h(x,y) i s c a l l e d the Green f u n c t i o n of D . For each x e D, G(x,*) i s the smallest non-negative super-harmonic f u n c t i o n u i n D s a t i s f y i n g Au = - 6 J x G(x,') should be thought of as the e l e c t r o s t a t i c p o t e n t i a l i n D that would a r i s e from a u n i t p o i n t charge at x i f 3D were made of e l e c t r i c a l l y conducting m a t e r i a l , provided 3D i s connected. One can show that G i s j o i n t l y continuous on D x D and G(x,y) = G(y,x) f o r a l l x,y e D . C l e a r l y i f n >_ 3 then E n i s a Green r e g i o n and the Green f u n c t i o n G of lR n i s given by G(x,y) = $(x,y) . I t i s a l s o c l e a r that any open subset of a Green region i s a Green r e g i o n . In p a r t i c u l a r , any open subset of E n , where n >_ 3, i s a Green r e g i o n . A moment's thought w i l l show that an open set D i n ]R i s a Green r e g i o n i f f D ^ E . 2 One can show that i f D i s an open set i n E then D i s a 2 Green r e g i o n i f f E \D i s not a p o l a r s e t . This i s Myrberg's theorem -see 8.33 of Helms [1]. (A p o l a r set i s a set contained i n the set of " p o l e s " ( i e . , i n f i n i t i e s ) of a superharmonic f u n c t i o n . P o l a r sets are the " s m a l l " sets of p o t e n t i a l theory.) 2 One can a l s o show that i f D i s an open set i n E (or f o r that matter, i n K.n) then D i s a Green r e g i o n i f f there i s a non-constant - 6 -non-negative superharmonic f u n c t i o n i n D . This i s part of 8.33 of Helms [1], but i t ' s not hard to give a proof using only the m a t e r i a l i n the f i r s t s i x chapters of Helms; i e . , the m a t e r i a l up to and i n c l u d i n g the chapter on Green p o t e n t i a l s . 1.10. Let D be a Green region i n ]R n w i t h Green f u n c t i o n G . I f y i s a measure on the o - f i e l d of B o r e l sets of D, we de f i n e Gy: D —»- [0,°°]. by Gy(x) = G(x,y)dy(y) I f Gy i s f i n i t e at at l e a s t one p o i n t of each component of D then Gy i s superharmonic i n D, y i s f i n i t e on compact sets (and so can be regarded as a Schwartz d i s t r i b u t i o n i n D) and AGy = -y . In t h i s case we say Gy i s the p o t e n t i a l of y, and we d e s c r i b e y as a measure i n D such that Gy i s a p o t e n t i a l . I f y i s a measure i n D such that Gy i s a p o t e n t i a l then the greatest harmonic minorant of Gy i n D i s zero. Gy i s thus the smallest non-negative superharmonic f u n c t i o n i n D having y as Riesz measure. I f u i s a superharmonic f u n c t i o n i n D which has a subharmonic minorant, and i f y i s the Riesz measure of u, then Gy i s a p o t e n t i a l and u = Gy + h where h i s the grea t e s t harmonic minorant of u . This i s known as the Riesz decomposition theorem. 1.11. Let D be an open set i n H n . I f u i s a non-negative superharmonic f u n c t i o n i n D and E i s - 7 -a subset of D then the r£duite of u over E r e l a t i v e to D (which we s h a l l denote by red(u,E,D)) i s defined to be the infimum of the set of non-negative superharmonic f u n c t i o n s i n D which majorize u on E . The balayage of u over E r e l a t i v e to D (which we s h a l l denote by bal(u,E,D)) i s defined to be the lower r e g u l a r i z a t i o n of u . One can show that bal(u,E,D) i s superharmonic i n D and the set where bal(u,E,D) d i f f e r s from red(u,E,D) i s a p o l a r subset of the boundary of E . We note that the more usual n o t a t i o n s f o r red(u,E,D) and bal(u,E,D) are and r e s p e c t i v e l y . Of course, i f D i s not a Green r e g i o n then re"duite and balayage are t r i v i a l n o t i o n s , s i n c e then every non-negative superharmonic f u n c t i o n i n D i s constant. 1.12. The main reference we recommend f o r b a s i c p o t e n t i a l theory i s the book of Helms [ 1 ] . Other u s e f u l references are the books of Du P l e s s i s [ 1 ] and B r e l o t [ 1 ] . In p a r t i c u l a r , Du P l e s s i s uses the n a t u r a l and i n t u i t i v e language of d i s t r i b u t i o n theory, whereas Helms avoids i t . - 8 -2. THE INTEGRAL REPRESENTATION OF BALAYAGE Throughout t h i s s e c t i o n , n i s a p o s i t i v e i n t e g e r , D i s a Green r e g i o n i n TRU, and G i s the Green f u n c t i o n of D . I f E i s a subset of ]R n and y i s a measure i n D such that Gy i s a p o t e n t i a l then bal(Gy,E,D) i s the p o t e n t i a l of a measure i n D which we s h a l l denote by bal(y,E,D) . Thus bal(Gy,E,D) = G bal(y,E,D) . In t h i s s e c t i o n we s h a l l prove the f o l l o w i n g i n t e g r a l represent-a t i o n formula f o r balayage: I f u i s any non-negative superharmonic f u n c t i o n i n D then f o r every x i n D, bal(u,E,D)(x) = u ( y ) b a l ( 6 ,E,D)(dy) D X This was proved by B r e l o t [ 1 ] . Our method of proof d i f f e r s from B r e l o t ' s i n that i t r e q u i r e s no appeal to the theory of the D i r i c h l e t problem; i t uses only the c l a s s i c a l p o t e n t i a l theory contained i n the f i r s t seven chapters of the book of Helms [ 1 ] . We begin w i t h a weak v e r s i o n of the "domination p r i n c i p l e " . 2.1. P r o p o s i t i o n : Let E be a closed subset of D and l e t y be a measure i n D such that Gy i s a p o t e n t i a l s y l i v e s on E, and y does not charge polar subsets of the boundary of E. Suppose v i s a non-negative superharmonic f u n c t i o n i n D such - 9 -that v >_ Gy on E\Z where Z i s some polar s et. Then v >_ Gy throughout D . Proof: vAGy i s a p o t e n t i a l i n D so vAGy = Gv f o r some measure v i n D . Now Gv >_ Gy on E\Z and we wish to show Gv >_ Gy throughout D . Let x be i n D . For 0 < r < distance(x^R n\D) l e t a^ be the uniform u n i t d i s t r i b u t i o n on the closed b a l l of rad i u s r centred at x, and l e t g = b a l ( a ,E,D) . r r Now B l i v e s on E and Ga = Gg on E\P where P i s some r r r polar subset of the boundary of E . Al s o B r does not charge p o l a r s e t s , by 2.2 below. Hence Gvda = Ga dv r r GB dv r f Gv dB Gy dB GB r dy Ga dy r Gy da L e t t i n g r go to 0, we obt a i n Gv(x) > Gy(x) • - 10 -2.2. Lemma: Let u be a superharmonic f u n c t i o n i n an open set V i n H n , and l e t y be the Riesz measure of u . Then y does not charge p o l a r subsets of the set where u i s f i n i t e . Proof: By a standard argument we can reduce to the case where V i s a Green region (even an open b a l l ) and u i s a p o t e n t i a l i n V . Then u = G^u . I t s u f f i c e s to show that i f K i s any compact sub-set of {u < 0 0 } then y does not charge polar s e t s . Now G y K V K i s f i n i t e so by 6.21 of Helms [ 1 ] , given e > 0 there i s a compact se t C £ K such that y(K\C) < e and G y i s continuous on V I f P i s a polar set i n V then there i s a measure y w i t h compact support i n V such that G^y = 0 0 o n P n K . Now V d y c = G y y c dy < °°, so y c ( { G v y = «>})= 0 Hence the y -outer measure of P i s l e s s than e . K As e > 0 was a r b i t r a r y , y does not charge p o l a r s e t s . K • 2.3. Lemma: Let E be a closed subset of D and l e t u be a p o t e n t i a l i n D which i s f i n i t e on the boundary of E . Let h be the greatest harmonic minorant of u i n the open set D\E . Let v = h i n D\E u on E Then v = reM(u,E,D) . Proof: Let w = r£d(u,E,D) . Then w £ u and w i s harmonic i n D\E so w <_ v . Let be a sequence of open b a l l s such that - 11 -D\E = uB. i and V i , { j: B. = B.} i s i n f i n i t e . Let u = u and l e t PI(u . ; B .) i n B . 1 1 i u i+1 u. l i n D\B. l Then each u. i s superharmonic i n D, and u. 4- v . l l Hence v, the lower r e g u l a r i z a t i o n of v, i s superharmonic i n D and i s , i n f a c t , the p o t e n t i a l of a measure v which l i v e s on the closed set E and (by 2.2) does not charge polar subsets of the boundary of E . Now w >_ v on E and {w < w} i s a polar set. Thus by the domination p r i n c i p l e 2.1, w >_ v throughout D . Now i n D\E, w = w and v = v . A l s o , on E, w = u = v . Thus v = w . Remark: I f u i s not f i n i t e on the boundary of E the co n c l u s i o n of the above lemma may f a i l . For instance suppose E = {x} and u = G(x,*) f o r some x e E, and n (the dimension of lR n) i s at l e a s t 2 . Then v = u but red(u,E,D) = ° ° l r v ; . be p o t e n t i a l s i n D which are f i n i t e on the boundary of E . Then Proof: The greatest harmonic minorant of a sum of two superharmonic • 2.4. C o r o l l a r y : Let E be a closed subset of D and l e t u ,u b a l ( u + u = baKu^E.D) + bal(u 2,E,D) . - 12 -fun c t i o n s (each of which has a subharmonic minorant) i s the sum of t h e i r greatest harmonic minorants - see 5.22 of Helms [ 1 ] . Hence red(u^ + U £ , E, D) = red(u 1 }E,D) + r£d(u2,E,D) . I t f o l l o w s that bal(u 1+u 2,E,D) and b a l ( u ,E,D) + bal(u 2,E,D) d i f f e r at most on a po l a r s e t . But two superharmonic f u n c t i o n s which are equal except on a po l a r set are equal everywhere. • The f o l l o w i n g r e s u l t i s proved i n chapter 8 of Helms [1] using the D i r i c h l e t problem theory developed i n that chapter. To sub-s t a n t i a t e our c l a i m that the r e s u l t s of t h i s s e c t i o n can be e s t a b l i s h e d without t h i s part of the theory, we i n c l u d e a proof. 2.5. Lemma: Let Z be a po l a r subset of D and l e t x e D\Z . Then there i s a f i n i t e measure u i n D such that Gu = 0 0 on Z but Gy(x) < °° . Proof: By the d e f i n i t i o n of a polar set (Helms [ 1 ] , p. 126) there i s an open set V <=_ R n and a superharmonic f u n c t i o n v i n V such that Z _c V and v = 0 0 on Z . Let v be the Riesz measure of v . Let (B_^) be a sequence of open b a l l s such that (D\{x}) n V = uB. . I For each i , l e t v. be the measure on the B o r e l s e t s of D defined I by v^(A) = v(AnB_^) . Each i s a f i n i t e measure i n D s a t i s f y i n g Gv. (x) < 0 0 . A l s o , i n B. , AGv. = -v = Av . Hence i n B., Gv. and i l i l i - 13 -v d i f f e r by a harmonic f u n c t i o n . In p a r t i c u l a r , {Gv = °°} n B. = {v = •»} n B. . 1 1 1 Now choose a sequence (a.) of p o s i t i v e r e a l numbers such that Y a.v. (D) < . 1 1 00 1 and y a.Gv.(x) < . 1 1 i 00 The d e s i r e d measure y can be taken to be £ i • 2.6. C o r o l l a r y : Let u be a non-negative superharmonic f u n c t i o n i n D and l e t E be any subset of D . Then bal(u,E,D) i s the smallest non-negative superharmonic f u n c t i o n v i n D such that v >_ u on E except f o r a polar set. Proof: Let w = bal(u,E,D) . Then w i s a non-negative super-harmonic f u n c t i o n i n D and E n {w<u} i s a p o l a r set. Let v be a non-negative superharmonic f u n c t i o n i n D such that Z = E n {v<u} i s a p o l a r set. Let x e D\Z . By 2.5 there i s a f i n i t e measure u i n D such that Gy = 0 0 on Z but Gy(x) < 0 0 . For any e > 0, v + eGy ^ u on E, so v + eGy >_ w . L e t t i n g e go to 0, we f i n d that v (x) >_ w(x) . Hence v >_ w on D\Z As Z i s p o l a r , v >_ w throughout D . • 2.7. C o r o l l a r y : Let ( u^) ^ e a n i n c r e a s i n g sequence of non-negative superharmonic f u n c t i o n s i n D whose supremum u i s not i d e n t i c a l l y i n f i n i t e on any component of D, and hence i s superharmonic. Let - 14 -(E_^) be an i n c r e a s i n g sequence of subsets of D, w i t h union E . Then b a l ( u ,E ,D) + bal(u,E,D) . Proof,: Let w = bal(u,E,D) . C l e a r l y bal(u_^,E^,D) increases to a superharmonic f u n c t i o n v <_ w . I f x e E and v(x) < u(x) then f o r some i , x e E^ and v(x) < u_^(x) . Thus E n {v<u} £ u (E. n {bal(u.,E.,D) < u.}) . i Hence E n {v<u} i s a p o l a r s e t , so v >_ w by 2.6. • 2.8. Lemma: Let u be any non-negative superharmonic f u n c t i o n i n D . Then u i s the l i m i t of an i n c r e a s i n g sequence of bounded p o t e n t i a l s i n D whose Riesz measures are f i n i t e . Proof: Let be a sequence of open r e l a t i v e l y compact subsets of D which increases to D . For each i , l e t u. = bal(iAu,D. ,D) l l Then each u_^  i s a p o t e n t i a l i n D, bounded by i , and s a t i s f y i n g u. = uAi i n D. . Hence u. i u . A l s o , i f u. i s the Riesz l l l l measure of u. then u.(D) = u.(D.) < 0 0 . i x i i n 2.9. Theorem: Let u be a non-negative superharmonic f u n c t i o n i n D and l e t E be any subset of D . Then f o r each x e D, bal(u,E,D)(x) = u ( y ) b a l ( 6 ,E,D)(dy) . Proof: I . Assume u i s a f i n i t e p o t e n t i a l w i t h Riesz measure u . a) Assume E i s closed i n D . Let a = y E B = yD\E - 15 -Then u = Ga + GB and Ga and G3 are f i n i t e p o t e n t i a l s so by 2.4, bal(u,E,D) = bal(Ga,E,D) + bal(GB,E,D) . F i r s t consider Ga . By the domination p r i n c i p l e 2.1, bal(Ga,E,D) = Ga . Al s o bal(G6 x,E,D) = G8^ on E\Z where Z i s p o l a r ; hence bal(G6 ,E,D) = G6 almost everywhere with respect to a, by 2.2. X X Thus Ga(y)bal(6 ,E,D)(dy) X G bal(<$x,E,D) (z)a(dz) G6 x(z)a(dz) = Ga(y)6 x(dy) = Ga(x) Now consider GB . i ) F i r s t suppose x I E or n = 1 . Let V be the open set D\E and l e t G' be the Green f u n c t i o n of V. Let h (resp. k) be the greatest harmonic minorant of GB (resp. GS^) i n V . As GB and G6 are f i n i t e on the boundary ( i n D) of E, X and bal(G8,E,D) i n V k = bal(G6 x,E,D) i n V, by 2.3. ( I f n = 1, every superharmonic f u n c t i o n i s f i n i t e . ) Now G'B = G8 - h i n V . But G'B(x) = G' (x,z)g(dz) G(x,z) - k(z)3(dz) V = G8(x) - bal(G6 ,E,D)(z)B(dz) . - 16 -Thus bal(GB,E,D)(x) = bal(G6 ,E,D)(z)B(dz) X G bal(6 ,E,D)(z)B(dz) GB(y)bal(6 x,E,D)(dy) i i ) Now suppose x e E and n _> 2 . Then {x} i s p o l a r so bal(v,E\{x),D) = bal(v,E,D) f o r each non-negative superharmonic f u n c t i o n v i n D, by 2.6. For each i e U l e t B^ be the open b a l l of ra d i u s 2 centred at x and l e t E = E\B^ . Then each E. i s closed i n D and E. + E\{x} . Hence, by 2.7, l i ba l ( v , E ,D) + bal(v,E\{x},D) f o r each non-negative superharmonic f u n c t i o n v i n D . Now by i ) , bal(GB,E ,D)(x) = ba l ( G 6 x , E i , D ) ( z ) 8 ( d z ) f o r each i . L e t t i n g i go to i n f i n i t y , we o b t a i n bal(GB,E,D)(x) = bal(G6 x,E,D)(z)g(dz) G8(y)bal(6 x,E,D)(dy) Combining our r e s u l t s so f a r we have bal(Gy,E,D)(x) = Gy(y)bal(6 x,E,D)(dy) f o r a l l x e D, where Gy i s f i n i t e and E i s closed i n D . b) Now assume that E i s merely a countable union of closed subsets of D . Let (E^) be an i n c r e a s i n g sequence of closed subsets of D - 17 -whose union i s E . Then f o r each i , bal(Gy,E ,D)(x) = G y ( y ) b a l ( 6 x , E i , D ) ( d y ) , by a ) . L e t t i n g i go to i n f i n i t y , we o b t a i n bal(Gy,E,D)(x) = Gy(y)bal(6 x,E,D)(dy) . c) Now l e t E be an a r b i t r a r y subset of D . Let us impose the a d d i t i o n a l assumption that y(D) < °° . F i x e > 0 . Let v = e + bal(Gy,E,D) w = e + b a l ( G 6 x , E , D ) S = {v > Gy} T = {w > G6 } x Then E i s contained i n S to w i t h i n a p o l a r set so f o r any non-negative superharmonic f u n c t i o n f i n D, b a l ( f ,E,D) <_ b a l ( f ,S,D) . The same statement holds w i t h S replaced by T . A l s o , S = u {v>r} n {Gy <_ r} reQ Now f o r each r , {v>r} i s open and {Gy <_ r} i s closed i n D . Thus S i s a countable union of closed subsets of D . S i m i l a r l y , so i s T . Hence - 18 -bal(Gy,E,D)(x) < bal(Gy,T,D)(x) Gy(y)bal(6 x,T,D)(dy) bal(G6 x,T,D)(z)y(dz) w dy bal(G6 ,E,D)dy = ey(D) + <_ sy (D) + = ey(D) + = ey(D) + bal(Gy,S,D)(x) (by b) bal(G6 x,S,D)dy Gy(y)bal(6 x,S,D)(dy) (by b) < ey(D) + v = ey(D) + e + bal(Gy,E,D)(x) As y(D) < «° and e > 0 was a r b i t r a r y , bal(Gy,E,D)(x) Gy (y)bal(5 x,E,D)(dy) < bal(Gy,E,D)(x) I I . Now l e t us consider the general case. By 2.8, there i s an i n c r e a s i n g sequence (u-^) °^ f i n i t e p o t e n t i a l s i n D such that u. + u and each u. has f i n i t e Riesz measure. By I . c ) , f o r each 1 1 i we have b a l ( u ,E,D)(x) = u (y ) b a l ( 6 ,E,D)(dy) 1 X - 19 -L e t t i n g i go to i n f i n i t y and applying 2.7 on the l e f t and the monotone convergence theorem on the r i g h t we ob t a i n at l a s t bal(u,E,D)(x) = u(y)bal(S x,E,D)(dy) • 2.10. C o r o l l a r y : Let E be any subset of D and l e t y be a measure i n D such that Gy i s a p o t e n t i a l . Then f o r each- x e D, bal(Gy,E,D)(x) = bal(G6 x,E,D)dy Proof: bal(Gy,E,D)(x) = Gy(y)bal(6 x,E,D)(dy) G b a l ( 6 x , E , D ) ( z ) y ( d z ) bal(G6 x,E,D)dy 2.11. C o r o l l a r y : Let E be any subset of D . Then f o r a l l x,y e D, • bal(G6 ,E,D)(x) = bal(G<5 ,E,D)(y) y x Proof: Take y = & i n 2.10. y • 2.12. C o r o l l a r y : Let E be any subset of D and l e t u,v be non-negative superharmonic f u n c t i o n s i n D . Then bal(u+v,E,D) = bal(u,E,D) + bal(v,E,D) . Proof: This f o l l o w s immediately from the l i n e a r i t y of the i n t e g r a l . • - 20 -2.13. C o r o l l a r y : Let E be any subset of D . Let x e D and l e t y be a measure i n D such that bal(u,E,D)(x) = u(y)y(dy) f o r every non-negative superharmonic f u n c t i o n u i n D of the form u = G(z,«) f o r z e D . Then y = bal(6 x,E,D) . Proof: For any z e D, G Y(z) = G ( z , y ) Y ( d y ) = bal(G(z,-),E,D)(x) G(z,y)bal(6 x,E,D)(dy) = G ba l ( 6 ,E,D)(z) . That i s , Y a n d bal(6 x,E,D) have the same Green p o t e n t i a l r e l a t i v e to D . • 2.14. Lemma: Let f be a twice c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n w i t h compact support i n D . Then f i s e x p r e s s i b l e as the d i f f e r e n c e of two non-negative bounded continuous superharmonic f u n c t i o n s i n D . Proof: Let g = -Af and l e t u = Gg +, v = Gg . Then u and v are f i n i t e continuous superharmonic f u n c t i o n s i n D, by 6.22 of Helms [ 1 ] . Now A(u-v) = -g = Af i n D . Hence there i s a harmonic f u n c t i o n h i n D such that u - v = f + h . Now u and v are - 21 -bounded on the s u p p o r t of g and hence a r e bounded on D by the d o m i n a t i o n p r i n c i p l e 2.1. Thus h i s a l s o bounded on D . L e t M = sup h . Then f = (u+M) - (v+M-h), and u + M , v + M - h a r e bounded n o n - n e g a t i v e c o n t i n u o u s superharmonic f u n c t i o n s i n D . • Remark: U s i n g D i r i c h l e t problem t h e o r y , one can show t h a t the h i n the above p r o o f must be 0, but we don't need t h i s h e r e . 2.15. Theorem: L e t E be any subset of D . L e t y be a measure i n D such t h a t Gy i s a p o t e n t i a l . I f A i s any B o r e l subset of D, then: a) x>—>• b a l ( 6 x > E , D ) (a) i s a B o r e l f u n c t i o n i n D b) b a l ( y , E , D ) ( A ) = y ( d x ) b a l ( 6 x , E , D ) ( A ) P r o o f : a) b a l ( 6 ,E,D)(.D) = 1 bal(6 ,E,D)(dy) = b a l ( l , E , D ) (x) <_ 1 f o r each x e D; i n p a r t i c u l a r , b a l ( 6 x > E , D ) i s a f i n i t e measure i n D . Suppose $ i s a t w i c e c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n w i t h compact support i n D . Then <Ky)bal(6x,E,D)(dy) i s a B o r e l f u n c t i o n i n D by 2.14 combined w i t h 2.9. By a monotone c l a s s argument, i t f o l l o w s t h a t x b a l ( 6 x , E , D ) (A) i s a B o r e l f u n c t i o n i n D f o r any B o r e l subset A of D . - 22 -b) By a) we can define a measure v on the B o r e l subsets of D by v(A) = u(dx)bal(6 x,E,D)(A) . Then f o r any z e D, Gv(z) = v(dy)G(y,z) U(dx) bal(6 x,E,D)(dy)G(y,z) bal(G6 z,E,D)du = bal(Gy,E,D)(z), where we have a p p l i e d 2.9 and 2.10. Thus Gv = G bal(u,E,D), so v = bal(y,E,D) . • - 23 -3. THINNESS 3.1. D e f i n i t i o n . Let D be a Green region i n ]Rn . Let E £ D and l e t x e D . We s h a l l say E i s t h i n at x r e l a t i v e to D i f f there i s a non-negative superharmonic f u n c t i o n u i n D such that bal(u,E,D)(x) < u(x) . A l s o , we s h a l l use the f o l l o w i n g n o t a t i o n s : fringe(E,D) = {x e E: E i s t h i n at x r e l a t i v e to D} base(E,D) = {x e D: E i s not t h i n at x r e l a t i v e to D} . 3.2. Theorem: Let D be a Green region i n ]R n . Then there i s a bounded p o t e n t i a l v i n D such that f o r every E £ D, base(E,D) = (bal(v,E,D) = v} . Proof: Let be a sequence of open r e l a t i v e l y compact subsets of D such that the range of 1 S a n open base f o r D . For v — i each i , l e t v. = bal(l,V.,D) . Let v = )2 v. . Then v i s a l l l bounded p o t e n t i a l i n D . Suppose E £ D and x e D\base(E,D) . We'll show that bal(v,E,D)(x) < v(x) . W e l l , there i s a non-negative superharmonic f u n c t i o n u i n D such that bal(u,E,D)(x) < u(x) . Let c be a number s t r i c t l y between bal(u,E,D)(x) and u(x) . Now {u > c} i s open i n D and contains x . Thus f o r some j , x e V £ {u > c} . Then u >_ c bal(l,V^,D) i n D . Hence bal(u,E,D) > c bal(v.,E,D) . Now cv.(x) = c > b a l ( u , E , D ) ( x ) , so - 1 3 v.(x) > b a l ( v . ,E,D) (x) . Since 2 _ : l b a l ( v . ,E,D) + £ 2~\. i s a 3 3 3 1 non-negative superharmonic f u n c t i o n i n D which i s greater than or - 24 -equal to v on E except f o r a p o l a r s e t , < v ( x ) , by 2.6. • 3.3. C o r o l l a r y : Let D be a Green r e g i o n i n ]R n Let E c D Then fringe(E,D) i s a p o l a r set and base(E,D) i s a countable i n t e r s e c t i o n of open s e t s . Proof: Let v be as i n 3.2. Then fringe(E,D) = E n {bal(v,E,D) < v}, and so i s a p o l a r set by 7.40 of Helms [1]. Also D\base(E,D) = {bal(v,E,D) < v } , and so i s a countable union of closed subsets of D as v and bal(v,E,D) are lower semicontinuous. f u n c t i o n G . Let y be a measure i n D such that Gy i s a p o t e n t i a l . Let E be any subset of D . Then bal(y,E,D) l i v e s on base(E,D) . Proof: Let v be as i n 3.2. Consider any x e D . Let w = bal(v,E,D). Then bal(w,E,D) = w by 2.6. Thus • 3.4. C o r o l l a r y : Let D be a Green r e g i o n i n H n , w i t h Green w(y)bal(S ,E,D)(dy) = w(x) = v ( y ) b a l ( 6 ,E,D)(dy), by 2.9. A l s o , w(x) i s f i n i t e and w <_ v . Thus b a l ( 6 ,E,D)({w < v}) = 0 . - 25 -As t h i s i s true f o r a l l x e D, we have bal(y,E,D)({w < v}) = 0, by 2.15. But {w < v} = D\base(E,D> . • 3.5. C o r o l l a r y : Let D be a Green region i n ]R n, l e t E £ D, and l e t u,v be non-negative superharmonic f u n c t i o n s i n D . Then the f o l l o w i n g are e q u i v a l e n t : a) u <_ v on E\Z f o r some polar set Z b) u <_ v on base(E,D) . Proof: a) => b) . Let f = bal(u,E,D) and l e t g = bal(v,E,D) . Then f <_ g by 2.6. But by the d e f i n i t i o n of base(E,D), f = u on base(E,D) and g = v on base(E,D) . Hence u <_ v on base(E,D) . b) => a) . Let Z = fringe(E,D) . Then Z i s a p o l a r set by 3.3, and u <_ v on E\Z . • Now we are going to show that i f D and D' are Green regions i n ]R n, E £ H n , and x e D n D' then E n D i s t h i n at x r e l a t i v e to D i f f E n D' i s t h i n at x r e l a t i v e to D' . 3.6. P r o p o s i t i o n : Let D be a Green r e g i o n i n JRn and l e t E £ D . a) I f x 6 D\E then E i s t h i n a t x . b) I f n = 1, x e D, and E i s t h i n at x then x i E . - 26 -Proof: a) Let G be the Green f u n c t i o n of D, l e t u = G(x,*)» and l e t v = bal(u,E,D) . Let W be a connected open subset of D such that W n E = 0 and x e W . Then v i s harmonic i n W but u i s not. Thus u - v i s a non-negative superharmonic f u n c t i o n i n W which i s s t r i c t l y p o s i t i v e at some point of W and hence at a l l p o i n t s of W . In p a r t i c u l a r , v(x) < u(x) . Thus E i s t h i n at x . b) Let u be a non-negative superharmonic f u n c t i o n i n D . Then red(u,E,D) = bal(u,E,D) since only the empty set i s polar i n dimension one. Thus u = bal(u,E,D) on E . But i n one dimension, a l l super-harmonic f u n c t i o n s are continuous. Hence u and bal(u,E,D) agree on the c l o s u r e of E i n D . • 3.7. Lemma: Let D be a Green r e g i o n i n ]R n, where n >_ 2 . Let W be an open subset of D, and l e t x e W . Then there i s a f i n i t e continuous non-negative superharmonic f u n c t i o n w i n D such that w <_ w(x) i n D and w(x) > sup w . D\W (Remark: This r e s u l t i s f a l s e i f n = 1 and x belongs to an unbounded component of D .) Proof: Let m = sup G(x,*) where G i s the Green f u n c t i o n of D . D\W ( I f D\W i s empty, l e t m = 0 .) Then m i s f i n i t e , by 5.8 of Helms [ 1 ] . Let w = G(x,«)A(m+l) . Then w <_m + 1 = w(x), and sup w < m + 1 . D\W • - 27 -3.8. Theorem. Let D be a Green r e g i o n i n ]R n, where n >_ 2 . Let E £ D and l e t x e D . Then the f o l l o w i n g are e q u i v a l e n t : a) E i s t h i n at x r e l a t i v e to D . b) There i s an open subset W of D, a non-negative super-harmonic f u n c t i o n g i n D and a constant a such that x e W, g(x) < a, and g >_ a on W n (E\{x}) . Proof: Let F = E\{x} . a) => b) Let u be a non-negative superharmonic f u n c t i o n i n D such that bal(u,E,D)(x) < u(x) . Now Z = F n (bal(u,E,D) < u} i s a p o l a r set so by 2.5, there i s a non-negative superharmonic f u n c t i o n v i n D such that v = °° on Z but g(x) < u(x) where g = v + bal(u,E,D) . Let a be a number s t r i c t l y between g(x) and u(x) and l e t W = {u > a} . Then x e W and g(x) < a . A l s o , since g^_u on F, g > a , on W n F . b) => a) By 3.7, there i s a f i n i t e non-negative superharmonic f u n c t i o n w i n D such that w <_ w(x) i n D and w(x) > sup w . D\W We'll show that bal(w,E,D)(x) < w(x) . Let m = sup w . D\W ( I f D\W i s empty, l e t m = 0 .) Since w(x) > m, there i s a p o s i t i v e r e a l number b such that w(x) - ba >_ m . Let - 28 -v = w(x) + b(g-a) . Then v i s superharmonic i n D . Now v >_ w(x) - ba >_ m, so v >_ 0 i n D and v >_ w on D\W . Also v >_ w(x) on W n F as g - a >_ 0 there. Hence v >_ w on W n F as w(x) >_ w i n D . Thus v >_ w on F . Now {x} i s p o l a r , as n >_ 2 . Thus v >_bal(w,E,D) i n D, by 2.6. But v(x) = w(x) + b(g(x) - a) < w(x) . • 3.9. Theorem: Let E be any subset of IR n and l e t D,D' be Green regions i n E.n . Suppose x £ D n D' . Then E n D i s t h i n at x r e l a t i v e to D i f f E n D' i s t h i n at x r e l a t i v e to D' . Proof: C l e a r l y we need only prove the forward i m p l i c a t i o n . I f n = 1, the r e s u l t f o l l o w s from 3.6. Hence assume n >_ 2 and E n D i s t h i n at x r e l a t i v e to D . Then by 3.8, there i s an open subset W of D, a non-negative superharmonic f u n c t i o n g i n D, and a number a such that x e W, g(x) < a, and g >_ a on W n (E\{x}) . Let G (resp. G 1) be the Green f u n c t i o n of D (resp. D') . Let U be an open r e l a t i v e l y compact neighbourhood of x i n W n D' . Let u be the Riesz measure of g . Then u(U) < 0 0 so we can de f i n e a f i n i t e measure v i n D' by v(A) = y(AnU) f o r A e B o r e l D' . Let g' = G'v . Now i n U, Ag' = -v = Ag . Thus there i s a harmonic f u n c t i o n h i n U such that g' = g + h i n U . Now g'(x) = g(x) + h(x) < a + h(x) . Choose e > 0 such that g'(x) < a + h(x) - e . Let W be an open neighbourhood of x i n U such that h >_ h(x) - e i n W . - 29 -Let a' = a + h(x) - e . Then x e W, g' (x) < a' , and g' >_ a' on W n (E\{x}) . Thus E n D' is thin at x relative to D', by 3.8. • Note that the above theorem implies that thinness is a local property. 3.10. Definition. Let E £ ]Rn and let x e TRU . We shall say E is thin at x i f f there is a Green region D in TRn such that x e D and E is thin at x relative to D . Also, we shall use the following notations: fringe(E) = {x e E: E is thin at x} base(E) = {x e ]Rn: E is not thin at x} . 3.11. Corollary: Let E £ ]Rn . a) If x e ]Rn and E is thin at x then for any Green region D in H n with x e D, E n D is thin at x relative to D • b) For any Green region D in H n , fringe(EnD,D) = D n fringe(E) and base(EnD,D) = D n base(E) . c) fringe(E) i s a polar set and base(E) is a countable intersection of open subsets of E.n . d) If n = 1, fringe(E) is empty and base(E) = E . Proof: a) and b) follow immediately from 3.9. c) If n >_ 3, 3Rn i t s e l f i s a Green region. If n = 1 or 2, ]Rn can be written as the union of two Green regions. Now apply 3.3. - 30 -d) apply 3.6. • 3.12. Let us round out t h i s s e c t i o n w i t h some remarks on the f i n e topology. We don't a c t u a l l y need the f i n e topology i n what f o l l o w s , but i t provides a h e l p f u l p e r s p e c t i v e . Let D be an open subset of ]R n . Then the f i n e topology on D i s the weakest topology i n D which makes a l l superharmonic f u n c t i o n s i n D continuous. I t i s stronger than the usual topology of D, w i t h e q u a l i t y i f f n = 1.. One can show that the f i n e topology on D i s a l s o generated by the f u n c t i o n s of the form Uy|D, where y ranges over measures on 3Rn w i t h compact support contained i n D . I t f o l l o w s that the f i n e topology on D i s equal to the r e s t r i c t i o n to D of the f i n e topology on H n . I f D i s a Green region w i t h Green f u n c t i o n G then the f i n e topology on D i s a l s o generated by the f u n c t i o n s of the form Gy, where y ranges over measures w i t h compact support i n D . I f D i s any open subset of ]R n and S i s any c o l l e c t i o n of superharmonic f u n c t i o n s i n D which i s l a r g e enough to generate the f i n e topology on D, and i f u + v e S f o r a l l u,v e S, then the sets of the form W n {u < c} (W open £ D, u e S , c e H ) c o n s t i t u t e a base f o r the f i n e topology on D . Using t h i s i n co n j u n c t i o n w i t h 3.8, one f i n d s that f o r E £ H n and x e ]R n (where n >_ 2 ) , E i s t h i n a t x i f f x i s not a f i n e l i m i t p o i n t of E . Also i f V £ H n (where n i s a r b i t r a r y ) , V i s f i n e l y open i f f ]Rn\V i s t h i n at each point of V . - 31 -4. THE STRONG FORM OF THE DOMINATION PRINCIPLE In t h i s s e c t i o n we are going to prove the s o - c a l l e d strong form of the domination p r i n c i p l e , f i r s t f o r a Green r e g i o n , and then f o r 2 E. (where we use the l o g a r i t h m i c p o t e n t i a l ) . For the case of a Green 2 re g i o n , t h i s r e s u l t i s due to B r e l o t [ 2 ] . For E , i t may be new. We a l s o give a proof of the domination p r i n c i p l e i n E 1 . Of course t h i s i s q u i t e easy, and the a d j e c t i v e " s t r o n g " i s superfluous i n t h i s case, owing to the f a c t that i n dimension one a l l superharmonic f u n c t i o n s are f i n i t e and continuous. 4.1. Theorem ( B r e l o t ' s strong domination p r i n c i p l e ) Let D be a Green r e g i o n i n E n w i t h Green f u n c t i o n G . Let y be a measure i n D such that Gy i s a p o t e n t i a l . Let E be any subset of D . Then the f o l l o w i n g are eq u i v a l e n t : a) y(D\base(E,D)) = 0 . b) Whenever v i s a non-negative superharmonic f u n c t i o n i n D such that v >_ Gy on E\Z where Z i s a p o l a r s e t , then v >_ Gy throughout D . c) Whenever v i s a p o t e n t i a l i n D such that v >_ Gy on E, then v >_ Gy throughout D . Proof: a) —> b) I f x e base(E,D) then f o r any non-negative superharmonic f u n c t i o n u i n D, - 32 -u ( y ) b a l ( 6 ,E,D)(dy) x bal(u,E,D)(x) (by 2.9) = u(x) u(y)6 (dy), x so b a l ( 6 ,E,D) = 6 x by 2.13. Hence bal(y,E,D) = y by 2.15. Thus bal(Gy.E,D) = Gy . Hence v >_ Gy throughout D by 2.6. b) => c) c) i s j u s t a s p e c i a l case of b) . c) => a) Let f = bal(Gy,E,D) . Then f = Gv where v = bal(y,E,D) Let Z = { f < G y } n E . Then Z i s a p o l a r s e t . Now v(D\base(E,D)) = 0 by 3.4. Suppose y(D\base(E,D)) 4 0 . Then y 4 v, so Gy 4 Gv . As Gv <_ Gy and Gv, Gy are superharmonic, t h i s i m p l i e s that {Gv < Gy} i s not a p o l a r s e t . Hence f o r some x e D\Z, we have Gv(x) < Gy(x) . By 2.5, there i s a p o t e n t i a l g i n D such that g = °° on Z but Gv(x) + g(x) < Gy(x) . Let v = Gv + g . Then v i s a p o t e n t i a l i n D and v >_ u on E, but v(x) < u ( x ) . • 4.2. Theorem. Let D be a Green region i n R n w i t h Green f u n c t i o n G . Let y be a measure i n D such that Gy i s a p o t e n t i a l , and l e t v be a non-negative superharmonic f u n c t i o n i n D w i t h Riesz measure v . Then the f o l l o w i n g are e q u i v a l e n t : a) v >_ Gy almost everywhere w i t h respect to y, and y(Z) £ v(Z) f o r every B o r e l p o l a r set Z £ D . b) v >^  Gy everywhere i n D . - 33 -Proof: a) => b) Let E = {v ^Gu} and l e t P = f r i n g e ( E ) . Then y(D\E) = 0 . Also P i s a p o l a r set (and i s B o r e l by 3.3, s i n c e f r i n g e ( E ) = E\base(E,D)) so y(Z) <_ v(Z) f o r every B o r e l set Z £ P . Let a = Mn\p > 6 = y p . Then 6 <_ v, so there i s a (unique) measure y i n D such that 8 + y = v . Now Gy = Ga + G8 and v = GB + Gy + h where h i s the g r e a t e s t harmonic minorant of v i n D . Let w = Gy + h . Then w >_ Ga on E, except p o s s i b l y on the p o l a r set {GB = 0 0) . A l s o , a(D\(E\P)) = 0 so a(D\base(E,D)) = 0 . Thus w >^  Ga throughout D, by 4.1. Hence v >_ Gy throughout D . • b) => a) Obviously v >_ Gy y - a.e. For the proof that y charges p o l a r sets l e s s than v, look ahead to 10.8. • 2 We now take up the proof of the strong domination p r i n c i p l e i n 3R 4.3. Lemma: Let y be a measure on such that i s a p o t e n t i a l . Then: a) l i m (U^(x) - yCR 2)1>"(x)) = 0 . | | x | |-*» b) I f y has compact support, or i f y(dx) = f ( x ) d x where 2 f i s non-negative and l o c a l l y Lebesgue i n t e g r a b l e on H and tends to zero at i n f i n i t y , then l i m (U M(x) - y ( R 2 ) $ ( x ) ) = 0 . | |x| |-*» - 34 -Proof: a) U y(x) - yOR 2)$ (x) (x-y) - $ (x)du(y) Now i f x > 1 and | |x-y| | >_ 1 then (x-y) - $ (x) = l o § 1 x-y H Thus $ (x-y) - $ (x) —*• 0 as | |x| | —>• «>, f o r each f i x e d y e TR.^ . The d e s i r e d r e s u l t now f o l l o w s from the Lebesgue dominated convergence theorem, once we have e s t a b l i s h e d the f o l l o w i n g c l a i m ( f o r the assumption that U y i s a p o t e n t i a l i s equivalent to the f i n i t e n e s s of l o g + | | x | | d y ( x ) , by 1.6). 2 Claim: For a l l x,y £ H , I $ (x-y) - (x) I ± Y n ( l o g 2 + l o g + l |y| F i r s t note that f o r any r >_ 0, l o g ( l + r ) <_ l o g 2 + l o g + r . (Consider the two cases 0 <_ r <_ 1 and r >^  1 .) F i r s t suppose | | x | | •> 1 and || x-y | | >_ 1 . Then $~(x-y) - $ (x) = j- l o g Thus i f | |x-y| | >_ | |x| | then 1 x-y 1 |x| - 35 -(x-y) - $ (x) 2TT l o g 1 x-y 1 |x | 1 27 l o g 1 _l_ y 1 T X < ^ l o g d + ||y||) < |^ (log 2 + l o g + | | y | | ) w h i l e i f I I x-y I I <_ ||x|| then |$~(x-y) - $ ( x ) | = ^  l o g 1*1 1 x-y | 1 (x-y) + y = o - l°g —V\ —F\ 2TT ° x-y i 2 7 l o § 1 + |y| 1 x-y - 27 l o g ( 1 + I' yl < ^  ( l o g 2 + log +||y||) Now suppose ||x-y|| £ 1 . Then |$"(x-y) - <2>~(x)| = $~(x) l o g + | |x| | <_j^ l o g + ( | |x-y| | + | |y| - 17 l o g ( 1 + I'yl x < 1 . Then <_ ( l o g 2 + l o g + | |y F i n a l l y suppose |$ (x-y) - $ ( x ) | = <3> (x-y) = ^ l o g + | | x - y | | l ^ l o g + ( | | x | | + ||y||) < i ^ l o g d + ||y||) < ^  ( l o g 2 + l o g + | | y | - 36 -The c l a i m i s now e s t a b l i s h e d . b) I f u has compact support K then U y ( x ) = 0 f o r a l l x e 1^ whose d i s t a n c e from K i s a t l e a s t 1 . On the o t h e r hand, i f y(dx) = f ( x ) dx where f s a t i s f i e s the s t a t e d c o n d i t i o n s then c l e a r l y U y ( x ) —>- 0 as | |x| | —*- » . Now U y = U y - U y and $ = $ + - $ , and $ + ( x ) = 0 f o r | |x| | >_ 1 . Hence U y ( x ) - yQR )$(x) —• 0 as 4.4. C o r o l l a r y : L e t y be a measure on such t h a t U y i s • a p o t e n t i a l . Then l i m i n f ( U y ( x ) - yQR )*(x)) > 0 . The p r o o f i s c l e a r . 4.5. Lemma: L e t y be a measure on ]R 2 such t h a t U y i s a p o t e n t i a l . F o r each p o s i t i v e r e a l number r l e t B denote the 2 open b a l l of r a d i u s r c e n t r e d a t 0 i n ~R and l e t u = » r y' = U 7 • Then: K\B r 2 a) F o r a l l x e H and a l l p o s i t i v e r e a l numbers r , u ^ O O > - ^ y O R 2 \ B r ) i o g ( l + -LiiLL ) + $ ( y ) d y ( y ) | y| |>r b) F o r a l l p o s i t i v e r e a l numbers e and a l l p o s i t i v e i n t e g e r s k, t h e r e e x i s t s a p o s i t i v e r e a l number r ^ such t h a t f o r y y r r0 — r < °° w e n a v e U + e >_ U on B^ r . - 3 7 -P r o o f : a) C o n s i d e r any r i n (0,°°) . Then f o r any x i n R and any y i n E 2 \ B , r x-y j | < — r y + M 1 + so $(x-y) = - 27 l o g j |x-yI i " 27 log 1 + x Thus f o r a l l x i n TR , y r U r ( x ) = $ ( x - y ) d y ( y ) E 2 \ B > - ^ y O R 2 \ B r ) l o g 1 + + $ ( y ) d y ( y ) y !>r b) F o r any x i n TR and r i n (O, 0 0), y y' U y ( x ) = U r ( x ) + U r ( x ) > U r ( x ) - y O R 2 \ B r ) l o j 1 + 1_ 2IT y >r l o g | | y | | d y ( y ) , by a ) . Now choose r ^ i n ( l , 0 0 ) so t h a t ^yOR 2\B )iog(i+k) + ^ log||yI|dy(y) y | | > r r The c o n c l u s i o n of b) then f o l l o w s . • - 38 -2 4.6. Lemma: Let K be a compact non-polar subset of ]R . Then 2 there i s a non-zero measure A on R which i s supported by K, A 2 such that U i s bounded above on TR 2 Proof: Let D be a Green region i n R c o n t a i n i n g K and l e t G be the Green f u n c t i o n of D . (We can take D to be an open b a l l contain-ing K, f o r instance.) Let u = bal(l,K,D) . Then u i s a p o t e n t i a l i n D . Let A be the Riesz measure of u . Then A i s supported by 2 K . We s h a l l denote the obvious extension of A to a measure on R by the same l e t t e r A . Now there i s a harmonic f u n c t i o n h i n D such that = u + h i n D . Since u i s bounded i n D and h, by c o n t i n u i t y , i s bounded i n a neighbourhood of K, i s bounded i n a A 2 neighbourhood of K . A l s o , U i s continuous on ]R \K . I t now A 2 f o l l o w s from 4.3(b) that U i s bounded above on R 2 Now here i s the strong domination p r i n c i p l e f o r R 4.7. Theorem: Let u be a non-zero measure on R 2 such that U y 2 i s a p o t e n t i a l . Let E be a subset of ]R and l e t v be a super-2 harmonic f u n c t i o n on R such t h a t : 2 a) l i m i n f (v(x) - pQR )*(x)) > -« b) v >_ U y on E\P where P i s a po l a r set c) uQR 2\base(E)) = 0 . Then v > U P on a l l of R 2 . Proof: The proof proceeds by reducing to the case of a Green r e g i o n . For each r > 0 l e t denote the open b a l l of rad i u s r centred 2 at 0 i n ]R and l e t u denote u_ . Choose e > 0 . By r B r 4.5(b), there e x i s t s r ^ i n (0,°°) such that f o r a l l r i n [ r ^ , 0 0 ) - 39 -u _ U y + -| > U r on B . 2 — r 2 Next, as u ^ 0 but yQR \base(E)) = 0, base(E) ± 0 . Hence E i s not a p o l a r s e t . (Combine 2.6 and 3.11). Now base(E) contains E\fringe(E) and, by 3.11, f r i n g e ( E ) i s p o l a r . Thus base(E) i s not p o l a r . Hence there e x i s t s r ^ i n [ r ^ , 0 0 ) such that base(E) n B i s not p o l a r . Now base(E) n B i s a n a l y t i c ; i n f a c t r l r i i t i s a countable i n t e r s e c t i o n of open s e t s , by 3.11. Thus there i s a compact non-polar set K contained i n base(E) n B , by 6.23, r l 7.32, and 7.33 of Helms [ 1 ] . By 4.6, there i s a non-zero measure X 2 X e 2 on TR. supported by K such that U <_ y on TR . Choose a i n (0,1] . Then U a X = aU A <_ -| on TR2 . Thus f o r a l l r i n [r 0,«0, y _ U y + e >• U r ' a on B , — r where y = y + aA . As a,A. ^  0, there e x i s t s r„ i n [ r , ,°°) r,a r 2 1' such that f o r a l l r i n [ r ^ , 0 0 ) , y r a0R 2) > yOR 2) • Then f o r any r i n t ^ , 0 0 ) , 2 l i m i n f (v(x) - y QR ) * ( x ) ) = +°° . i i i i r ,a I l x l Choose r , i n [r9,°°) . Let y = M • Then there e x i s t s r , i n 2 [r„,°°) such that on 1R \B we have J r4 2 v > y(]R )$ and yOR2)* + e > U Y where the second estimate f o l l o w s from 4.3(b), since y has compact - 40 -support. Now choose r,. i n (r^ ,°°) and consider the Green region B , which we s h a l l denote by B . Let r 5 h = U Y on 8B PI(U Y;B) i n B (Note that the support of y i s a compact subset of B .) Then h i s continuous on B, and h|B i s the greatest harmonic minorant of U i n B . Now v + e >_ h on 8B and v + E - h i s lower semi-continuous on B and superharmonic i n B . Hence v + e - h i s a non-negative superharmonic f u n c t i o n i n B . Now v + £ - h ^ U y + e - h on (EnB)\P and U y + e - h >_ U Y - h on B . Thus v + e - h ^ U T - h on (EnB ) \P . Now base(EnB ,B) r 3 r 3 r 3 contains base(E) n B by 3.11. Hence y(B\base(EnB ,B)) = 0 . r 3 r 3 Y Y Therefore v + e - h > _ U - h throughout B by 4.1, since U - h Y i s the Green p o t e n t i a l of y r e l a t i v e to B . Thus v + e >_U i n B . As t h i s i s true f o r a l l r,. i n [r^,°°), v + e >_ U Y on ]R2. Now l e t t i n g r ^ go to <*>, we o b t a i n v + e >_ u y + a ^ on H 2 . Next, l e t t i n g a decrease to zero we f i n d that v + e >_ U y on ]R2 . As e > 0 was a r b i t r a r y , i t f o l l o w s that v >_ U y on H 2 , and we are done. • 4.8. Theorem: Let u,v be measures on H 2 such that U y, U V are p o t e n t i a l s and u 4 v . Let c be a r e a l number. Then a) and b) below are e q u i v a l e n t . - 41 -a) U V + c >_ U y on K b) i ) U V + c >_ U y almost everywhere with respect to y 2 i i ) y(Z) <_ v(Z) f o r a l l B o r e l p o l a r subsets of H 2 2 i i i ) V0R ) £ y(R ) . Proof: a) => b) i ) i s obvious. The proof of i i ) w i l l be def e r r e d . 2 See 10.8. i i i ) For each x i n H l e t y be the uniform u n i t x 2 i i d i s t r i b u t i o n on (y e E : j |y-x|| = 1} . Then U V + c dy x Y U X dv + c = U V(x) + c and Thus V1 d y x = U y(x) + c U V + c > U y 2 2 Hence vQR ) £ yOR ) • This f o l l o w s from 4.3(a) .. b) = > a) Let E = {U V + c >_ U y} and l e t P = f r i n g e ( E ) . Then P i s a B o r e l p o l a r set by 3.11(c). Thus y(Z) <_ v(Z) f o r a l l B o r e l sets contained i n P . Let 8 = y p , a = y - B , y = v - & . Then 2 ct 6 y a,B,Y are ( p o s i t i v e ) measures on H , U , U , U are p o t e n t i a l s , 2 Y a aQR \base(E)) = 0, and U + c >_ U on E except p o s s i b l y on the B 2 2 p o l a r set {U = °°} . Al s o yOR) <_aCJR ) so l i m i n f ( ( U Y ( x ) + c) - a ( R 2 ) $ ( x ) ) >_ c, I l x | |-^°° - 42 -by 4.4. Thus, i f a ^ 0, we may conclude that U + c >_U on 2 6 R by applying 4.7. Then by adding U to both sides of t h i s v y 2 i n e q u a l i t y we o b t a i n U + c >_ U on R On the other hand, suppose a i s zero. Then y = B <_ v but 2 2 v 0 R ) < y Q R ) so y = v . But we are assuming y f v . • 4.9. C o r o l l a r y . Let y be a measure on R 2 such that U y i s a p o t e n t i a l and y ^ 0 . Let c be a r e a l number. Suppose U y <_ c y - a.e. Then U y < c everywhere. Proof: Take v = 0 i n 4.8. • F i n a l l y we g i v e a proof of the domination p r i n c i p l e f o r R F i r s t of a l l , r e c a l l (1.5) that i f y i s a measure on R then U y i s a p o t e n t i a l i f f x dy(x) i s f i n i t e . Now we prove a lemma. 4.10. Lemma. Let y be a measure on R such that U y i s a p o t e n t i a l . Let 5 be the centre of mass of y . ( I f y = 0 j u s t l e t £ be any r e a l number.) Consider the f u n c t i o n f = y(R)$(£>*) - U y . Then f >_ 0, f i s i n c r e a s i n g on f i s decreasing on [£,°°), and f (x) —• 0 as [x [ —^ - «> . Proof: By the choice of £, y(IR)5 (-°°,C] . Then y dy(y) . Suppose x i s i n f (x) = - j yOR)(?-x) + \ 1 2 x-y|dy(y) x - y - |x-y|dy(y) ( x - y ) l ( _ T O ) X ] ( y ) d y ( y ) - 43 -Hence f i s non-negative and i n c r e a s i n g on . Also f (x) —>- 0 as x —• by the Lebesgue dominated convergence theorem. The i n t e r v a l i s t r e a t e d s i m i l a r l y . • 4.11. C o r o l l a r y . Let y,v be measures on TR such that U y, U V are p o t e n t i a l s , and l e t c be a r e a l number. Suppose U V + c > U y . Then: a) vCJR) <_ uQR) b) i f vQR) = uQR) then c >_ 0 and y and v have the same centre of mass. Proof: By 4.10, l i m (U y(x) - y(IR)$(C,x)) = 0 | x | -**> and l i m (U V(x) - vOR)$(n,x)) = 0 |x|-**> where E, (resp. n) i s the centre of mass of y (resp. v) . The c o r o l l a r y f o l l o w s immediately from t h i s . • Now here i s the domination p r i n c i p l e f o r ]R . 4.12. Theorem. Let y be a non-zero measure on E. such that U y i s a p o t e n t i a l and l e t v be a superharmonic f u n c t i o n on R such that - 44 -a) l i m i n f (v(x) - y ( E ) $ ( x ) ) > -» | x | -*» b) v > U y almost everywhere w i t h r e s p e c t to y Then v > U y everywhere on TR . P r o o f : A f u n c t i o n on TR i s superharmonic i f f i t i s f i n i t e and concave. Note t h a t such a f u n c t i o n i s a u t o m a t i c a l l y c o n t i n u o u s . L e t E = {v ^ U y } and l e t W = H\E . Then W i s open. A l s o y(W) = 0 so U y i s harmonic i n W . S i n c e we a r e i n d i m e n s i o n one, t h i s j u s t amounts to s a y i n g t h a t on each component of W, the graph of U y i s a s t r a i g h t l i n e . By the c o n t i n u i t y of U y , t h i s a c t u a l l y h o l d s on the c l o s u r e of each component of W . L e t p e W and l e t C be the component of W c o n t a i n i n g p . Then C = (a,b) where a e {-°°} u E, b e E u {<*>}, and a < p < b . A l s o E 4 0 s i n c e y 4 0, so a t l e a s t one of a and b i s f i n i t e Case 1. a and b b o t h f i n i t e . Then v ( a ) >_U y(a) and v ( b ) >^U y(b) . A l s o v i s concave, and U y i s a s t r a i g h t - l i n e f u n c t i o n on [a,b] . Hence v >_ U y on [a,b] . In p a r t i c u l a r , v ( p ) >. U P ( p ) . Case 2. a = b f i n i t e . By a) and 4.10, l i m i n f ( v ( x ) - U y ( x ) > -°° . x->-°° Hence v - U y i s bounded below on (-ro,b] . For each x e (-°°,p] , t h e r e i s a unique number c ( x ) e [0,1] such t h a t p = (1 - c ( x ) ) x + c ( x ) b - 45 -As x —>• -<*>, c ( x ) —>- 1 . Now v(p) 1 (1 - c ( x ) ) v ( x ) + c ( x ) v ( b ) 1 (1 - c ( x ) ) ( v ( x ) - U U ( x ) ) + U y ( p ) , s i n c e U y i s a s t r a i g h t - l i n e f u n c t i o n on (-°°,b] and v ( b ) _> U y ( b ) . L e t t i n g x go to we o b t a i n v ( p ) > D y ( p ) . Case 3. a f i n i t e , b = 0 0 . T h i s i s s i m i l a r to case 2 . • 4.13. C o r o l l a r y . L e t u,v be measures on TR such t h a t U y , U*V a r e p o t e n t i a l s , and l e t c be a r e a l number. Assume a l s o t h a t P ^ 0 . Then a) and b) below a r e e q u i v a l e n t . a) U V + c >_ U P on a l l of TR b) vQR) <_ uQR), and U V + c >_ U y almost everywhere w i t h r e s p e c t t o u . P r o o f : a) ==> b) A p p l y 4.11(a). b) = > a) By 4.10, we can take v t o be U V + c i n 4.12. • - 46 -5. BROWNIAN MOTION PRELIMINARIES In t h i s s e c t i o n we s h a l l l a y down the n o t a t i o n and terminology that we s h a l l need f o r Brownian motion. Due to the l a r g e number of l e t t e r s used up by t h i s n o t a t i o n , we reserve the r i g h t to use these l e t t e r s f o r other purposes l a t e r on. However, whenever they are used without e x p l a n a t i o n t h e i r meanings w i l l be as defined i n t h i s s e c t i o n . 5.1. For 0 < t < °°, p t w i l l denote the f u n c t i o n on ]R n defined by , s f 1 *n/2 -|| x | | 2 / 4 t Then: a) p s a t i s f i e s the heat equation: 8 7 P t = A p t ,n b) p i s a p r o b a b i l i t y d e n s i t y on ]R w i t h mean 0 and varian c e 2nt: p t > 0 P t ( x ) d x = 1 xp t ( x ) d x = 0 x| p t ( x ) d x 2nt c) ( p t) i s a con v o l u t i o n semigroup: p g ( x - y ) p t ( y ) d y - 47 -5.2. The p o i n t a t i n f i n i t y f o r H n w i l l be denoted by 9, and ]R n o w i l l denote the space ]R n u {3} where 8 i s c o n s i d e r e d as an i s o l a t e d p o i n t . 5.3. F o r 0 < t < °°, P w i l l denote t h e f u n c t i o n on H n x B o r e l t 0 0 d e f i n e d by P t ( x , A ) = \ P t ( y - x ) d y i f x 4 3 AfHRn 1 A ( 3 ) i f x = 3 Then (P ) i s a t e m p o r a l l y homogeneous Markov t r a n s i t i o n f u n c t i o n on . P (x,A) s h o u l d be thought of as the p r o b a b i l i t y t h a t i f o t a p a r t i c l e s t a r t s a t x a t a c e r t a i n time and moves a c c o r d i n g to Brownian motion, i t w i l l be found i n the s e t A t u n i t s of time l a t e r . Note t h a t a p a r t i c l e s t a r t i n g a t 3 does not go anywhere. Now suppose 0 <_ s < t ^ < ... < t ^ and A^,. . . ,A^ e B o r e l A l s o l e t x e ]R n . We wish t o d e f i n e P (x,A ,...,A, ), O S y t ^ , • • . , t ^ 1 K which i s to be thought of as the p r o b a b i l i t y t h a t i f a p a r t i c l e s t a r t s a t x a t time s, i t w i l l be i n A. a t time t . f o r 1 x i = l , . . . , k . W e l l , c l e a r l y we sho u l d have ^ t / ^ V • p V s ( x ' V and P s . j_ (x,A ,. . . ,A. ) , t ^ , . . . , J-P (x,da )P (a ,A ,...,A.) ' 1 1' 2' k A l - 48 -so by r e c u r s i o n we are lead to de f i n e V V P (x,da ) A ' 1 P (a da ) A 2 C1' C2 1 P t - t ( a k - l ' d 5.4. We d e f i n e ft to be the set of a l l f u n c t i o n s u>: [0,°°] —*• E„ such t h a t : i s continuous on [0,°°) a) (« ) = 3 (Then i f co(t) = 3 f o r some t e [0,°°), io i s i d e n t i c a l l y equal to 3 .) We w r i t e CJ f o r the element of P. which i s i d e n t i c a l l y equal 3 to 3 . For each t e [0,°°] we de f i n e B : fi-*lR3 by Bfc(u)) = u ( t ) , and we l e t be the a - f i e l d on Q generated by {B g: 0 <_ s <_ t} We a l s o w r i t e 8^  f o r 8^  . CO I f y i s any a-f i n i t e measure on B o r e l R^ then there i s a o unique B^-outer measure P y on ft such that i f Ap,A^,...,A^ are i n B o r e l E ^ and 0 < t n < ... < t. then 3 l k P y ( B Q e A Q, B e A^-.-.B , V w ( d x ) P 0 ; t 1 , . . . , t . ( s t » A r - " » A k ) • •v 1 k (In p a r t i c u l a r , t a k i n g A^ = ... = A^ = R^ we have P M ( B Q e A Q) = y - 49 -P P i s c a l l e d the Wiener measure on Q w i t h i n i t i a l measure y . E y w i l l denote the i n t e g r a l w i t h r e s p e c t t o P y . C u s t o m a r i l y , one t a k e s y t o be a p r o b a b i l i t y measure, i n which case P y i s a l s o a p r o b a b i l i t y measure. However, we s h a l l have o c c a s i o n t o c o n s i d e r y's which a r e n o t p r o b a b i l i t y measures, and which i n some c a s e s a r e even i n f i n i t e . P y can a l s o be d e f i n e d as the unique 8^-outer measure on Q, such t h a t f o r A e B o r e l ]R n and 0 < s < t < ° ° we have P y ( B Q e A) = y(A) P y ( B t e A | B ° ) = P s ; t ( B s , A ) The second e q u a l i t y h e r e i s known as the Markov p r o p e r t y . I f y = S then P y i s a l s o denoted by P X . I f H e B° then x x \—y P X ( H ) i s a B o r e l f u n c t i o n on and P y ( H ) = y ( d x ) P X ( H ) f o r any y . F o r any t e [0,°°] we d e f i n e 8^ t o be the a - f i e l d on Q g e n e r a t e d by ^ B s : t £. s and we d e f i n e 0 t: —>• by ( 0 t c j ) ( s ) = u)(s+t) . The 9{.'s a r e c a l l e d t r a n s l a t i o n o p e r a t o r s . S i n c e B • B = B , , 6„ i s ( ^ B , B^)-measurable. Now here i s s t s+t t t another v e r s i o n o f the Markov p r o p e r t y : F o r any n o n - n e g a t i v e B -measurable f u n c t i o n <j> on (and any o - f i n i t e measure y on B o r e l ]Rg) and f o r 0 <_ t < °°, we have B. Ey(4> • e t |B°). = E - 50 -Now we i n t r o d u c e the a - f i e l d s By, By, B, and . We take By to be the c - f i e l d of a l l P y - m e a s u r a b l e s u b s e t s o f ft . S i n c e P y i s o - f i n i t e on B^  ( i n d e e d , on Bg) and o u t e r r e g u l a r w i t h r e s p e c t to B°, By = { H e f t ; L c H c M and P U(M\L) = 0 f o r some L,M e B°}, Next, we l e t By = { H e By: P y(HAF) = 0 f o r some F i n B°}, F i n a l l y we l e t 1 . . 8 " y where the i n t e r s e c t i o n s r u n over a l l a - f i n i t e measures y on B o r e l . (We would get the same t h i n g i f we j u s t l e t t h e i n t e r -s e c t i o n s r u n over p r o b a b i l i t y measures.) These a - f i e l d s a r e a l l c l o s e d under the S o u s l i n o p e r a t i o n , and B y + = By, B = B^  . For any A c we l e t o D A = i n f { t >_ 0: B t e A} T A = i n f { t > 0: B t e A} . i s c a l l e d the debut o f A and T^ i s c a l l e d the h i t t i n g time of A . I f A i s a n a l y t i c (or more p a r t i c u l a r l y , B o r e l ) then and T • a r e (B ) - s t o p p i n g t i m e s . T h i s i s the r e a s o n f o r i n t r o d u c i n g the completed a - f i e l d s . I f we d i d not do t h i s we would have D , T A A - 51 -b e i n g (8^)-stopping times f o r A c l o s e d , and ( B ^ + ) - s t o p p i n g times f o r A open, but f o r A m e r e l y B o r e l , we would be a t a l o s s . F o r each t , and each f i n i t e y, the map 8 i s ( y B,8 V)-measurable, where y8 = {H £ Q: L c H c M f o r some L,M e °8 w i t h P P(M\L) = 0} and v = yP^ . I t f o l l o w s t h a t 6fc i s (8,8)-measurable. I f $ i s u x a n o n - n e g a t i v e 8 -measurable f u n c t i o n on Q, then (j) i s P -measurable f o r y - a.a. x and x t — E (<j>) i s measurable w i t h r e s p e c t to the c o m p l e t i o n of y, and E % ) = y(dx)E X((j.) . I f $ i s B-measurable, then x i — * • E (tj>) i s u n i v e r s a l l y measurable. In terms of B and B t > the Markov p r o p e r t y can be s t a t e d as f o l l o w s : F o r any n o n - n e g a t i v e B-measurable f u n c t i o n on fi, any a - f i n i t e measure y on B o r e l E.^, and any t e [0,°°) a Ey(<|) • 8 t |B t ) = E t(<f>) The system (ti,B,B^,B^,B) w i l l be c a l l e d s t a n d a r d Brownian  motion ( i n n d i m e n s i o n s ) . 5.5. Consider a system (A,M,Mt,Xt,P) where: A i s a set M i s a o-field of subsets of A (M )_ ^  i s an increasing family of sub-o-fields of M t 0<t<°° - 52 -f o r each t e [0,°°], X t i s an M t~measurable map of A i n t o f o r each co e A, t 1—>• X t(to) i s c o n t i n u o u s on [0,°°) X^ i s i d e n t i c a l l y e q u a l t o 9 P i s a measure on M which i s a - f i n i t e on MQ i f A e B o r e l and 0 < s < t < 0 0 then 0 P ( X t e A|M8) = P s ; t ( X s , A ) . Such a system w i l l be c a l l e d a g e n e r a l i z e d Brownian motion p r o c e s s (GBMP f o r s h o r t ) . I f i n a d d i t i o n P ( A ) = 1, we s h a l l s i m p l y say t h a t the system i s a Brownian motion p r o c e s s (BMP f o r s h o r t ) . I f (A,M,M t,X t,P) i s a GBMP and T i s an ( M F C + ) - s t o p p i n g time then ( A , M , M T + t + , X T + t , P ) i s a l s o a GBMP. T h i s i s known as the s t r o n g Markov p r o p e r t y . For s t a n d a r d Brownian motion, a v e r y u s e f u l v e r s i o n of the s t r o n g Markov p r o p e r t y i s : i f <f> i s a n o n - n e g a t i v e B-measurable f u n c t i o n on Q, T i s a (8 t)-stopping time and u i s n 9 a a - f i n i t e measure on B o r e l JR™ t h e n B„ Ey(<j> . eT|BT) = E T(<f>) . (Note t h a t 6 T i s (B,B)-measurable and B^ i s measurable from B to the u n i v e r s a l c o m p l e t i o n of B o r e l .) o Suppose (A,M,M t,X t,P) i s a GBMP . L e t u be the " d i s t r i b u t i o n " of X Q : u(A) = P ( X n e A) f o r A e B o r e l . - 53 -Suppose y i s o - f i n i t e . For each w e A, the map t *• X t(uj) i s an element of ft . Thus we have a n a t u r a l map of A i n t o ft; l e t us c a l l i t 4> . For each H e 8°, i^ - 1[H] e M and P ( * _ 1 [ H ] ) = P y(H) . Al s o B • IJJ = X . Thus we may regard (A,M,M ,X t,P) as an "enlargement" of (ft,B°,8°,Bt,Py) . In t h i s manner we can prove things about GBMP's by f i r s t proving them about standard Brownian motion, where we can use t r a n s l a t i o n operators and v a r i o u s other aspects of the s t r u c t u r e of standard Brownian motion which give us more "handles" f o r computations. 5.6. Theorem. Let u e C 2 ( R n ) and suppose u and a l l i t s f i r s t two p a r t i a l s are bounded on E. N . Let y be a p r o b a b i l i t y measure t on R n Au(B )ds . Then s For 0 <_ t < «>, l e t M = u(B t) -(M t) i s a martingale over (ft,8,B t,P y) . Sketch of Proof: Using 5.1(a) and i n t e g r a t i n g by p a r t s , one f i n d s that E y(M t) = udy f o r a l l t . Now apply the Markov property. • 5.7. C o r o l l a r y : Let u e C OR ) and l e t y be a p r o b a b i l i t y rt .n ,,2^n ry: t (.n measure on TR Suppose that f o r a l l t e (0,°°), we have For 0 <_ t < °°, l e t M = u(B t) - Au(B )ds s TR E u(x) p t ( x ) d y ( a ) d x < 0 0 and rt 0 ^  n J v(x)p (x)dy(a)dx < °°, E E - 54 -where v = |u|, | |grad u||, and |Au | , r e s p e c t i v e l y . Then (Mfc) i s a m a r t i n g a l e over ( f i , B , 8 t , P y ) . 2 S k e t c h o f P r o o f : Cut u by C f u n c t i o n s w i t h compact s u p p o r t s , a p p l y 5.6 to the c u t f u n c t i o n s , and take a l i m i t . • 5.8. C o r o l l a r y : L e t u e C 20R n) . F o r 0 <_ t < °°, l e t ft M t = u ( B t ) - A u ( B g ) d s . Suppose t h e r e a r e numbers a,B e [0,°°) and y e [1,2) such t h a t II I I Y v ( x ) <_ a e 3 ' ' X ' I f o r a l l x e ]R n, where v = |u|, ||grad u||, and |Au | , r e s p e c t i v e l y . Then (M t) i s a m a r t i n g a l e over (f2,B,8t ,P a) , f o r any a e E. n . 5.9. Theorem: L e t a e R n and l e t T be a ( 8 t ) - s t o p p i n g time s a t i s f y i n g P 3 ( T < °°) = 1 . Then: a) E 3 ( | | B T - a | | 2 ) < 2 n E a ( T ) . b) E a ( T ) i s f i n i t e i f f tf = { | | B _ - a | | 2 : 0 < t < »} i s P - u n i f o r m l y i n t e g r a b l e . I n t h i s c a s e , we have e q u a l i t y i n a ) . P r o o f : A l l m a r t i n g a l e s w i l l be over (Q,B,E^,Pa) . L e t u(x) = ||x - a | | 2 , f o r x e ]R n . Then Au = 2n, so ( u ( B t ) - 2nt) i s a m a r t i n g a l e , by 5.8. Hence E a ( | | B g - a|| 2) = 2 n E a ( S ) f o r any bounded ( B t ) - s t o p p i n g time S, by the o p t i o n a l sampling theorem. P a r t a) now f o l l o w s from F a t o u ' s lemma. Moreover, i f H i s u n i f o r m l y - 55 -i n t e g r a b l e then t a k i n g S = T A t and l e t t i n g t go to i n f i n i t y , a i i i 12 a we f i n d t h a t E ( | | B^ - a | | ) = 2nE (T) < °° . C o n v e r s e l y , suppose E a ( T ) < « . Then E a ( | | B r p t - a l l 2 ) < E 3 ( T ) f o r a l l t , so 2 { | | B ^ A - a||: 0 <_ t < °°} i s L -bounded and hence u n i f o r m l y i n t e g r a b l e . Now each component of ( B t - a) i s a m a r t i n g a l e , by 5.8. Thus B T A t - a = E a ( B T - a|5 T A t) f o r a l l t , by the o p t i o n a l sampling theorem. But then f o r a l l t , l|B T, t-a|| 2<EM|B T-a|| 2|B T A t), by Jensen's i n e q u a l i t y . Thus H i s u n i f o r m l y i n t e g r a b l e . • 5.10. C o r o l l a r y : L e t y be a p r o b a b i l i t y measure on H n and l e t T be a (B t)-stopping time s a t i s f y i n g P y ( T < °°) = 1 . Then: a) E y ( | | B T - B Q | | 2 ) < 2 n E y ( T ) . b) E V ( T ) i s f i n i t e i f f H = { | | B - B | | : 0 <_ t < »} i s P y - u n i f o r m l y i n t e g r a b l e . In t h i s c a s e, we have e q u a l i t y i n a ) . (Note t h a t 5.10 h o l d s even i f y does not have f i n i t e v a r i a n c e . ) 5.11. C o r o l l a r y : L e t y be a p r o b a b i l i t y measure on ]R n and l e t A be a B o r e l s u b s e t of ]R n . L e t T = i n f { t ^ 0 : B t i A} . Then E y ( T ) < ^ - ( d i a m e t e r ( A ) ) 2 . — zn 5.12. Dynkin's f o r m u l a : L e t D be an open s u b s e t o f E. N and l e t 2 u be bounded and c o n t i n u o u s on D, and C i n D, w i t h Au bounded i n D . L e t R = i n f { t >_ 0: B f c I D}, l e t a e D, and - 56 -l e t T be a ( B t ) - s t o p p i n g time such t h a t T <_ R and E (T) < Then (Note t h a t i f D i s bounded then E d ( T ) < » f o l l o w s from T <_R .) T h i s i s e s s e n t i a l l y Theorem 2, paragraph 1, c h a p t e r 4 of Rao [ 1 ] . We p o i n t out t h a t the p r o o f Rao g i v e s i s c o r r e c t f o r a e D, but s e r i o u s l y f l a w e d f o r a e 9D . However, t h i s d i f f i c u l t y can be overcome. - 57 -6. PRELIMINARIES ON BROWNIAN MOTION AND POTENTIAL THEORY In t h i s s e c t i o n we s t a t e the r e s u l t s that we s h a l l need on the connections between Brownian motion and p o t e n t i a l theory. These r e s u l t s are a l l well-known, but i t i s a l i t t l e d i f f i c u l t to give convenient references f o r them. For the general theory of Markov processes and p o t e n t i a l theory, the reader may consult the fundamental papers of Hunt [ 1 ] , [ 2 ] , [ 3 ] , or the books of Meyer [2] or Blumenthal and Getoor [ 1 ] . In these works, the p o t e n t i a l theory i s defined i n terms of the process, and the connection between Brownian motion and c l a s s i c a l p o t e n t i a l theory i s mentioned, but not proved. For the theory of Brownian motion and c l a s s i c a l p o t e n t i a l theory, the l e c t u r e notes of Rao [1] are e x c e l l e n t , but some r e s u l t s which we need f o r B o r e l sets are proved only f o r compact s e t s . Throughout t h i s s e c t i o n , D i s an open subset of ]R n and R = i n f { t >_ 0: B I D} . 6.1. Theorem: Let u be a non-negative superharmonic f u n c t i o n i n D . Let S,T be (B t ) - s t o p p i n g times, w i t h S <_ T . Then f o r any x e D, u(x) l E X ( u ( B s ) l { s < R } ) > E X ( u ( B T ) l { T < R } ) . Thus i f u(x) i s f i n i t e then (u(B ) 1 , ,) i s a supermartingale t t t < K . r over ( f t , B , ( 8 t ) , P x ) . 6.2. Theorem: Let u be superharmonic i n D . Let ft^ = { CJ e ft: t *—> u(B t(w)) i s continuous on [0,R(<JJ)) and f i n i t e on (0,R(co))} . - 58 -Then: a) e S o u s l i n B ^ ; hence e B . b) For any x e D, P x ( f i \ n ) = 0 . (This i s due to Doob [1].) ( I f x e H^\D then {B = x} c {R = 0} so b) i s a c t u a l l y t r u e f o r a l l x e . ) 6.3. Theorem: Let A be an a n a l y t i c subset of lR n and l e t u be a non-negative superharmonic f u n c t i o n i n D . Then f o r a l l x e D, b a l ( u , A n D, D)(x) = E X ( u ( B T ) l { T < R } ) , where T = T . A 6.4. Theorem: Let A be an a n a l y t i c subset of H n , and l e t T = T A . Then f o r each x e ]R n, A i s t h i n at x i f f P X(T > 0) = 1. Thus base(A) = {x e ]R n: P X(T > 0) = 0} and f r i n g e ( A ) = {x e A: P X ( T > 0) = 1} . A l s o , we have P X ( B T i base(A), T < °°) = 0 f o r a l l x e ]R n . 6.5. Theorem: Let A be an a n a l y t i c subset of ]R n and l e t T = T.. A Then the f o l l o w i n g are e q u i v a l e n t : a) A i s p o l a r . b) P X(T < °°) = 0 f o r at l e a s t one x e ]R n . c) P X(T < ») = 0 f o r a l l x e ]R n . Moreover, i f n = 1 or 2 then e i t h e r P (T < <») = 0 f o r a l l x e H n , or P X(T < ») = 1 f o r a l l x e IR n . - 59 -6.6. C o r o l l a r y : Suppose n = 1 or 2 . Then the f o l l o w i n g a r e e q u i v a l e n t : a) D i s a Green r e g i o n . b) P X ( R < °°) = 1 f o r some x e D, or D i s empty. c) P X ( R < °°) f o r a l l x e D . 6.7. Theorem: L e t E be any su b s e t o f TRn, and l e t x e H n . Then the f o l l o w i n g a r e e q u i v a l e n t : a) E i s t h i n a t x . b) There i s a G s e t H c ]R n w i t h E £ H and P X ( T U > 0) = o H c) There i s a B o r e l s e t Y £ ]R n w i t h E £ Y and P X ( T y > 0) d) There i s an a n a l y t i c s e t A £ R n w i t h E £ A and P X ( T A > 0) = 1 . A 6.8. P r o p o s i t i o n : Suppose n >_ 3 . L e t ft = {_ e ft: l i m ] |B (to) | | = °°} . Then ft e B ° , and P X ( f t 1 ) = 1 f o r any x e ]R n . ( T h i s may be deduced from 6.3 w i t h D = ]R n, u s i n g the s t r o n g Markov p r o p e r t y and the f a c t t h a t the p o t e n t i a l of a measure w i t h compact support goes to z e r o a t i n f i n i t y . ) 6.9. Theorem: Suppose D i s a Green r e g i o n , and G i s the Green f unct i o n o f D . Then f o r any x e D and any A & B o r e l D, f G(x,y)dy = A P X ( B e A, t < R)dt 0 - 60 -Sketch of Proof: I t ' s enough to prove the theorem w i t h 1 replaced ________________________ CO by <f>, where <j> i s a non-negative C f u n c t i o n w i t h compact support i n D . Let u = G<f> . Then u(x) —>• 0 as x —>• z e 3D, except f o r a p o l a r set of z's, by 8.31 of Helms [ 1 ] . A l s o , i f n >_ 3 then u(x) —»- 0 as | |x| | —>- » . Thus by 6.5, 6.6, and 6.8, u(B t) —> 0 as t + R, P X - a.s. f o r any x e D . Now approximate D by an i n c r e a s i n g sequence of r e l a t i v e l y compact open subsets and apply Dynkin's formula 5.12. • - 61 -7. EMBEDDING MEASURES IN BROWNIAN MOTION IN A GREEN REGION, USING  NON-RANDOMIZED STOPPING TIMES. 7.1. Throughout t h i s s e c t i o n D i s a Green region i n ]R n, G i s the Green f u n c t i o n of D, and R = i n f { t >_ 0: Bfc I D} . 7.2. I f y i s a measure i n D and T i s a (8^)-stopping time then l e t y^ be the measure on B o r e l D defined by y T ( A ) = P y ( B T e A, T < R) . The measure y T i s thus obtained by l e t t i n g y d i f f u s e under Brownian motion up to time T, where only Brownian paths which stay i n D f o r the whole i n t e r v a l of time [0,T] c o n t r i b u t e to y^ . Given y, i t i s n a t u r a l to ask what measures v can be w r i t t e n i n the form v = y^ . I f T i s allowed to be "randomized" (see 9.2) then Rost [1] has shown that one gets p r e c i s e l y the measures v s a t i s f y i n g Gv <_Gy . However, i f we r e q u i r e T to be a genuine ( B t ) - s t o p p i n g time then there i s at l e a s t one a d d i t i o n a l c o n s t r a i n t on v . For example, i f we take n = 3, D = H n , y=<5, v = y 6 + -j y where y i s the uniform u n i t d i s t r i b u t i o n on the u n i t sphere centred at 0 i n ]R n, then v can be reached from y using a randomized stopping time (the obvious one!) but not w i t h a genuine stopping time as b) of the f o l l o w i n g r e s u l t shows. 7.3. P r o p o s i t i o n . Let y be a measure i n D such that Gy i s a p o t e n t i a l , and l e t v = y^ where T i s some (B )-stopping time. Then: a) Gv <_ Gy (so v i s f i n i t e on compact subsets of D) - 62 -b) For every B o r e l p o l a r subset Z of D, v(Z) = y(Z n E ) , where E = {x e D: P X(T = 0) = 1} (Note that E i s u n i v e r s a l l y measurable.) Proof: a) Let x e D . Then Gv(x) = v(dy)G(y,x) = E y ( G ( B _ , x ) l { T < R } ) y ( d z ) E ^ ( G ( B _ , x ) l { _ < R } ) y(dz)G(z,x) (by 6.1) = Gy(x) b) v(Z) - P y ( B T e Z, T < R) = P y(B_ £ Z, T = 0, T < R) (by 6.2) y ( d x ) P X ( B Q e Z, T = 0) y ( d x ) l z ( x ) P * ( T = 0) = y(Z n E ) , x, s i n c e f o r every x, P (T = 0) = 0 or 1 • S i m i l a r l y , we can show: - 63 -7.4. P r o p o s i t i o n . Let u be a measure i n D such that Gy i s a p o t e n t i a l . Let S,T be (B t)-stopping times w i t h S <_ T . Then Gy g >_ Gy^ 7.5. Lemma. Let y be a measure i n D such that Gy i s a p o t e n t i a l . Let A be an a n a l y t i c subset of 3Rn and l e t T = T^  Let v = y T . Then f o r a l l x i n D, Gv(x) = E X ( G y ( B T ) l { T < R } ) Proof: E X ( G y ( B T ) l { T < R } ) y ( d y ) E X ( G ( y , B T ) l { T < R } ) y(dy)bal(G(y,-),A n D,D)(x) y(dy)bal(G(x,«),A n D,D)(y) y ( d y ) E y ( G ( x , B T ) l { T < R } ) = E y ( G ( x , B T ) l { T < R } ) v(dz)G(x,z) = Gv(x) . (by 6.3) (by 2.11) • To put i t another way, y T = bal(y,A n D,D) . 7.6. C o r o l l a r y . Let y be a measure i n D such that Gy i s a p o t e n t i a l . Let U be a B o r e l (or j u s t c o a n a l y t i c ) subset of D, and l e t T = . Suppose D\U i s t h i n at each p o i n t of U . - 64 -(That i s , suppose U i s f i n e l y open.) Then y^,(U) = 0 . P r o o f : As we have j u s t o b s e r v e d , y T = bal(y,D\U,D) . As D\U i s t h i n a t each p o i n t o f U, base(D\U,D) £ D\U . Now a p p l y 3.4. • 7.7. C o r o l l a r y : L e t y be a measure i n D such t h a t Gy i s a p o t e n t i a l , and l e t v be a superharmonic f u n c t i o n i n D, and suppose Gy >_ v i n D . L e t U be a B o r e l ( o r j u s t c o a n a l y t i c ) r e l a t i v e l y compact s u b s e t of D, and l e t T = . Suppose t h e r e i s a f u n c t i o n h which i s harmonic i n some open s e t c o n t a i n i n g the c l o s u r e of U, such t h a t Gy >_ h >_ v on U . Then Gy^ >_ v i n D . P r o o f : By 7.5, G y T ( x ) = Gy(x) f o r any x e D such t h a t P X ( T > 0) = 0 . That i s , G y T = Gy on base(D\U,D) (see 6.4). Suppose x e U . Then P (T < R) = 1 s i n c e U i s r e l a t i v e l y compact i n D . A l s o , f o r e v e r y • co e {BQ = x} and e v e r y t £ [ 0 , T ( c o ) ) , Gy(B t(a))) >_h(B t(u>)) . Hence, by 6.2, Gy(B^) >_ h(B T> P X - a.s. Thus G y T ( x ) >_ E X(h(B^,)) . But by Dynkin's f o r m u l a 5.12, and by the h a r m o n i c i t y of h, E X ( h ( B T ) ) = h(x) . Thus G y T ( x ) >_ v ( x ) . We have now shown t h a t Gy^, >_ v on D \ f r i n g e ( D \ U ) . But by 3.11, f r i n g e ( D \ U ) i s p o l a r . Hence Gy^ >^  v throughout D . • - 65 -7.8. Lemma: L e t y be a measure i n D such t h a t Gy i s a p o t e n t i a l . L e t N = {v: v i s a measure i n D and Gv <_ Gy} Then f o r any compact s e t K c_ D, sup{v(K): v e N} < 0 0 . P r o o f : L e t K be a compact subset of D . L e t V,W be open r e l a t i v e l y compact s u b s e t s of D such t h a t K £ V c w . L e t y' = bal(y,W,D) . Then y' l i v e s on W, so y'(D) < » . L e t v = b a l ( l , V , D ) . Then v i s a p o t e n t i a l i n D . L e t X be the R i e s z measure of v . Then X l i v e s on V . Now f o r any v e N, v(K) = l d v = GXdv = GydX = GXdv K GvdX Gy'dX GXdy' < y'(D) 7.9. Lemma: L e t y be a measure i n D such t h a t Gy i s a p o t e n t i a l . L e t (T_^) be a sequence of (B^)-stopping times and suppose T_^  —y T p o i n t w i s e on Q . Then: a) T i s a (B^.)-stopping time b) F o r any B o r e l f u n c t i o n t(>: D j<j>|Gy < 0 0, we have N>Gy_ •' j c) F o r any c o n t i n u o u s f u n c t i o n D, we have H such t h a t <f>Gy_ . <(> w i t h compact support i n - 66 -d>dv T. 1 P r o o f : a) i s a s t a n d a r d r e s u l t , and f o l l o w s from the r i g h t - c o n t i n u i t y of (Bt) . b) I t s u f f i c e s t o c o n s i d e r <J> >_ 0 . F o r t e [0,°°] and CA) e ft, l e t f R(o») Z t(o») = t A R ( _ ) <j>(B_(_))ds I t f o l l o w s from 6.9 t h a t f o r any (B ) - s t o p p i n g time S, ( * ) J * G y s = E y ( Z s ) . Now f o r w e { Z Q < » } , the map t —* Z (to) i s f i n i t e and c o n t i n u o u s on [0,°°] . A l s o Z f c <_ Z^ and E " ( Z 0 ) - <j)Gp < 0 0 . Thus b) f o l l o w s from the Lebesgue dominated convergence theorem. c) From b) i t f o l l o w s t h a t i f <j> i s C w i t h compact s u p p o r t i n D then 'S'dPn 4>dyn By 7 . 7 , sup y_ (K) < 0 0 f o r any compact s e t K _£ D . The i i a s s e r t i o n c) then f o l l o w s by an a p p r o x i m a t i o n argument. The above p r o o f seems a l i t t l e roundabout. I f y i s f i n i t e then a more d i r e c t p r o o f can be g i v e n , u s i n g 6.6 and 6.8. • - 67 -7.10. Lemma: L e t u be a measure i n D . C o n s i d e r a s e t U o f the form U = V n { v < c } , where V i s open i n D, v i s s u p e r -harmonic i n D, and c i s a r e a l number . L e t (T ) be a sequence of (B^)-stopping times and suppose 1\ —*• T p o i n t w i s e on ft . Suppose a l s o t h a t u_ (U) = 0 f o r a l l i . Then i P - ( U ) = 0 . P r o o f : Suppose u_(U) 4 0 . Then t h e r e i s an open s e t V' which i s r e l a t i v e l y compact i n V and a r e a l number c' < c such t h a t u-(U') 4 0, where U ' = V ' n { v < c ' } . L e t f be a n o n - n e g a t i v e f i n i t e c o n t i n u o u s f u n c t i o n i n D such t h a t f = 1 on V and f = 0 o u t s i d e some compact s u b s e t of V . L e t g be a non-n e g a t i v e f i n i t e c o n t i n u o u s f u n c t i o n on (-00 , 0 0] such t h a t g = 1 on (-°°.c'] and g = 0 on [c,°°] . L e t <f>(x) = f ( x ) g ( v ( x ) ) f o r x e D . Now cf> = 0 o u t s i d e U, so f o r a l l i , E y ( * ( B _ H ) - 0 . l l L e t A = {x e D: f o r a l l i , EX(<t>(B_ ) l r - „ i ) = 0} . Then (A i s T. {T.<R> u n i v e r s a l l y measurable and) u(D\A) = 0 . L e t x e A . By 6.6 and 6.8 (**t) i s P - a.s. u l t i m a t e l y o u t s i d e any compact s u b s e t of D . Thus on {T >_ R}, P X - a.s. •<V 0 = * ( BT ) : L{T<R} • On t h e o t h e r hand on {T < R}, P X - a.s. ^Tn{T.<R} - * ( BT ) : L{T<R} - 68 -s i n c e t *- ^ (B^) i s P A - a.s. c o n t i n u o u s on [0,R) by 6.2. Thus • ( BT. ) : L{T.<R} ^ V ^ R } 1 l P - a.s. on . By the Lebesgue dominated convergence theorem, i t f o l l o w s t h a t E X ( < K B T ) 1 { T < R } ) - 0 . As t h i s i s t r u e f o r a l l x e A, and as u l i v e s on A, E y ( * ( B T H { T < R } ) = 0 . Now <j> = 1 on U' . Hence u (U') = 0 . As t h i s i s a c o n t r a d i c t i o n , i t must be t r u e t h a t u (U) = 0 . • Remark. 'The s e t s of the form V n {v < c} (V open £ D, v s u p e r -harmonic i n D, c e H) c o n s t i t u t e a base f o r the f i n e t o p o l o g y of D . A l s o , any c o l l e c t i o n of f i n e l y open s e t s has a c o u n t a b l e sub-c o l l e c t i o n whose u n i o n d i f f e r s from the u n i o n of the whole c o l l e c t i o n by o n l y a p o l a r s e t . (See remark f o l l o w i n g V.1.17 o f Blumenthal and Getoor [1].) Thus, u s i n g the " f i n i t e n e s s p a r t " of 6.2, we see t h a t U i n the statement of the above lemma can be r e p l a c e d by any B o r e l f i n e l y open s e t . We s h a l l not need t h i s , though. At l a s t we can prove the main r e s u l t o f t h i s s e c t i o n , which i s a l s o one of the main r e s u l t s of t h i s t h e s i s . - 69 -7.11. Theorem: Let u be a measure i n D such that Gy i s a p o t e n t i a l . Let v be a measure i n D such that Gv <_ Gy and y(Z) <_ v(Z) for every Borel set Z £ {Gv = °°} . Then there i s a ( B t )-stopping time T such that v = y_ . (Note well that T i s not randomized.) Proof: Let S be the set of a l l ( B t )-stopping times S such that Gy >_ Gv . One can then show that a ( B )-stopping time T s a t i s f i e s y_ = v i f f T e S and whenever T <_ S e S then T = S P y - a.s. on {T < R} . (For the "only i f " part, use the f a that for any non-negative Borel function f i n D and any (8 )-stopping time S, we have fR (Gy s)f = E P ( f(B )dt) SAR This follows from 6.9 and the strong Markov property.) One can also show that S does contain such "maximal" elements. What i s more, t h i s approach i s reasonably constructive since we don't need Zorn's lemma to produce a "maximal" T; the p r i n c i p l e of dependent choice s u f f i c e s . Nevertheless we s h a l l describe another way of producing a suit a b l e T . We prefer t h i s second approach f o r i t s e x p l i c i t n e s s , i n s p i t e of the f a c t that i t i s not quite as s l i c k as the method outlined above. Let 1/ be a countable open base f o r D co n s i s t i n g of r e l a t i v e l y compact subsets of D . Let G be the weakest topology on D which i s stronger than the usual topology of D and which - 70 -makes Gv c o n t i n u o u s . L e t U be the c o l l e c t i o n o f s u b s e t s of D o f the form V n {Gv < c} where V e V and c i s a p o s i t i v e r a t i o n a l . Then U i s c o u n t a b l e . A l s o , U i s an open base f o r the t o p o l o g y G . ( T h i s a s s e r t i o n i s not used below; indeed we do n o t e x p l i c i t l y make use of the t o p o l o g y G, but i t g i v e s a p e r s p e c t i v e on the p r o o f . ) L e t ^ - ^ i > i ^ e a sequence i n U i n which each element of U o c c u r s i n f i n i t e l y many t i m e s . F o r each i , l e t H = i n f { t > 0: B t I V±} . L e t T Q = 0, and f o r i >. 1 l e t T._^ + IL • 6 T i f t h i s s t o p p i n g time i - 1 . . c i s i n i l e t T. = I T. , o t h e r w i s e , i - i (Note t h a t T. , + H. • 6„, = i n f { t > T. 1 : B„ I U.} .) Then i - i l T. 1 i - i t l i - i (T^) i s an i n c r e a s i n g sequence of s t o p p i n g times i n S . L e t T = l i m T. . Then T e S by 7.9. That i s , GX >_ Gv, where i-x» X = v T . We s h a l l show t h a t X = v . By 7.3(b), X charges p o l a r s e t s l e s s than u . Hence X(Z) <_ v(Z) f o r ev e r y B o r e l s e t Z c {Gv = <*>} . L e t A = {GX > Gv} . We c l a i m X(A) = 0 . Suppose X(A) > 0 . Then t h e r e i s a p o s i t i v e r a t i o n a l c such t h a t X({GX > c} n {c > Gv}) > 0 . But {GX > c} i s open i n D, and hence i s a c o u n t a b l e u n i o n of elements of 1/ . Thus t h e r e e x i s t s V e V such t h a t GX > c on V and X(U) > 0, where U = V n {c > Gv} . L e t I - { i _> 1: U = U} . Then I i s i n f i n i t e . B e f o r e p r o c e e d i n g f u r t h e r , l e t us n o t e t h a t i f X,Y a r e (B ) - s t o p p i n g times then the s t r o n g Markov p r o p e r t y i m p l i e s t h a t ( u x ) Y = U X + Y . Q • Now i f i e I then Gy_ > c > Gv on X i - 1 U. = U so by 7.7, G ( y T ) > Gv i n D ; hence 1-1 1 T. = T. . + H. • 6_ . Hence by 7.6, y_ (U) = 0 f o r i e I . l l - l l T. n T. l - l l But then by 7.10, u (U) = 0 . That i s , A(U) = 0 . As t h i s i s a c o n t r a d i c t i o n , we must have A (A) = 0 . Thus GA <_ Gv A - a.e., and, by 2.2, A(Z) <_ v(Z) f o r e v e r y B o r e l p o l a r subset Z of D . Hence GA <_ Gv, by the d o m i n a t i o n p r i n c i p l e 4.2. T h e r e f o r e GA = Gv . From t h i s i t f o l l o w s t h a t A = v . • As we s h a l l see s h o r t l y , the above theorem i s not the b e s t p o s s i b l e r e s u l t o f i t s t y p e . I b e l i e v e the f o l l o w i n g i s the b e s t p o s s i b l e r e s u l t , but I am u n a b l e to prove i t a t the p r e s e n t time. 7.12. C o n j e c t u r e : L e t y,v be measures i n D, and suppose Gy i s a p o t e n t i a l . Then the f o l l o w i n g a r e e q u i v a l e n t : a) There i s a ( B f c ) - s t o p p i n g time T such t h a t y_ = v . b) Gy >_ Gv, and t h e r e i s a B o r e l s e t A £ D such t h a t v(Z) = y(ZnA) f o r a l l B o r e l p o l a r s e t s Z £ D . (Note a) = > b) h o l d s by 7.3.) Now l e t us c o n s i d e r an example t h a t shows t h a t 7.11 i s not b e s t p o s s i b l e . - 72 -7.13. Example: Suppose D = ]Rn where n = 3 . (We make these assumptions to keep the computations from getting too messy.) Let u = 6 . Let ( r j ) ^ e a s e c l u e n c e of p o s i t i v e r e a l numbers decreasing to 0, such that there e x i s t m,M e ( l , 0 0 ) with r. m < — ^ — < M for a l l j . — r — 3 + 1 Let S. = {x e D: I I x l I = r.} 3 1 1 1 1 3 H. = {x e D: I I x l I < r.} J 3 A. = { x e D : r.,_ < l l x l l <r.} . 3 3 + 1 1 1 1 1 - j Let (s.) be a sequence i n [O,00) with £s. £ 1, and l e t v be 3 j 3 the s p h e r i c a l l y symmetric measure on D which l i v e s on u S_. and assigns mass s^ . to S^. f o r each j . Suppose v does not p i l e up too f a s t around 0, i n the sense that £js. < °° . Then there 3 J i s a (Bj.)-stopping time T such that u = v . Proof: F i r s t l e t us show that Gv grows very slowly at 0 compared to Gy . For each i , l e t v. = v„ . Then 3 S. 3 Gv(x) = < s. 4 i T r . 3 ^— for l l x l l < r, — r r for x > r . x 1 1 1 - 3 (It follows that Gv <_ Gy .) Thus for x e Afc, - 73 Gv(x) = 4lT k s. "3=0 r j l*lT j=k+l ^ so Gv(x) = v . I | x | j=0 J ' j j=k+l Gy(x) j r. .Jr.. j 0 0 r, A r . < I s. 1 L e t us c a l l the l a t t e r q u a n t i t y a ^ . Then oo oo c o r A r k=0 j=0 2 k=0 j CO /- CO \ j=0 J 1 k=j r j J j=0 J ^ £=0 F o r the remainder of the p r o o f , l e t us not make such s p e c i a l assumptions on v . L e t us assume, r a t h e r , t h a t v i s such t h a t Gv s a t i s f i e s the p r o p e r t i e s we have j u s t v e r i f i e d . E x p l i c i t l y , we s h a l l assume t h a t v i s a measure i n D such t h a t Gv <_ Gy and t h e r e a r e p o s i t i v e r e a l s ( a i . ) such t h a t £ a, < 0 0 and f o r k each k, Gv < a. G i n - k y Now f o r any E _c D, l e t C(E) denote the o u t e r c a p a c i t y of E r e l a t i v e t o D . (See Helms [ 1 ] , c h a p t e r 7.) F o r each j , l e t - 74 -be a c e r t a i n compact s u b s e t " o f S.. which w i l l be more e x p l i c i t l y s p e c i f i e d s h o r t l y , and l e t X^ = bal(y,E^.,D) . Now A l i v e s on E. . A l s o Gy = -,—— on E., so 1 4irr. i 3 VV = 4-r. = C(S.) ' J J Thus f o r x 6 H., J A (E ) C(E ) r \ ( v ) > J J 3 ___ VJA . v,A7 / / . _ \ <")_ -\ /-i/-c \ o_ j v ' - (ATr)(2r ) C(S ) 8TT_ C(E ) = C ( ^T ^ r T ^ N x l |Gy(x) C(E )||x|| = 2C(S ) r G y ( x ) ' Thus f o r x e A., 3 C(E.) G A j ( x ) ^ 2 C ( S * ) S G y ( x ) • Choose k so t h a t a. < ^ — f o r a l l i > k . Now the E.'s can J - 2M - j be chosen so C(E^)/C(S_.) i s any number between 0 and 1 . Thus f o r i > k we can choose E. so t h a t - 3 C(E.) 2C(S.)M " " j = a. L e t E = (D\KL ) u ( u E.) . L e t A = b a l ( y , E , D ) . Then GA = Gy j__k 2 on DXH^., and f o r j k, GA •> GA_. >_ a^.Gy >_ Gv on A . Hence GA > Gv . Now A = y T T, where U = T_, by 7.5. A l s o — U L C(E.) j _ k L ( S j ; j>k 2 - 75 -so E i s t h i n a t 0 . (See p r o o f o f theorem 10.21 of Helms [1] ~ t h i s i s the W i e n e r - B r e l o t t e s t f o r t h i n n e s s . ) Thus P°(U > 0) = 1, so Ujj({0}) = 0 . (Once (B^) l e a v e s 0 i t cannot come back to i t , because {0} i s p o l a r . ) Thus n e i t h e r p^ nor v charges p o l a r s e t s . Thus, by 7.11, t h e r e i s a (8 t)-stopping time V such t h a t ( V v = v ' But then v = p^, by the s t r o n g Markov p r o p e r t y , where T i s the (8 ) - s t o p p i n g time U + V • 9 . t u Q The p r o o f of the above example i s a r a t h e r d e l i c a t e " b a l a n c i n g a c t " . We have to take the E j ' s b i g enough so t h a t bal(Gp,E,D) >_ Gv . On the o t h e r hand, i f we took them too b i g then E would not be t h i n a t 0, and bal(p,E,D) would j u s t be p, so t h a t 7.11 would not a p p l y . - 76 -8. EMBEDDING MEASURES IN BROWNIAN MOTION IN I T OR ]R_, USING  NON-RANDOMIZED STOPPING TIMES. 8.1. I f y i s a measure on H n and T i s a (B^)-stopping time t h e n , f o r the purposes of t h i s s e c t i o n , y^ w i l l denote the measure on B o r e l ]R n d e f i n e d by y^,(A) = P y ( B T e A, T < °°) . (Of c o u r s e the c o n d i t i o n "T < °°" i s s u p e r f l u o u s ; we i n c l u d e i t j u s t f o r emphasis.) As i n the p r e v i o u s s e c t i o n , we c o n s i d e r the q u e s t i o n : What measures v can be w r i t t e n i n the form v = y^ ?" Of c o u r s e i f n >_ 3, t h i s i s a s p e c i a l case of the q u e s t i o n c o n s i d e r e d i n the p r e v i o u s s e c t i o n . However, i f n = 1 or 2 then n -K i s not a Green r e g i o n , so the above q u e s t i o n i s then not subsumed under the one of the p r e v i o u s s e c t i o n . I f T i s a l l o w e d to be randomized (and y,v a r e f i n i t e ) then Rost [2] has g i v e n a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r v to be e x p r e s s i b l e i n the form v = y T ; R o s t ' s c r i t e r i o n i s i n terms o f a - e x c e s s i v e f u n c t i o n s . Skorohod [1] showed t h a t ( i n the case n = 1) i f y = &^  f o r some x e 1R and i f v i s a p r o b a b i l i t y measure on H w i t h f i n i t e v a r i a n c e and mean e q u a l to x, then t h e r e i s a randomized (8 )-s t o p p i n g time T such t h a t E (T) < °° and v = u . (Then of x 1 1 c o u r s e E (T) = y v a r i a n c e ( v ) ; the f a c t o r o f - j a r i s e s because we have n o r m a l i z e d Brownian motion to get the c l e a n e s t form f o r Dynkin's f o r m u l a 5.12.) Note t h a t by 4.10, a p r o b a b i l i t y measure v on H has c e n t r e of mass e q u a l t o x i f f U V i s a p o t e n t i a l and U V <_ U y - 77 -where y = . Of c o u r s e , t h i s r e s u l t o f Skorohod's i s the g r a n d -f a t h e r o f a l l t h e s e embedding r e s u l t s . Dubins [1] and Root [ l j gave p r o o f s of Skorohod's r e s u l t which produce non-randomized T's . Doob (see Meyer [3]) has noted t h a t i f y,v a r e any p r o b a b i l i t y measures on H t h e n t h e r e i s a ( B ^ ) - s t o p p i n g time T such t h a t y_ = v . T h i s somewhat d i s q u i e t i n g r e s u l t l e a d s us to r e s t r i c t the c l a s s of s t o p p i n g times t h a t w i l l be " a l l o w e d " , as f o l l o w s . 8.2. D e f i n i t i o n . L e t T be a ( B f c ) - s t o p p i n g time, and l e t y be a measure on H n such t h a t U y i s a p o t e n t i a l . We s h a l l say t h a t T i s y - s t a n d a r d i f f whenever R,S a r e ( B t ) - s t o p p i n g times and y R V S y R y S R < S < T then U and U a r e p o t e n t i a l s and U > U 8.3. D i s c u s s i o n of 8.2. I f n >_ 3 then any T i s y - s t a n d a r d , by y T 7.4. Suppose n <_ 2 and T i s y - s t a n d a r d . Then U i s a p o t e n t i a l and i s l e s s than or e q u a l to U y . Hence y_ has t o t a l mass g r e a t e r than or e q u a l to t h a t o f y, by 4.8 ( i f n = 2) or 4.11 ( i f n = 1) . Hence i n d i m e n s i o n one or two, i f T i s y - s t a n d a r d then T i s P y - almost s u r e l y f i n i t e . Suppose n = 1, y = 6^, T = i n f { t > 0: B t = 1}, and v = y_ . Then v = &1 . Thus T i s 2 not y - s t a n d a r d . Now suppose n = 2 , S = {x e H : | | x | | = r } , y i s the u n i f o r m u n i t d i s t r i b u t i o n on S-, T = i n f { t > 0: B t e S^}, and v = y_ . Then T i s P y - a.s. f i n i t e by 6.5, s i n c e S^ i s not a p o l a r s e t . C l e a r l y t h e n , v i s the u n i f o r m u n i t d i s t r i b u t i o n on S^ . Now - 78 -irV) = w h i l e U V ( x ) = 2TT 2TT log 2 for ]Ix < 2 l o g I|x I f o r x > 2 0 f o r | | x | | <_ 1 1 2TT log I|x|| for | |x j | > 1 Thus T i s not u-s t a n d a r d . 8.4. Lemma. L e t y,v be measures on H n such t h a t U y , U V a r e p o t e n t i a l s . Then $ ( x , y ) d y ( x ) d v ( y ) < °° Hence U y d v and U Vdy b o t h make sense, and they a r e e q u a l . A l s o , they a r e not e q u a l t o -°° P r o o f : I f n ^ 3, we have n o t h i n g t o p r o v e . I f n = 1 or 2 the n y and v a r e f i n i t e . I f n = 1 then by 4.10, l i m (yOR)$a,y) -I y I _ H x j $ ( x , y ) d y ( x ) ) = 0, and by 1.5, then by 4.3, $(£,y)dv(y) i s f i n i t e ; a l s o $ = -$ . i f n = 2 2 -l i m (yOR )$ (y) -Ixl $ ( x , y ) d y ( x ) ) = 0, and by 1.6, ( y ) d v ( y ) i s f i n i t e . A l s o , f o r any n, y s e t s . - 79 -$ ( x , y ) d y ( x ) i s c o n t i n u o u s , and so i s bounded on compact • 8.5. Theorem: L e t y be a measure on E n (where n = 1 or 2) such t h a t U y i s a p o t e n t i a l . L e t T be a (8^)-stopping time such t h a t E y ( T ) < °° . Then T i s y - s t a n d a r d , and E y ( f ( B f c ) d t ) = ( U y - U T ) f f o r any n o n - n e g a t i v e B o r e l f u n c t i o n f on E P r o o f : F i r s t l e t f be a non-negative compactly supported C f u n c t i o n on E n , and l e t u = U f . Then u i s C°°, grad u = U g r a d f , and Au = - f . L e t M = u(B t> + f (B ) d s , f o r 0 <_ t < » . By 0 5.8, (M t) i s a m a r t i n g a l e over (ft,8,8 t,P ) f o r any a e I T But u + c* i s bounded, where c = (See 4.3 f o r n = 2 and 4.10 f o r n = 1 .) Thus E y(|M f c|) < » f o r a l l t . Hence (M ) i s a m a r t i n g a l e over (ft,8,8 t,P y) . Of c o u r s e , (M ) has r i g h t -c o n t i n u o u s ( i n f a c t , c o n t i n u o u s ) p a t h s . Now u(x) = g(x) - c$ ( x ) , where g i s bounded on E ; a l s o $ (x) <_ | |x - y| | + $ (y) f o r a l l x,y i n E y ( $ ( B Q ) ) = From 5.10, t o g e t h e r w i t h the f a c t t h a t dy < «>, i t then f o l l o w s t h a t {u(B_ ) : 0 < t < °°} TAt — i s P y - u n i f o r m l y i n t e g r a b l e . (Here we use the f a c t t h a t E y ( T ) < °° .) A l s o TAt fT f ( B )ds < s — f ( B )ds and EM ( f ( B )ds) < b E y ( T ) < o ° , where b = sup f . Thus {M_ A t: 0 <_ t < °°} i s P y - u n i f o r m l y - 80 -i n t e g r a b l e . Hence E^CHj.) = E y ( M Q ) , by the o p t i o n a l sampling theorem. That i s , U f d u T - E y ( f ( B )ds) = U fdy . In p a r t i c u l a r , Urdy^, i s f i n i t e . Hence (as f can be non-zero) yT U i s a p o t e n t i a l . Now interchanging orders of i n t e g r a t i o n (which i s j u s t i f i e d by 8.4) we o b t a i n rT E y ( f ( B )ds) = 0 y yT (U y - U ) f . As t h i s holds f o r a l l compactly supported non-negative C f u n c t i o n s f on R n, we f i n d that U y >_ U ^, and then a monotone c l a s s argument shows that the formula holds f o r a l l non-negative B o r e l f u n c t i o n s f on H n . Now c l e a r l y we can apply the above argument to any (B )-stopping time which i s <_ T, s i n c e such a stopping time w i l l a l s o have f i n i t e P y- e x p e c t a t i o n . I t then f o l l o w s that i f R,S are (8 t)-stopping times and R <_ S <_ T then yR yS U , U are p o t e n t i a l s , and U R > u s That i s , T i s y-standard. • 8.6. Terminology. A measure y on ]R w i l l be c a l l e d good i f f y has compact support and U Y i s f i n i t e and continuous. Note that - 81 -n i f <j) i s a compactly s u p p o r t e d bounded B o r e l f u n c t i o n on H and y(dx) = <j>(x)dx, then y i s good. A l s o , by 4.6 of t h i s work t o g e t h e r w i t h 6.21 o f Helms [ 1 ] , any compact n o n - p o l a r s u b s e t o f ]R n c a r r i e s a non-zero good measure. 8.7. Theorem: L e t y be a measure on ]R n, where n = 1 or 2, n y ' and l e t ( y . ) . T be a n e t of measures on H such t h a t U i s a l l e i p o t e n t i a l f o r a l l i e I , and cfidy. <j>dy f o r a l l compactly s u p p o r t e d c o n t i n u o u s f u n c t i o n s <f> on H Then a) U ^ 1 dy U y dy f o r a l l good measures y on ]R n Now suppose a l s o t h a t the n e t ( U dv) converges t o a f i n i t e l i m i t f o r some non-zero measure v on 3Rn such t h a t U V i s a p o t e n t i a l . Then b) U y i s a p o t e n t i a l . u i c) The net (U ) converges u n i f o r m l y on compact s e t s t o U + C, where C i s some f i n i t e n o n - n e g a t i v e c o n s t a n t , d) U 1 dy U y - C dy f o r a l l good measures y on ]R n . (Remark: C need not be z e r o ; see 8.15 f o r a n a t u r a l example of t h i s . ) n y P r o o f : I f y i s a good measure on H then U + i s c o n t i n u o u s and, as n <_ 2, i s a l s o compactly s u p p o r t e d . P a r t a) f o l l o w s i mmediately from t h i s , upon i n t e r c h a n g i n g o r d e r s of i n t e g r a t i o n . Now suppose v i s as i n t h e statement of the theorem. Then f o r some 1^, sup ^ 0 U_ dy < Hence l i m ( s u p y.({x e H n : ||x|| >^r})) = 0 - 82 -But t h e n , s i n c e we a l s o know t h a t the net (j<|>dy^) converges t o a f i n i t e l i m i t f o r each c o n t i n u o u s compactly s u p p o r t e d f u n c t i o n $ on ]R n, t h e r e e x i s t s i ^ >_ i ^ such t h a t (*) sup y . (R n) < 0 0 ^ 1 Thus u0R n) < °° and | f dy,^ fd y f o r a l l bounded c o n t i n u o u s f u n c t i o n s f on H Next, i t i s easy t o show t h a t (**) Thus by Fat o u ' s lemma, u. U y < l i m i n f U 1 U_dv <_ l i m i U X d v Hence U ydv < 0 0 . T h e r e f o r e , as v 4 0, U y i s not i d e n t i c a l l y i n f i n i t e . Hence, by 1.5 ( i f n = 1) or 1.6 ( i f n = 2 ) , U_ i s f i n i t e everywhere, and U y i s a p o t e n t i a l . A l s o , f o r any x i n Rn, U V d i f f e r s from vOR n)$ (x,') by a bounded c o n t i n u o u s f u n c t i o n . I t f o l l o w s t h a t the n e t (U_ (x)) converges t o a f i n i t e l i m i t f o r each x i n ]R n . But f o r any x,y i n ]R n, | u \ x ) - U ^ ( y ) | < y.0R n)||x - y| I . y i Thus by (*), {U_ : i >_ i^} i s e q u i c o n t i n u o u s ( i n d e e d , u n i f o r m l y u. L i p s c h i t z ) so i n f a c t (U ) converges u n i f o r m l y on compact s u b s e t s of R n . L e t u be i t s l i m i t , which i s f i n i t e and c o n t i n u o u s on H n . L e t u = U y - u_ (-°°,00]-valued on H n . A l s o Then u i s l o w e r - s e i n i c o n t i n u o u s and U 1 dy udy f o r a l l good measures - 8 3 -Y on 1 R In p a r t i c u l a r , t h i s convergence h o l d s f o r y of the CO form y(dx) = <j>(x)dx, where <j> i s C and compactly s u p p o r t e d on ]R n . Thus y = -Au . ( C l e a r l y u i s l o c a l l y Lebesgue i n t e g r a b l e , and so d e f i n e s a Schwartz d i s t r i b u t i o n on H n .) Hence t h e r e i s a harmonic f u n c t i o n h on H n such t h a t u = v almost everywhere w i t h r e s p e c t t o Lebesgue measure on ]R n, where v = U Y - h . By a v e r a g i n g over a b a l l o f r a d i u s 6, c e n t r e d a t x, and l e t t i n g 6-1-0, one e a s i l y checks t h a t a c t u a l l y u(x) = v ( x ) f o r a l l x e E. n . Now by (**) , U Y <_ u_ . Thus h >_ 0 . But a n o n - n e g a t i v e harmonic f u n c t i o n on H n i s c o n s t a n t . ( T h i s i s P i c a r d ' s theorem - 1 . 1 1 of Helms [ 1 ] ; t r u e f o r any n .) Thus C i s j u s t the c o n s t a n t v a l u e of h . • 8.8. Lemma. L e t y,a be measures on H n , where n = 1 o r 2, such t h a t U Y , U 0 a r e p o t e n t i a l s . Suppose U Y - C >_ U A , where C e [0,°°) . Suppose a l s o t h a t yQR n) = aOR n) . Then C = 0 . P r o o f : I f n = 1, t h i s f o l l o w s e a s i l y from 4.10. Suppose n = 2 . L e t y be the u n i f o r m u n i t d i s t r i b u t i o n on the sphere o f r a d i u s 1 2 c e n t r e d a t 0 i n TR. . Then U Y * Y - C = ( U Y - C)*y > U ° * Y = U A * Y , 2 where * denotes c o n v o l u t i o n . L e t m = y(R ) . Then m = ( u * y ) 0 R 2 ) = ( a * y ) 0 R 2 ) . A l s o U Y * Y = U Y * y = - $ ~ * u = - U Y and - 84 -U ° * Y = -u° . Thus by 4.3, as ||x| |U P Y ( x ) - m*(x) | 0, and |u°* Y(x) - m*(x) | 0 From t h i s i t f o l l o w s t h a t C must be 0 . • 8.9. Lemma: Suppose M i s a f a m i l y of measures on E.n, where n = 1 or 2, such t h a t a) For each u e M, U y i s a p o t e n t i a l . b) sup{pOR n): u e M} < °° . c) i n f { U y ( x ) : y e M , x e K} > -» f o r some compact n o n - p o l a r s e t K c n n . Then l i m sup y({x eE. n: | |x| | >_r}) = 0 . r-x» yeM P r o o f : By 4.6, t h e r e i s a non-zero measure y c a r r i e d by K such t h a t U Y i s bounded above on ]Rn . L e t L = i n f { U y ( x ) : y e M , x e K} Then f o r any y e M , U Ydy = U Mdy > Ly(K) Y Then from b ) , the upper boundedness of U , and the f a c t t h a t U Y ( x ) — • - c o as | j x | | —> °°, i t f o l l o w s t h a t l i m sup y ( { x eE. n: ||x|| >_r}) = 0 . r-x» yeM Q - 85 -8.10. Lemma: L e t u be a f i n i t e measure on R n . L e t T be a c o l l e c t i o n of (B^)-stopping times such t h a t i f T e T and S i s a ( B t ) - s t o p p i n g time s a t i s f y i n g S <_ T, then S e T . Suppose t h a t l i m sup u_({x e ] R n : ||x|| >_r}) = 0 . r-x° TeT Then l i m sup P y ( T > t ) = 0 . t-*» TeT P r o o f : F o r each n a t u r a l number i , l e t = i n f { t _L 0: I l B t l I _. i ^ ' Then each R± i s P y - a.s. f i n i t e , by 5.11. I f T e T then T A R. e T and P y ( T _> R.) = y _ A R ({X e ]R n: | |x| | >_ i } ) . Thus i l i m sup P y ( T >_ R.) = 0 . TeT 1 F i x e > 0 . Then f o r some i , sup P y ( T >_R.) __ f" • Next, f o r TeT 1 1 some t e [ 0, °°) , Then P y ( R ± > t ) < | sup P y ( T >_ t ) <_ £ TeT • 8.11. Theorem: Let u be a measure on ]R n (where n = 1 or 2) such t h a t U y i s a p o t e n t i a l . L e t T be a s e t of good measures on H n such t h a t whenever a, B a r e measures on ]R n such t h a t U a , a r e p o t e n t i a l s and U dy >_ U dy f o r a l l y e f , then u " > U £ Suppose (T_^) i s a sequence o f y - s t a n d a r d ( B t ) - s t o p p i n g times c o n v e r g i n g p o i n t w i s e on ft to a f u n c t i o n T : ft -*- [0, °>] . (Note t h a t then T i s a (B^)-stopping time because (B^) is right-continuous.) - 86 -C o n s i d e r t h e f o l l o w i n g s t a t e m e n t s : a) There i s a measure a on ]R n , such t h a t U ° i s a p o t e n t i a l yT and aQR n) = P QR n) , w i t h U 1 >_ U a f o r a l l i . y T b) U i s a p o t e n t i a l and T i f y T U dy + D dy f o r a l l measures y e V c) T i s y - s t a n d a r d . Then a) => b) => c) P r o o f : a) => b ) . L e t H = {H : H i s a (B ) - s t o p p i n g time and H <_ T. f o r some i} . y H a Z 1 Then U > U f o r a l l H e H s i n c e each T. i s y - s t a n d a r d . T h e r e f o r e — x l i m sup y„({x e !Rn : | |x| | >_ r}) = 0 , by 8.9. Hence r+~ Heh1 H l i m sup P y(T^. >_ t ) = 0 , by 8.10. T h e r e f o r e T i s P y - a.s. t-*» i l — f i n i t e , so f o r each bounded c o n t i n u o u s f u n c t i o n <j> on H , <f> dy_ •+ i 4> dy_ , s i n c e ( B t ) i s c o n t i n u o u s on [0, °°) . L e t v be the u n i f o r m u n i t d i s t r i b u t i o n on t h e s u r f a c e o f t h e u n i t sphere c e n t r e d a t 0 . Then P r o > T. -U 1 dv = -U V dy T. x U V dy T. x U dv U a dv . - 86a -Thus t h e sequence ( T. U 1 dv) i s bounded. By a sub-subsequence argument, u s i n g 8.7 and 8.8, we o b t a i n t h e statement b) . b) => c ) . C o n s i d e r any bounded ( 8 t ) - s t o p p i n g time Q . We have <f>dp T.AQ l TAQ n H T ± A Q y f o r a l l bounded c o n t i n u o u s f u n c t i o n s <j> on ]R , and U > U f o r a l l i . Thus, by t h e same method we used i n c o n c l u d i n g t h e p r o o f of (a => b) , we f i n d t h a t f o r a l l measures y e f , T.AQ U 1 dy TAQ , U v dy . PT.AQ Y T . Now f o r each i , U 1 >_ U 1 as T_^  i s y - s t a n d a r d . I t f o l l o w s t h a t U ^ ^ dy > f dy f o r a l l y e T , whence U ^ >_ U In p a r t i c u l a r , i f R, S a r e ( B t ) - s t o p p i n g times s a t i s f y i n g R < S then y y y T T TARAt „ TASAt T T T u > u > u y y T f o r a l l t e [0, °°) . ( A l s o U >_ U , so t h e t o t a l mass of y T i s >_ t h e t o t a l mass of y , whence t h e s e two t o t a l masses must i n f a c t be e q u a l and thus T must be f i n i t e P y - a . s . ) . I f i n a d d i t i o n S <_T , we have / R A t > / s A t > - 87 -f o r a l l t e [0, °°) . L e t t i n g t -*• 0 0 and a p p l y i n g 8.7 and 8.8 we o b t a i n U R > U S . • 8.12. C o r o l l a r y : L e t y be a measure i n H n , where n = 1 o r 2, such t h a t U y i s a p o t e n t i a l . L e t T be a ( 8 t ) - s t o p p i n g y T time. Then T i s y - s t a n d a r d i f f U i s a p o t e n t i a l and y y U T A t > U T f o r a l l t e [0,°°) . P r o o f : (==>) f o l l o w s immediately from 8.2, the d e f i n i t i o n o f " y - s t a n d a r d " . (<==) T a k i n g t = 0 and a p p l y i n g 4.8 ( i f n = 2) or 4.11 ( i f n = 1) we f i n d t h a t y T ( R n ) = yOR n) . For each t , T A t i s y - s t a n d a r d , by 8.5. Now a p p l y 8.11, w i t h a = y and T. = T A i . T • 8.13. Theorem: L e t y be a measure on H n , where n = 1 or 2, such t h a t U y i s a p o t e n t i a l . L e t T be a (B^)-stopping time. L e t m be Lebesgue measure on H n . Then: a) F o r any measure v on ]R n, U y - U V i s d e f i n e d m - a.e., and i t s m - i n t e g r a l over any compact subset of H n makes sense, though i t may be -H>° . b) T i s y - s t a n d a r d i f f f o r each compact s u b s e t K of ]R n, u T y U - U dm i s f i n i t e and e q u a l t o E ( K 1 K ( B )ds) 0 - 88 -P r o o f : a) U M i s everywhere d e f i n e d , does not assume the v a l u e and i s m - i n t e g r a b l e over any compact su b s e t of 3Rn . Now v v e i t h e r the same i s t r u e of U , or e l s e U i s i d e n t i c a l l y -°° on ]R n except p o s s i b l y f o r a s e t of m-measure 0 where i t i s u n d e f i n e d . T h i s p r o v e s a ) . M T A i y T b) (=>) As T i s y - s t a n d a r d , U + U except p o s s i b l y on a p o l a r s e t , by 8.7. A l s o U y - U T A ± dm K = E y ( T A i l K ( B s ) d s ) f o r any i , and any compact su b s e t K of R. L e t t i n g i go to °°, we o b t a i n the d e s i r e d e q u a l i t y by the monotone convergence y y T theorem. The f i n i t e n e s s i s c l e a r , s i n c e U and U a r e p o t e n t i a l s . (<=) F o r any t e [0,°°) and any compact s e t K £]R", r M - i 'K U y - U T A t dm = E y ( < E y ( (•TAt l K ( B s ) d s ) l K ( B s ) d s ) (by 8.5) u K T I T - U dm ; K l^TAt P I J » P<j» hence U >_ U m - a.e. A l s o U i s a p o t e n t i a l by the y T A t y T f i n i t e n e s s assumption. T h e r e f o r e U >_ U everywhere. Now a p p l y 8.12. - 89 -Remark: Note t h a t E y ( T 1 (B )ds) i s t h e P y - e x p e c t e d amount of 0 K s time t h a t (B f c) spends i n K up t o time T . Observe the a n a l o g y between (*) o f the p r o o f of 7.9, and the f o r m u l a e s t a b l i s h e d i n the above theorem. 8.14. C o r o l l a r y : L e t y be a measure on E. n, where n = 1 o r 2, such t h a t U y i s a p o t e n t i a l . L e t R,S be (8 t)-stopping t i m e s , and c o n s i d e r the (8 ) - s t o p p i n g time T = R + S • 8 . t R Suppose R i s y - s t a n d a r d and S i s y_,-standard. Then T i s K y - s t a n d a r d . P r o o f : A p p l y 8.13 i n c o n j u n c t i o n w i t h the s t r o n g Markov p r o p e r t y . • 8.15. In o r d e r t h a t a (8 t)-stopping time T be y - s t a n d a r d , i t i s not s u f f i c i e n t t h a t fT E y ( 1„(B ) d s ) 0 K s be f i n i t e f o r each compact s e t K _c ]R n . T h i s i s shown by the f o l l o w i n g example. L e t n = 2 . L e t y be the u n i f o r m u n i t d i s t r i b u t i o n on {x e H : ||x|| = 2} . L e t A = {x e TR : ||x|| = 1}, and l e t T = i n f { t > 0: B^ e A} . As A i s not p o l a r , T i s P P -a.s. f i n i t e (because n < 3 ) , by 6.5. Then c l e a r l y v = y_ i s the u n i f o r m u n i t d i s t r i b u t i o n on A . Thus U V ( x ) = 0 f o r | | x | | <_ 1 - ^ l o g | | x | | f o r ||x|| > 1 - 90 -w h i l e U M ( x ) = - ^ l o g 2 f or _ 1_ x|| < 2 ^ l o g | | x | | f o r ||x|| > 2 Thus U M ^_U V, so T cannot be y - s t a n d a r d . Now f o r i = 3,4,5,... l e t A± = {x e H 2 : ||x|| = 1} and l e t = i n f { t > 0: B t e k± u A} Then each T^ has f i n i t e P y - e x p e c t a t i o n , by 5.11, and so i s y-s t a n d a r d , by 8.5. Now v. E y^, c o n s i s t s o f a u n i f o r m d i s t r i b u t i o n i o f mass e. on A. p l u s a u n i f o r m d i s t r i b u t i o n of mass 1 - E. on X I 1 A, where e^ e (0,1) i s determined by the e q u a t i o n l o g i + (1 - s ^ l o g 1 = l o g 2 T h i s f o l l o w s from Dynkin's f o r m u l a 5.12, and the f a c t t h a t l o g | | • | | 2> i s harmonic i n H^\{0} . Thus E. = 2 l l o g 1 Now one e a s i l y computes t h a t v. U 1 4- v _ l o g 2 2TT ' A l s o , i t i s c l e a r t h a t T. + T l (The f a c t t h a t e. x 0 can then be used t o g i v e a n o t h e r p r o o f t h a t T i s P y - a.s. f i n i t e . ) I t 2 f o l l o w s t h a t f o r any compact s e t K £ H , E y ( l K ( B s ) d s ) = K ( Uy - ( D T _ l o g _ 2 ) ) f which of c o u r s e i s f i n i t e . We a l s o remark t h a t the sequence o f measures (v^) y i e l d s a n a t u r a l example i n which the C of theorem 8.7 i s non-zero; C = _ l o g 2 2TT h e r e . • - 91 -8.16. Lemma: L e t H be a bounded c o - a n a l y t i c subset of E. n . L e t T = i n f { t > 0: _ t I H} . Then f o r any measure y on TRn such t h a t U y i s a p o t e n t i a l , T i s y - s t a n d a r d , and U T ( x ) = E X ( U y ( B _ ) ) f o r a l l x e R n . P r o o f : I f v i s any measure on E. n such t h a t U V i s a p o t e n t i a l then E V ( T ) < » by 5.11, so T i s v - s t a n d a r d by 8.5 ( i f n £ 2) . In p a r t i c u l a r , t h i s i s t r u e i f v = y, and a l s o i f v = 6^ f o r any x e H n . Hence E X ( U P ( B _ ) ) = E X ( 4 > ( y , B _ ) ) y ( d y ) , where the r e q u i r e d i n t e r c h a n g e i n o r d e r of i n t e g r a t i o n i s j u s t i f i e d by 8.4 ( i f n <_ 2) . We a l s o have U T ( x ) = E y ( $ ( x , B _ ) ) y ( d y ) Now we c l a i m t h a t f o r a l l x,y e ]R n, E ( $ ( y , B _ ) ) = E y ( $ ( x , B _ ) ) . L e t V be a bounded open s u b s e t of ]R n such t h a t H c V Case 1. x e IR n\V . Then P X ( T = 0) = 1, so E X ( $ ( y , B _ ) ) = *(y,x) = *(x,y) . subcase a. y e E. n\V . Then E y ( $ ( x , B _ ) ) = $(x,y) . subcase b. y e V . L e t W be an open s u b s e t of V such t h a t H £ W and W £ V . Then $(x,*) i s bounded and c o n t i n u o u s on W, - 92 -and harmonic i n W . A l s o T <_ i n f { t >_ 0: B t £ W} . Thus by Dynkin's f o r m u l a 5.12, E y ( $ ( x , B _ ) ) = $(x,y) . Case 2. y e ]R n\V . T h i s i s s i m i l a r t o case 1. Case 3. x,y e V . For each z i n V, l e t h ( z , ' ) be the g r e a t e s t harmonic minorant of $(z,«) i n V . L e t G be the Green f u n c t i o n of V . Then G = $ - h on V x V . A l s o T < i n f { t > 0: B t I V} P Z -a.s. f o r each z e V . Thus E x ( G ( y , B _ ) ) A l s o = b a l ( G ( y , - ) , V \ H , V ) ( x ) (by 6.3) = b a l ( G ( x , - ) , V \ H , V ) ( y ) (by 2.11) = E y ( G ( x , B _ ) ) (by 6.3) . E X(h(y,B_).) = h(y,x) (by 5.12) = h ( x , y ) = E y ( h ( x , B _ ) ) (by 5.12) E X ( $ ( y , B _ ) ) = E y ( $ ( x , B _ ) ) i n t h i s c a s e t o o , and the c l a i m i s e s t a b l i s h e d . Hence • - 93 -8.17. C o r o l l a r y : L e t H be a bounded c o - a n a l y t i c subset of H , and l e t y be a measure on H n such t h a t U y i s a p o t e n t i a l . L e t T = i n f { t > 0: B I H} . Suppose v i s superharmonic on R.n, v <_ U y , and t h e r e i s a f u n c t i o n h which i s harmonic i n some open s e t c o n t a i n i n g the y y T c l o s u r e of H, such t h a t v <_ h <_ U i n H . Then v <_ U P r o o f : I f n _^  3, t h i s i s j u s t a s p e c i a l case of 7.7. I f n = 1 or 2, we need o n l y emulate the p r o o f of 7.7, u s i n g 8.16 i n p l a c e of 7.5. • 8.18. Lemma: L e t y be a measure on H n . L e t H be a c o -a n a l y t i c subset of H n such t h a t ]R n\H i s t h i n a t each p o i n t of H . L e t T = i n f { t > 0: B I H} . Then y_(H) = 0 . P r o o f : L e t ( v^) be a n i n c r e a s i n g sequence of bounded open s u b s e t s of E. n whose u n i o n i s ]R n . For each i , l e t R. = i n f { t > 0: B_ i V.} and l e t y. be the r e s t r i c t i o n of y l — t I I y. to B o r e l V^ , . Then by 7.6, P ^"(B- e H, T < R ) = 0 f o r a l l i . L e t t i n g i — • ~, we o b t a i n y_(H) = 0 . ^ 8.19. Lemma: L e t (A,F,Q) be a a - f i n i t e measure space. L e t T be a c o l l e c t i o n of [0,°°]-valued F-measurable f u n c t i o n s on A . - 94 -Suppose the l i m i t of each i n c r e a s i n g sequence i n T b e l o n g s to T, and T 4 0 • Then T c o n t a i n s Q - e s s e n t i a l l y maximal elements. P r o o f : By the H a u s d o r f f m a x i m a l i t y p r i n c i p l e , T c o n t a i n s a maximal c h a i n C . As Q i s a - f i n i t e , some c o u n t a b l e subset CQ of C has the same Q - e s s e n t i a l supremum as C . Then T = sup CQ belongs to T and T i s a Q - e s s e n t i a l l y maximal element of T . However, the f u l l axiom of c h o i c e i s not r e a l l y needed to prove t h i s lemma. Here i s a p r o o f which uses o n l y the p r i n c i p l e of dependent c h o i c e . L e t f be a s t r i c t l y p o s i t i v e F-measurable f u n c t i o n on A such t h a t fdQ < 0 0 . D e f i n e g: T —> [0,°°) by g(T) = (1 - e T ) f d Q L e t TQ e T and l e t M Q = s u p { g ( T ) : T Q <_ T e T} . I f i _> 1 and T Q,...,T ,MQ,...,M have been d e f i n e d , l e t M = s u p { g ( T ) : T <_ T e T} and choose ^ e T such t h a t T <_ T± and g ( T ± ) _> M ± - 2 _ 1 . Then ( T ^ i s an i n c r e a s i n g sequence i n T . L e t . T = l i m T. . Then T e T . I f S e T and i - y o o T <_ S then f o r any i , g(S) <_ <_ g(T ) + 2 _ 1 <_ g(T) + 2" 1; hence g(S) <_ g(T) . Thus S = T Q - a.e. Hence T i s Q - e s s e n t i a l l y maximal i n T . • 8.20. Theorem: L e t u,v be measures on ]R n such t h a t U y, U V a r e p o t e n t i a l s . Suppose a) U y > U V . b) u(Z) <_ v(Z) f o r every B o r e l s e t Z £ {U V = »} . c) yOR n) = v O R n ) , on n >_ 3 . Then t h e r e i s a y - s t a n d a r d ( 8 t ) - s t o p p i n g time T such t h a t y T = v - 95 -P r o o f : We c o u l d prove t h i s i n much the same way as we proved 7.11. For the sake of v a r i e t y though, l e t us g i v e a somewhat d i f f e r e n t p r o o f . L e t T be the s e t of a l l y - s t a n d a r d ( 8 ^ ) - s t o p p i n g times T such t h a t U >_U . Then 0 e T, so T 4 0 . A l s o the l i m i t of any i n c r e a s i n g sequence i n T b e l o n g s to T, by 8.11 ( i f n <_ 2) or 7.9 ( i f n >_ 3 ) . Thus, by 8.19, T c o n t a i n s a P M -e s s e n t i a l l y maximal element T . L e t A = y T . We s h a l l show t h a t A = v . Of c o u r s e , we have o n l y to show t h a t U A <_ U V . L e t A = {U A > U V} . We c l a i m A(A) = 0 . Suppose n o t . Then t h e r e e x i s t s a number c and a bounded open s e t W such t h a t A(H) > 0, where H = W n {U* > c > U V} . L e t S = i n f { t > 0: B I H} . AS v Then S i s A-standard by 8.16, and U >_ U V by 8.17. A l s o , by the s t r o n g Markov p r o p e r t y , A = y , where T' = T + S • 0 . In the case n <_ 2, T' i s y - s t a n d a r d by 8.14. Thus T' e T . Now T <_T' . Hence P P ( T 4 T') = 0 . T h e r e f o r e u T = u T , . But u T ( H ) > 0 w h i l e , by 8.18, y^,, (H) = 0 . T h i s c o n t r a d i c t i o n shows t h a t we must have A(A) = 0 . Thus A v U -<_U A - a . e . I f Z i s a B o r e l p o l a r s e t then - 96 -A(Z) = A(Z n {U X <_ U V}) = A(Z n {U X = 00} n { n X <_ u V}) (by 2.2) = A(Z n {U V = 0 0 } ) <_ y(Z n {U V = »}) (by 6.5) 1 y(z) . Thus by the d o m i n a t i o n p r i n c i p l e , <_UV . (See 4.2 i f n > 3, 4.8 i f n = 2, or 4.13 i f n = 1 ; note t h a t i n the case n <_ 2 we are assuming t h a t uQR n) = vOR n) .) 8.21. Remark: Baxter and Chacon [1] have proved the f o l l o w i n g theorem: " L e t y,v be p r o b a b i l i t y measures on ]R n such t h a t y v U , U are p o t e n t i a l s . Suppose a) U y > U V . b) l i m ( U P ( x ) - U V ( x ) ) = 0 . I I x I I "><J0 c) U V i s f i n i t e and c o n t i n u o u s on TRU . Then t h e r e i s a ( B t ) - s t o p p i n g time T such t h a t y_ = v . I f v has a f i n i t e second moment so does y, and T can be chosen so t h a t I xI I 2dv(x) = |x|| 2dy(x) + 2 n E M ( T ) I f n >_ 3, c o n d i t i o n b) can be dropped." Now by Lemma 5 of Bax t e r and Chacon [1], i f y,v a r e f i n i t e measures on ]R n such t h a t U y , U V are p o t e n t i a l s and U P >_ U V , then - 97 -IxI I^dv (x ) = | x| |^du(x) + 2n U y ( x ) - U V ( x ) d x A l s o , by 8.13, i f T i s y - s t a n d a r d then U y ( x ) - U P T ( x ) d x = E y ( T ) . ( I f n >_ 3, use 6.9 i n p l a c e of 8.13). Thus 8.20 i s a g e n e r a l i z a t i o n of the c i t e d theorem of Baxter and Chacon. In p a r t i c u l a r : I f n >_ 3, we no l o n g e r r e q u i r e y and v to have the same t o t a l mass; moreover, they need not even be f i n i t e measures. C o n d i t i o n b) of the theorem of Bax t e r and Chacon has been c o m p l e t e l y e l i m i n a t e d . C o n d i t i o n c) has been p a r t l y e l i m i n a t e d . No c o n t i n u i t y assumption on U V i s made. A l s o , the f i n i t e n e s s assumption on U V has been r e l a x e d ; f o r example, i t s u f f i c e s t h a t U V be f i n i t e y - a.e. Of cou r s e 8.20 i s not b e s t p o s s i b l e ( u n l e s s n = 1 ) . T h i s i s shown by 7.13, i n the case n = 3. I b e l i e v e the f o l l o w i n g r e s u l t i s "best p o s s i b l e " . 8.22. C o n j e c t u r e : L e t y,v be measures on ]R n such t h a t U y , U V a r e p o t e n t i a l s . Then t h e r e i s a y - s t a n d a r d ( 8 t ) - s t o p p i n g time T such t h a t y^ = v i f and o n l y i f the f o l l o w i n g 3 c o n d i t i o n s a r e s a t i s f i e d : a) U y > U V . b) There i s a B o r e l s e t A £ ]R n such t h a t v(Z) = y(Z n A) f o r every B o r e l p o l a r s e t Z £ R n . c) n > 3, or y ( R n ) = vCR n) . - 9 8 -9. RANDOMIZED STOPPING TIMES, AND ENLARGEMENTS OF PROBABILITY SPACES Suppose y i s the u n i t p o i n t mass a t 0 i n H n , and v i s the u n i t measure on 3Rn which has h a l f o f i t s mass a t 0 and the o t h e r h a l f u n i f o r m l y d i s t r i b u t e d on S = {x e H n : ||x|| = 1 } . I f v = u T f o r some (8^) - s t o p p i n g time T then P°(T>0) = 1 so {0} cannot be a p o l a r s e t , f o r i f i t were we would have P ° ( B T e {0}) = 0 , c o n t r a d i c t i n g v({0}) 4 0 . Thus we can o n l y have v = y^ i f n = 1 . But f o r any n , t h e r e i s an o b v i ous "randomized s t o p p i n g time" T such t h a t v = "y " . Namely, l e t 0 w i t h p r o b a b i l i t y 1 / 2 T w i t h p r o b a b i l i t y 1 / 2 T h i s m o t i v a t e s the f o l l o w i n g d e f i n i t i o n s . 9 . 1 . D e f i n i t i o n : L e t (A,F) and ( X , A ) be measurable spaces. A randomized random v a r i a b l e ( r r v , f o r s h o r t ) i n ( X , A ) (over (A,F)) i s a map x : A p r o b a b i l i t y measures on A , such t h a t f o r e v e r y A i n A , x(')(A) i s F-measurable. I f X i s a t o p o l o g i c a l space, we s h a l l speak s i m p l y o f r r v ' s i n X ; i t w i l l be u n d e r s t o o d t h a t A = B o r e l X , the a - f i e l d g e nerated by the open s e t s of X . 9 . 2 . D e f i n i t i o n : I f (A,F) i s a measurable space and ^t^Q<t«=° i s an i n c r e a s i n g f a m i l y of sub - a - f i e l d s of F then a randomized - 99 -( F t ) - s t o p p i n g time ( ( F ^ J - r s t , f o r s h o r t ) i s a r r v x i n [ O , 0 0 ] o v e r ( A , F ) such t h a t f o r 0 _ < t < ° ° , x ( ' ) ( [ 0 , t ] ) i s F -measurable. (That i s , i n t u i t i v e l y , the c o n d i t i o n a l p r o b a b i l i t y t h a t we s t o p by time t depends o n l y on the i n f o r m a t i o n t h a t we have by time t .) 9.3. Examples: L e t ( A , F ) be a measurable space. L e t ( X , A ) be another measurable space and l e t f : A •+ X . D e f i n e x : A -*• p r o b a b i l i t y measures on A by = 6 f ( _ ) ; t h a t i s , x(u)(A) = 1 ( f ( _ ) ) . F o r any A £ X , x A ( f ( ' ) ) 1 S F - m e a s u r a b l e i f f f X [ A ] e F . Thus x i s a r r v i n ( X , A ) i f f f i s a r v i n ( X , A ) ; t h a t i s , i f f f i s measurable. I n t h i s c ase we c a l l x the r r v a r i s i n g from the r v f . Now l e t ( F ) be an i n c r e a s i n g f a m i l y of sub - a - f i e l d s t 0_t<°° of F , l e t T be a r v i n [ O , 0 0 ] , and l e t x be the r r v i n [0,°°] a r i s i n g from T . For any t e [0,°=) , x ( - ) ( [ 0 , t ] ) i s F t - m e a s u r a b l e i f f {T <_ t} e F • Thus x i s a r s t i f f T i s a s t o p p i n g time. 9.4. L e t ( A , F ) and ( X , A ) be measurable s p a c e s . Then r e l a t i v e t o a measure Q on F , r r v ' s i n ( X , A ) may be more or l e s s i d e n t i f i e d w i t h measures on the p r o d u c t o - f i e l d F ® A , whose p r o j e c t i o n on A i s e q u a l to Q . T h i s i s made - 100 -p r e c i s e by the f o l l o w i n g r e s u l t , Theorem: a) I f x i s a n r r v i n (X ,A) then to x(^) (H(w)) i s F-measurable f o r any H i n F ® A . (Where H(„) = the s e c t i o n of H over _ .) b) I f x i s an r r v i n (X ,A) and Q i s a measure on F , and i f we d e f i n e y on F ® A by y(H) = Q(d_)x(o))(H(_)) then y i s a measure on F ® A and y(F><X) = Q(F) f o r F i n F . c) I f Q i s a s e m i f i n i t e measure on F and A i s c o u n t a b l y generated, and i f x a n d x ' a r e r r v ' s i n (X ,A) which g i v e r i s e to the same measure y on F ® A ( a c c o r d i n g to the fo r m u l a i n p a r t b ) ) then the s e t E = (X r" X M bel o n g s t o F and Q(E) = 0 . d) Suppose Q i s a o - f i n i t e measure on F , y i s a measure on F ® A such t h a t y(F*X) = Q(F) f o r a l l F i n F , A i s c o u n t a b l y g e n e r a t e d , and f o r each F i n F w i t h Q(F) f i n i t e , the measure A •-»• y(A*A) i s i n n e r r e g u l a r w i t h r e s p e c t to a semicompact c l a s s c o n t a i n e d i n A . Then t h e r e e x i s t s a r r v x i n (X ,A) such t h a t y(H) = Q(d_)x(u)(H(_)) - 101 -f o r a l l H e F ® A . Note t h a t x i s Q - e s s e n t i a l l y unique by c ) . (A semicompact c l a s s i s a c o l l e c t i o n C of s e t s , such t h a t c o u n t a b l e £ C , n = 0 i m p l i e s n = 0 f o r some f i n i t e = c0 .) We s h a l l not prove t h i s theorem h e r e . L e t us remark t h a t P a c h l [1] has r e c e n t l y shown t h a t i n d ) , the c o n d i t i o n t h a t A be c o u n t a b l y generated can be dropped, p r o v i d e d we a r e w i l l i n g to have each x ( w ) d e f i n e d o n l y on a sub - a - f i e l d A^ of A , where f o r each H e F ® A , H(co) i s i n A f o r Q -almost a l l co and co * * co >->• x(w) (H(co)) i s Q -measurable, where Q i s the c o m p l e t i o n of Q . I n the same paper, P a c h l showed t h a t the requirement of i n n e r r e g u l a r i t y w i t h r e s p e c t to a semicompact c l a s s i s , i n a sense, n e c e s s a r y . 9.5. L e t us a l s o remark t h a t f o r a s e p a r a t e d c o u n t a b l y generated measurable space (X , A ) , the f o l l o w i n g a r e e q u i v a l e n t : a) Every f i n i t e measure on A i s i n n e r r e g u l a r w i t h r e s p e c t to a semicompact c l a s s (depending on the measure). b) (X , A ) i s B o r e l i s o m o r p h i c to a u n i v e r s a l l y measurable subspace of TR. . c) (X , A ) i s B o r e l i s o m o r p h i c to a u n i v e r s a l l y measurable subspace of a P o l i s h space. d) (X , A ) i s u n i v e r s a l l y measurable i n every c o u n t a b l y s e p a r a t e d measurable space i n which i t i s B o r e l embedded. (A P o l i s h space i s a s e p a r a b l e c o m p l e t e l y m e t r i z a b l e t o p o l o g i c a l space.) - 102 -L e t us say t h a t (X,A) i s a u n i v e r s a l l y measurable space i f f (X,A) i s a measurable space which i s B o r e l i s o m o r p h i c t o a u n i v e r s a l l y measurable subset of 3R . 9.6. P r o p o s i t i o n : L e t (A,F,Q) be a measure space and l e t (F ) . be a r i g h t - c o n t i n u o u s i n c r e a s i n g f a m i l y of sub - a - f i e l d s t U ^  t "^00 of F • Assume Q i s a - f i n i t e on F^ . L e t x be a r r v i n [0,°°] over (A,F) . Then the f o l l o w i n g a r e e q u i v a l e n t : a) There i s an ( F t ) - r s t T such t h a t x = X Q " a.e. f dQ = 0 b) Whenever t e [0,°°), f e L 1 ( Q ) . such t h a t F f o r a l l F e F , and g e C[0,°°] such t h a t g = 0 on [t,°°] , then Q(du>) X(w)(ds)f(_)g(&) = 0 c) For each t e [0,°°) , x ( # ) ( [ 0 , t ] ) i s Q - a.e. e q u a l to an F^-measurable f u n c t i o n . P r o o f : a) => b) I f A e B o r e l [ 0 , t ] , then x ( . ) ( A ) i s F -measurable. b) => c) L e t t e [0,~) and l e t ^ = x(»)([0,t)) . Then \\) h dQ = ^ E(h|F t)dQ E(ip|F t)E(h|F t)dQ E(i|>|F )h dQ for a l l h e L 1(Q) . Hence i|> = E(i^ | Ffc) - 103 -(Note t h a t we can t a l k of c o n d i t i o n a l e x p e c t a t i o n s w i t h r e s p e c t t o F because Q i s a - f i n i t e on ^ t •) Now x ( ' ) ( [ 0 , t ] ) = l i m x ( * ) ( [ 0 , t + 2 ~ k ) ) . U s i n g the r i g h t -k-*>° c o n t i n u i t y o f (F^_) 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 . c) => a) For r r a t i o n a l , 0 < r < 0 0 we can choose f u n c t i o n s A r on A such t h a t : 0 < A < 1 — r — A i s F -measurable r r r < r ' => A < A , — r — r A r = x ( O ( [ 0 , r ] ) Q - a.s. Now f o r each to e A , t h e r e i s a unique p r o b a b i l i t y measure x(to) on [0,°°] such t h a t t e [0,°°) , x ( t o ) ( [ 0 , t ] ) = i n f A (to) . r>t r Then x i s the sought r s t . (The r i g h t c o n t i n u i t y o f (F^) i s used h e r e too.) • 9.7. D e f i n i t i o n : An enlargement o f a measure space (A,F,Q) i s a p a i r c o n s i s t i n g o f a measure space (M, G, R) and a (G,F)-measurable map \p : M + A such t h a t ip(R) = Q ( i . e . R ( ^ _ 1 [ F ] ) = Q(F) f o r a l l F e F) . 9.8. L e t (M,G,R,IJ0 be an enlargement of a a - f i n i t e measure space (A,F,Q) . L e t (X,A) be a u n i v e r s a l l y measurable space, and l e t f be a r v i n (X,A) over (M,G) . - 104 -D e f i n e g : M ->• A x X by g(p) = (Mp) . f (p)) • Then g i s (G, F ® A)-measurable. L e t p = g(R) on F ® A . Then the p r o j e c t i o n of u on A i s ijJ(R) = Q . Thus, by 9.4, t h e r e i s a Q - e s s e n t i a l l y unique r r v x i n (X,A) over (A,F) such t h a t V(H) = Q(d_)x(u)(H(_)) f o r a l l H e F ® A In t h i s way, a r v over an enlargement g i v e s r i s e to a r r v over the o r i g i n a l space. One e a s i l y checks t h a t i f (R ) . i s a _ _eA d i s i n t e g r a t i o n o f R w i t h r e s p e c t to , F , then x(^) = ^R^) f o r Q-a.a. _ e A . 9.9. Any r r v over a measure space a r i s e s , i n the f a s h i o n d e s c r i b e d i n 9.8, from a r v i n an enlargement. L e t us d e s c r i b e e x p l i c i t l y why t h i s i s s o. L e t (A,F) , (X,A) be measurable spaces and l e t x be a r r v i n (X,A) over (A,F) . L e t Q be a measure on F > and l e t us show how x a r i s e s from a r v over an enlargement of (A.F,Q) . W e l l , l e t M = A x x G = F ® A R(G) = Q(d_))xCw) (G(_)) f o r G e G = p r o j e c t i o n o f M on f = p r o j e c t i o n of M on X . Then (M,6,R,^) i s an enlargement of (A,F,Q) , f i s an r v i n (X,A) over (M,6) , and x a r i s e s from f i n the manner d e s c r i b e d i n 9.8. - 105 -9.10. Given two r r v ' s , they may be r e a l i z e d as r v ' s i n a common enlargement, by a p r o c e s s s i m i l a r to t h a t d e s c r i b e d i n 9.9. However, the j o i n t d i s t r i b u t i o n o f the two r v ' s i s not determined by the two r r v ' s a l o n e ; i t depends on the enlargement. T h i s i s why we cannot j u s t work w i t h r r v ' s i n g e n e r a l , b u t must a l s o c o n s i d e r enlargements. 9.11. D e f i n i t i o n . A f i l t e r e d measurable space i s a system ( A , F , F T ) » where ( A , F ) i s a measurable space and ^t^0<t<°° ^ S an i n c r e a s i n g f a m i l y o f sub - a - f i e l d s of F . A f i l t e r e d measure space i s a system ( A , F , F ,Q) , where ( A J F J F ^ ) i s a f i l t e r e d measurable space and Q i s a measure on F which i s a - f i n i t e on FQ . (The l a s t assumption i s made to ensure t h a t we can c o n s i d e r c o n d i t i o n a l e x p e c t a t i o n s w i t h r e s p e c t to any F .) 9.12. D e f i n i t i o n . An enlargement o f a f i l t e r e d measure space ( A , F , F T , Q ) i s a f i l t e r e d measure space (M,G,G^,R) t o g e t h e r w i t h a map ip such t h a t (M,G,R,ijj) i s an enlargement of the measure space (A »F,Q) and such t h a t i n a d d i t i o n , \JJ i s (G^, F^)-measurable f o r 0 £ t < °° . An o p t i o n a l enlargement of ( A , F , F T , Q ) i s an enlargement (M,G,Gt,R,iJ;) such t h a t f o r each t e [0,«) and each g e l^CM.G ,R) we have E(g|*,F) = E(g|i|;,F ) Q-a.s. We remark t h a t o p t i o n a l enlargements a r e e s s e n t i a l l y the " d i s t r i b u t i o n a l e n largements" of Bax t e r and Chacon [ 3 ] . These a u t h o r s i n t r o d u c e d t h i s n o t i o n to f a c i l i t a t e the d i s c u s s i o n o f randomized time changes. - 106 -Next we prove a r e s u l t which c l a r i f i e s the meaning o f the n o t i o n o f o p t i o n a l enlargement. 9.13. Theorem: L e t ( A , F , , Q ) be a f i l t e r e d measure space such t h a t (F ) i s r i g h t - c o n t i n u o u s and F = a(uF ) . L e t (M , G , G ,R,i|0 t t t t be an enlargement of (A,F,F._,Q) . Then the f o l l o w i n g a r e e q u i v a l e n t : a) (M , G , G t,R , i | j ) i s an o p t i o n a l enlargement. b) E v e r y ( G ^ ) - s t o p p i n g time T g i v e s r i s e , as i n 9.8, to a randomized ( F t ) - s t o p p i n g time T . c) Whenever (X i s a m a r t i n g a l e over (A , F F Q) t 0<t<°° t (Y^) E ( X t ° i | i ) i s a m a r t i n g a l e over (M , G , G t,R) . d) Whenever (E,E) i s a measurable space and ( X _ ) „ ' ' r x t 0<t < o° i s a Markov p r o c e s s i n (E,E) over ( A , F , F t > Q ) then (Y^) = (X^oijj) i s a Markov p r o c e s s over (K,C,G^,R) . Moreover, w i t h r e g a r d to a) => d ) , i f (^t) n a s a t r a n s i t i o n f u n c t i o n , then (Y^) has the same t r a n s i t i o n f u n c t i o n . P r o o f : a) => b) . L e t x be the r r v i n [0,°°] a r i s i n g from T as d e s c r i b e d i n 9.8. L e t t e [O, 0 0) . Then f o r any F e F , Q ( d _ ) x ( u ) ( [ 0 , t ] ) - = R ( ^ _ 1 [ F ] n { T < t}) . ' F Thus x ( * ) ( [ 0 , t ] ) = E ( l { _ < t ^ U , F ) , which i s Q - a.e. e q u a l to an F^-measurable f u n c t i o n by o p t i o n a l i t y . Hence, by 9.6, x i s Q - a.e. e q u a l to an ( F ^ ) - r s t T . b) => a) . L e t t e [0,°°) and l e t G e G . By c o n s i d e r i n g the ( G t ) - s t o p p i n g time T = t 1 £ + 0 0 L ^ . » one f i n d s t h a t E(l_|i | > ,F ) i s - 107 -F^-measurable mod Q . B e f o r e p r o c e e d i n g to the p r o o f of the r e m a i n i n g i m p l i c a t i o n s , we make the f o l l o w i n g o b s e r v a t i o n : L e t 0 <_ s < t < 0 0 . Then (*.) and (**) below a r e e q u i v a l e n t . (*) F o r any g i n ^(M.G^R) , E ( g | * , F ) = E(g|i |»,F ) Q - a.e. L S (**) F o r any f i n i-^A . F ,Q) , E ( f ° i H G ) = E ( f | F W • s s T h i s can be prov e d by an " o r t h o g o n a l i t y argument" such as was used i n the p r o o f o f (b => c) of 9.6. The e q u i v a l e n c e (a <=> c) f o l l o w s immediately from t h i s t o g e t h e r w i t h F = a(uF ) . t t a) => d ) . L e t f be a n o n - n e g a t i v e E-measurable f u n c t i o n on E L e t 0 <_ s < t < » . Then E ( f ( Y f c ) | 6 g ) = E ( f ( X t W | G , ) = E ( f ( X t ) | F s ) o ^ (by (*) => (**)) = E ( f ( X J | X - 1 ( E ) W ((X,) i s Markov) t s t = E ( f ( Y . ) | Y _ 1 ( E ) ) . t s Thus ) i s Markov. A s i m i l a r c a l c u l a t i o n shows t h a t any t r a n s i t i o n f u n c t i o n f o r ( X t ) i s a l s o a t r a n s i t i o n f u n c t i o n f o r ( Y t ) . - 107a -d) => a ) . L e t (E,E) be a measurable space and (X^) a p r o c e s s i n (E,E) ov e r (A,F,Ft,Q) such t h a t F = X _ 1(E) f o r 0 < t < » . t t -Then ( X t ) i s Markov. Hence, by h y p o t h e s i s , (Y ) = (X t°4>) i s a l s o Markov. L e t 0 <_ s < t < °° and l e t f be a no n - n e g a t i v e F^-measurable f u n c t i o n on A . S i n c e = X ^(E) , f = h ( X t ) f o r some no n - n e g a t i v e E-measurable f u n c t i o n h on E . Then E(f°^|G ) = E(h(Y )|G ) s t 1 s = E(h(Y )|Y _ 1(E)) ((Y,) i s Markov) t s t = E ( h ( X t ) | X 8 1 ( E ) ) o ^ = E ( h ( X t ) | F g ) o ^ . Thus, u s i n g (**) => (*) , and the f a c t t h a t F = o(uF ) , we t t see t h a t the enlargement i s o p t i o n a l . Of c o u r s e , we must show t h a t t h e r e i s a measurable space (E,E) and a p r o c e s s (X ) i n (E,E) - 108 -such t h a t F = X~ X ( E ) f o r a l l t . L e t E = Q x [0,~) E = the a - f i e l d o f ( F t ) - p r o g r e s s i v e l y measurable s e t s and l e t X t ( _ ) = (_,t) f o r _ e A , t e [0,°°) . T h i s does the t r i c k . • Remark: The r i g h t - c o n t i n u i t y o f ('F ) i s not needed f o r the e q u i v a l e n c e of a ) , c ) , and d) i n the above theorem; a l s o , the assumption t h a t F = o ( u F ) i s not needed f o r the e q u i v a l e n c e of t t a) and b ) . 9.14. Observe t h a t i f (M,G,G^,R,u)) i s an o p t i o n a l enlargement of (A, F , F Q) then ( M , G , G ,R,iJ.) i s an o p t i o n a l enlargement of ( A . F . F . + . Q ) • 9.15. P r o p o s i t i o n . L e t ( A , F , F , Q ) be a f i l t e r e d measure space, where (^t) 1 S r i g h t - c o n t i n u o u s , and l e t X be a r r v i n [O, 0 0] over ( A , F ) . L e t M = A x [0,»] G = F ® A , where A = B o r e l [0,°=] G F ® A f o r 0 < t < 0 0 , where t t t -A = {A e A : A n (t,»] = 0 or (t,»]} R ( G ) = Q(dw ) x(u)(G(u)) f o r G e G and l e t \p = p r o j e c t i o n o f M on A . Then the f o l l o w i n g a r e e q u i v a l e n t . a) There i s an ( F ) - r s t x such t h a t X = T Q-a.e. b) ( M , G , G ,R,IJJ) i s an o p t i o n a l enlargement of (A , ^ , ^ t » Q ) • c) ( M , G , G T + , R , i J ; ) i s an o p t i o n a l enlargement of ( A , F , F t , Q ) . - 109 -P r o o f : C l e a r l y (M , G , G t,R,iJj) i s an enlargement of ( A , F > F t » Q ) • As (F ) i s r i g h t - c o n t i n u o u s , the e q u i v a l e n c e o f b) and c) f o l l o w s from 9.14. b) => a) . L e t T = p r o j e c t i o n o f M on [0,°°] . Then {T <_ t} = A x [ 0 , t ] f o r 0 _< t < °° , so T i s a ( G ^ - s t o p p i n g time. Hence, by (a => b) of 9.13, t h e r e i s an (F ) - r s t T such t h a t R ( ^ _ 1 [ F ] n T _ 1 [ A ] ) = Q (dto) x (to) (A) f o r F e F and A e A . But then x = T Q - a.e. a) => b) . One e a s i l y checks t h a t i f t e [0,°°) and G e G then T(»)(G) i s F -measurable and E ( l J i J > . F ) = T(»)(G) Q - a.e. • t G Note t h a t i n the above p r o o f , T i s r e a l i z e d as the ( G ^ - s t o p p i n g time T i n the o p t i o n a l enlargement (M , G , G t,R,i(j) . In 9.18 below, we d e s c r i b e another way of r e a l i z i n g x i n an o p t i o n a l enlargement. 9.16. Lemma: L e t y be a p r o b a b i l i t y measure on [0,°°] , and l e t m be Lebesgue measure on the B o r e l s e t s of (0,1) . Then t h e r e a r e unique maps f , g : (0,1) ->• [0,°°] such t h a t : a) f and g a r e i n c r e a s i n g b) f i s l e f t - c o n t i n u o u s c) g i s r i g h t - c o n t i n u o u s d) y = f(m) = g(m) Moreover, i f h : (0,1) •+ [O, 0 0] i s i n c r e a s i n g and h(m) = y then f <_ h _< g and f o r a l l u i n (0,1) we have f (u) = h(u-) and g(u) = h(u+) . - n o -Sketch o f P r o o f : F o r 0 < u < 1 , l e t f ( u ) = sup{t e [0,»] : y ( [ 0 , t ] ) < u} (where sup 0 = 0 ) and l e t g(u) = i n f { t e [0,°°] : y ( [ 0 , t ) ) > u} . • 9.17. D e f i n i t i o n : L e t ( A , F , F T , Q ) be a f i l t e r e d measure space, where (F ) i s r i g h t - c o n t i n u o u s . L e t (X,A,m) be a measure space w i t h m(X) = 1 . L e t M = A * X G = F ® A G = (F F C ® A) (= n F 9 A) e>0 t + e f o r 0 _< t < °° R = Q <_. m i> = p r o j e c t i o n o f M on A . Then c l e a r l y (M,G,Gt,R,^) i s an o p t i o n a l enlargement o f (A,F,F^.Q ) ; w e c a l l i t the p r o d u c t enlargement o f ( A , F , F T , Q ) by (X,A,m) . 9.18. P r o p o s i t i o n : L e t ( A . F , F T , Q ) be a f i l t e r e d measure space, w i t h (F ) r i g h t - c o n t i n u o u s . L e t (X,A,m) be a measure space w i t h m(X) = 1 , and l e t ( H , G , G ^ , R , M ) ) be the p r o d u c t enlargement o f ( A , F , F T , Q ) by (X,A,m) . Then: a) I f T i s a ( G t ) - s t o p p i n g time then T(',x) i s an ( F t ) - s t o p p i n g time f o r each x e X , and T = T(',m) i s a randomized (F ) - s t o p p i n g time. - I l l -b) Suppose X = (0,1) , A = B o r e l X , and m = Lebesgue i measure on A . I f T i s a randomized ( F ^ - s t o p p i n g time, and i f f o r each to i n A we l e t T(to,») be the unique i n c r e a s i n g l e f t - c o n t i n u o u s map f of (0,1) i n t o [0,°°] such t h a t f(m) = x(to) (see 9.16) then x i s a ( G ^ - s t o p p i n g time. P r o o f : a) { T < t } € ' F ® A . Hence {T(«,x) < t} = {to e A: (to,x) e {T < t}} e F , and T ( * ) ( [ 0 , t ) ) = m({x e X : T(»,x) < t}) i s F t - m e a s u r a b l e . The d e s i r e d c o n c l u s i o n s now f o l l o w from the r i g h t - c o n t i n u i t y o f (F' ) . b) T ( t o,u) = sup{t e [0,~] : x ( t o ) ( [ 0 , t ] ) < u} . (Here sup 0 = 0.) As x ( t o ) ( [ 0 , t ] ) i s r i g h t - c o n t i n u o u s i n t , T(to,u) = sup{r e [0,°°) : r i s r a t i o n a l and i ( t o ) ( [ 0 , r ] ) < u} . Thus T(to,u) < t i f f t h e r e i s a r a t i o n a l r e [0, t) such t h a t x ( t o ) ( [ 0 , r ] ) _> u . That i s , {T < t} = r £ $ t ) ^ , u ) e M : u < T (to) ( [ 0 , r ])} . r r a t i o n a l As x(»)([0,r]) i s F^-measurable, {T < t} e F ^ ® A . Thus T i s a (G ) - s t o p p i n g time. 9.19. L e t (A,F,Q) be a measurable space and l e t E be a s e p a r a b l e normed sp a c e . I f Z i s a weak*-dense su b s e t of the u n i t b a l l of the * t o p o l o g i c a l d u a l E of E then the s m a l l e s t o - f i e l d on E making a l l the elements of Z measurable c o i n c i d e s w i t h the - 112 -a - f i e l d generated by the norm-open s u b s e t s o f E . Thus t h e r e i s no ambiguity about what we mean by an F-measurable f u n c t i o n from A i n t o E . We w r i t e L X(A , F,Q;E) f o r the space of a l l f u n c t i o n s f : A E such t h a t f i s F-measurable and f(w)||Q(du.) < 0 0 , equipped w i t h the seminorm f •* | |f | | = | | |f (oi) | |Q(d_) . We wish to d e s c r i b e the d u a l o f t h i s space. Now E need not be s e p a r a b l e , and E may not be the d u a l * o f E , so we have to be c a r e f u l about what we mean by a measurable * E - v a l u e d f u n c t i o n . We s h a l l say t h a t a f u n c t i o n g : A -»• E i s weak * - F-measurable i f f _ H- <x,g((~)> i s F-measurable f o r each x e E . In t h i s c a s e, u s i n g the s e p a r a b i l i t y o f E , we can show t h a t the n u m e r i c a l f u n c t i o n u> •->• | |g(aj)| | i s F-measurable. * A l s o , i f f : A -> E i s F-measurable and g : A E i s F-measurable, then _ •->• <f (_), g(t_)> i s F-measurable. T h i s can be p r o v e d by a p p r o x i m a t i n g f by c o u n t a b l y - v a l u e d F-measurable f u n c t i o n s . A l s o , i f g, g' : A ->• E a r e weak*-F-measurable then {g 4 g'} e F , s i n c e i t i s e q u a l to {||g - g'|| = 0} . Now , g d e f i n e s a c o n t i n u o u s l i n e a r f u n c t i o n a l T on L X(A , F,Q;E) by the f o r m u l a <f,r> = < f ( _ ) , g(_)>Q(d_) S i m i l a r l y g' d e f i n e s T' . I f Q i s s e m i f i n i t e , then - 113 -||r|| = the Q - e s s e n t i a l supremum o f ||g(»)|| , and r = I" i f f Q({g t g')) = 0 . On the o t h e r hand, i f Q i s c - f i n i t e and i f r i s any c o n t i n u o u s l i n e a r f u n c t i o n a l on l?~(1\,T,Q;E) then t h e r e i s a weak*-F-measurable f u n c t i o n g :A E such t h a t ||g(w)|| 1 I M l f o r a 1 1 w i n A f and <f,T> = < f ( „ ) , g(_)>Q(d_) f o r a l l f i n _ 1(A,F,Q;E) We s h a l l not prove t h i s r e s u l t h e r e . See Ionescu T u l c e a and I o n e s c u T u l c e a [ 1 ] , V I I . 4 , f o r a p r o o f i n the more g e n e r a l case i n which E i s not assumed s e p a r a b l e . In t h i s case the l i f t i n g theorem i s used, so one assumes Q to be complete. I n our case, i n which E i s s e p a r a b l e , t h i s i s u n n e c e s s a r y . ( L e t us a l s o remark t h a t i f E i s not assumed s e p a r a b l e , the F-measurable f u n c t i o n s i n t o E a r e d e f i n e d to have s e p a r a b l e range.) 9.20. N o t a t i o n : I f X and Y are s e t s and f , g a r e r e a l - v a l u e d f u n c t i o n s on X, Y r e s p e c t i v e l y , then f ® g w i l l denote the f u n c t i o n on X * Y d e f i n e d by (f®g)(x,y) = f ( x ) g ( y ) . 9.21. L e t (A,^,Q) be a o - f i n i t e measure space, and l e t K be a compact m e t r i z a b l e s p a c e . Then C ( K ) , the space of c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s on K , equipped w i t h the supremum norm, i s a s e p a r a b l e normed sp a c e . Thus C(K) can p l a y the r o l e o f E i n 9.19. - 114 -I f X i s a r e a l - v a l u e d f u n c t i o n on A x K such t h a t X(co,») i s c o n t i n u o u s on K f o r each to i n A and X(»,p) i s F-measurable f o r each p i n K (such an X i s c a l l e d a C a r a t h e o -dory f u n c t i o n ) then X i s F ® ( B o r e l K)-measurable and hence to •-»• X(to,») i s an F-measurable C ( K ) - v a l u e d f u n c t i o n . On the o t h e r hand, i f f i s an F-measurable C ( K ) - v a l u e d f u n c t i o n then f(»)(p) i s an F-measurable r e a l - v a l u e d f u n c t i o n f o r each p i n K , s i n c e the p o i n t - e v a l u a t i o n s a r e c o n t i n u o u s l i n e a r f u n c t i o n a l s on C(K) . Thus F-measurable C ( K ) - v a l u e d f u n c t i o n s on A can be i d e n t i f i e d w i t h Caratheodory f u n c t i o n s on A x K . Under t h i s i d e n t i f i c a t i o n , l?~(A,F,Q;C(K)) becomes i d e n t i f i e d w i t h the s e t of Car a t h e o d o r y f u n c t i o n s X on A x K s a t i s f y i n g sup |x(»,p) |dQ < °° . ' p e K I f f e L 1 ( A } F J Q ) and g e C(K) then f o g be l o n g s to t h i s s e t . L e t RRV(A,F,Q;K) be the space o f randomized random v a r i a b l e s i n K over (A,F) , equipped w i t h the weak t o p o l o g y i n d u c e d by the maps Q(dto) X ( t o ) ( d p ) h ( t o ) ( p ) where h ranges over L 1(A,F,Q;C(K)) . To save w r i t i n g , l e t us denote t h i s space by j u s t RRV, f o r the remainder o f 9.21. I f x e R R V > then h >->- <x>h> i s a c o n t i n u o u s p o s i t i v e l i n e a r - 115 -f u n c t i o n a l on L. 1(A , F,Q;C(K)) s a t i s f y i n g <x,f«l> = fdQ f o r a l l f e L^"(A,F,Q) . On the o t h e r hand, any f u n c t i o n a l w i t h t h e s e p r o p e r t i e s a r i s e s from a ( Q - e s s e n t i a l l y unique) x b e l o n g i n g to RRV. Thus { < X » # > : X e RRV} i s a weak*-closed s u b s e t of the u n i t 1 * b a l l of L (A , F,Q;C(K)) , and so i s weak*-compact by the Banach-A l a o g l u theorem. I t f o l l o w s t h a t RRV i s a compact space. A l s o , i f H has dense l i n e a r span i n L*~(A, F,Q;C(K)) then the maps o f the form X •->• <X>h> (heH) i n d u c e the o r i g i n a l t o p o l o g y of RRV. Hence i f F i s c o u n t a b l y generated mod Q, RRV i s p s e u d o - m e t r i z a b l e (and two elements of RRV a r e z e r o d i s t a n c e a p a r t i f f they a r e e q u a l Q - a . e . ) . Now the elements of any c o u n t a b l e s u b s e t of RRV are a l l randomized random v a r i a b l e s over ( A , F ' ) where F ' i s some c o u n t a b l y generated sub - o - f i e l d of F . I t f o l l o w s t h a t RRV i s s e q u e n t i a l l y compact, even when i t i s not p s e u d o - m e t r i z a b l e . 9.22. L e t us g i v e an example o f a sequence of random v a r i a b l e s which converges to a randomized random v a r i a b l e which i s not a random v a r i a b l e . L e t A = [0,1) F = B o r e l A Q = Lebesgue measure on F K = [0,1] . F o r i = 1,2,..., l e t X^ be the c h a r a c t e r i s t i c f u n c t i o n o f u [ ^ r - , ^j) . Then each X. i s a r v i n K over (A , F ) ; j - 1 2 1 2 1 1 j odd - 116 -l e t be the c o r r e s p o n d i n g r r v . L e t x be the r r v i n K over (A,F) d e f i n e d by Then u s i n g the f a c t t h a t the bounded c o n t i n u o u s f u n c t i o n s are dense i n 9.23. Now here i s a v e r s i o n o f a r e s u l t due to Bax t e r and Chacon [ 2 ] . The r e a d e r may a l s o f i n d the review o f t h i s a r t i c l e , by Meyer [ 4 ] , to be e n l i g h t e n i n g . Theorem. L e t (A,F,F ,Q) be a f i l t e r e d measure space, w i t h (Ffc) r i g h t - c o n t i n u o u s . L e t RST = RST(A,F,F t >Q) be the space o f randomized ( F ^ ) - s t o p p i n g t i m e s , endowed w i t h the topo l o g y i t i n h e r i t s as a subspace o f RRV = RRV(A,F,Q; [0,°°]) . Then: a) RST i s compact and s e q u e n t i a l l y compact. b) I f (x.) i s a n e t c o n v e r g i n g to x i n RST, and i f (Z ) t'0<t< CO i s a r e a l - v a l u e d c a d l a g q u a s i - l e f t - c o n t i n u o u s p r o c e s s s a t i s f y i n g sup |z | dQ < 0<t<°° 00 t h e n <x. ,Z> -»• <x,Z> ; t h a t i s , Q(du>) x i ( _ ) ( d t ) Z t ( _ ) - 117 -P a r t i a l P r o o f : a) S i n c e (F^) i s r i g h t - c o n t i n u o u s , we can a p p l y (b => a) of 9.6 to c o n c l u d e t h a t i f x b e l o n g s to the c l o s u r e of RST i n RRV then t h e r e i s a T i n RST such t h a t x = t Q - a.e. Thus RST i s compact and s e q u e n t i a l l y compact because RRV i s . (See 9.21). b) We remark t h a t " c a d l a g " stands f o r " c o n t i n u a d r o i t avec des l i m i t e s a gauche" ( r i g h t c o n t i n u o u s w i t h l e f t l i m i t s ) . I f ( Z t ) i s c o n t i n u o u s then the d e s i r e d c o n c l u s i o n f o l l o w s immediately from the d e f i n i t i o n o f the t o p o l o g y of RRV, and does not depend on the f a c t t h a t the T ^ ' s a r e randomized s t o p p i n g t i m e s . F o r the g e n e r a l case, i n which (Z^~) i s merely q u a s i - l e f t c o n t i n u o u s , the f a c t t h a t the T ^ ' s a r e randomized s t o p p i n g times, r a t h e r than a r b i t r a r y randomized random v a r i a b l e s i n [O, 0 0] , i s c r u c i a l . (To see t h i s , j u s t c o n s i d e r the non-randomized case.) S i n c e we s h a l l be concerned o n l y w i t h c o n t i n u o u s p r o c e s s e s , we r e f e r the r e a d e r to B a x t e r and Chacon [2] or Meyer [ 4 ] . 9.24. F o r the moment, l e t M denote the space o f measures y on [0,°°) s a t i s f y i n g y({0,°°)) £ 1 , endowed w i t h i t s u s u a l vague t o p o l o g y . M i s a compact m e t r i z a b l e space. F o r each y i n M , l e t F be the d i s t r i b u t i o n f u n c t i o n o f y : u F ( t ) = y ( [ 0 , t ] ) (0<t<») . The map y — * F t | i s a 1-1 map of M onto the s e t of i n c r e a s i n g - 118 -r i g h t - c o n t i n u o u s maps o f [O, 0 0) i n t o [0,1] . I f ( P ^ ) i s a sequence i n M and y e M then y. ->• y i n M i f f F ( t ) -> F ( t ) i u± y f o r each t i n [0,°°) a t which F^ i s c o n t i n u o u s , as i s shown i n most any s t a n d a r d t e x t on p r o b a b i l i t y t h e o r y . One can show t h a t t h i s r e s u l t h o l d s f o r n e t s as w e l l as sequences. Indeed, i f f o r each y e M we l e t U be the c o l l e c t i o n o f s e t s of the form y { v e M : |F ( t i ) - F v ( t i ) | < e f o r i = 0,...,k} where e > 0 , k e IN , and t - , . . . , t , a r e c o n t i n u i t y p o i n t s of F , O k y then (U ) w c o n s t i t u t e s a neighbourhood base f o r a H a u s d o r f f y yeM t o p o l o g y on M which i s weaker than the g i v e n t o p o l o g y o f M and which t h u s , by compactness, must be i d e n t i c a l to the g i v e n t o p o l o g y o f M . (Note t h a t we do not c l a i m t h a t the elements o f U a r e open — y o n l y t h a t they are neighbourhoods of y .) Now l e t H be the s e t of a l l i n c r e a s i n g r i g h t - c o n t i n u o u s maps o f [0,°°) i n t o [0,°°] . S i n c e [0,°°] may be i d e n t i f i e d w i t h [0,1] by means o f an o r d e r - p r e s e r v i n g homeomorphism, H may be made i n t o a compact m e t r i z a b l e space i n a n a t u r a l way. I f (IK) i s a n e t i n H and h e H , then h -»• h i n H i f f h ^ t ) h ( t ) f o r each t i n [0,°°) at which h i s c o n t i n u o u s . Now any h e H i s c o n t i n u o u s except a t c o u n t a b l y many t ' s i n [0,°°) . Thus i f (h^) i s a sequence i n H c o n v e r g i n g to h e H then h^ h p o i n t w i s e almost everywhere w i t h r e s p e c t to Lebesgue measure on [0,°°) . I t f o l l o w s t h a t i f <j> i s any bounded c o n t i n u o u s f u n c t i o n on [0,°°] and f any Lebesgue i n t e g r a b l e f u n c t i o n on [O, 0 0) then - 119 -f ( t ) 4 > ( h ( t ) ) d t i s a co n t i n u o u s r e a l - v a l u e d f u n c t i o n on H U s i n g the r i g h t - c o n t i n u i t y o f the h's we can deduce from t h i s t h a t f o r any t e [O, 0 0) , the e v a l u a t i o n map h i — > h ( t ) i s a B o r e l f u n c t i o n on H ; i n d e e d i t i s the p o i n t w i s e l i m i t of a sequence o f c o n t i n u o u s f u n c t i o n s on H . Now c o u n t a b l y many o f these e v a l u a t i o n maps s u f f i c e to s e p a r a t e the p o i n t s o f H . I t f o l l o w s from t h i s t h a t H = B o r e l H i s the s m a l l e s t o - f i e l d on H making a l l the e v a l u a t i o n maps measurable. F o r each t e [0,°°] l e t H be the a - f i e l d o f s u b s e t s of H generated by the s e t s of the form {h e H : h(a) e E} where a e [0,°°) and E e B o r e l [ 0 , t ] . C l e a r l y (H^) i s an i n c r e a s i n g f a m i l y of c o u n t a b l y generated sub-o-f i e l d s o f H and H = H . 9.25. D e f i n i t i o n . L e t (A,F,F ) be a f i l t e r e d measurable space and l e t H be as i n 9.24. A r i g h t - c o n t i n u o u s (F )-time change i s a map T : A H such t h a t f o r each a e [0,») , T(»)(a) i s an" ( F ^ ) - s t o p p i n g time. Observe t h a t T : A ->• H i s a r i g h t - c o n t i n u o u s (F )-time change i f f f o r a l l t e [0,») and a l l A e H , T _ 1 [ A ] e F f c . (Here Hfc i s as i n 9.24.) 9.26. P r o p o s i t i o n . L e t (A,F,F ,Q) be a f i l t e r e d measure space, ' where (F ) i s r i g h t - c o n t i n u o u s . L e t H, H, be as i n 9.24. L e t X be a r r v i n H ov e r (A,F) . A l s o , l e t M = A x H G = F ® H - 120 -( G t ) = ( ( F t ® H t ) + ) R(G) = Q(d_ ) x(u)(G(u))) f o r G e G ij. = p r o j e c t i o n o f M on A . Then the f o l l o w i n g a r e e q u i v a l e n t : a) (M ,G,G ,R,iJ.) i s an o p t i o n a l enlargement of (A ,F,F._,Q) b) F o r a l l t e [0,») and a l l A e H , x ( * ) ( A ) i s F f c -measurable mod Q . c) Whenever t e [0,°°) , k e ]N , a Q , . . . , a k e [0,°°) , e £ (0,°°) , (f>i e C([0,°°)) such t h a t $± = 0 on [t,°°) f o r i = 0,...,k , g i s the f u n c t i o n on H d e f i n e d by k g(h) = n F a.+e l <j>. ( h ( s ) ) d s ] , i=0 * U. I and f £ L ( A , F , Q ) such t h a t then fdQ = 0 f o r a l l F £ F , Q(dui) x(to)(dh ) f (u)g(h) = 0 P r o o f : F i r s t note t h a t ij. i s (G, F)-measurable, (G^, F^)-measurable f o r a l l t , and ^(R) = Q . a) => b) I f A eH then A x A e G f c . A l s o , E (_ A > < A | i j ; , F) = X(«)(A) Q - a.e. b) => c) F o r each i , = 0 on [t,°°) so h <—*• i(;^(h(s)) i s H^-measurable (i n d e e d H -measurable) f o r each s . I t f o l l o w s t h a t g i s H -measurable. Thus X(»)(dh)g(h) i s e q u a l Q - a.e. - 121 -to an F^-measurable f u n c t i o n . Hence Q(dto)f (to) x ( c o ) ( d h ) g ( h ) = 0 c) => b) F i r s t , f o r any g o f the s o r t d e s c r i b e d i n c) we can c o n c l u d e t h a t x ( * ) ( d h ) g ( h ) i s e q u a l Q - a.e. to an F -measurable f u n c t i o n . Then ( l e t t i n g e -+ 0) we f i n d t h a t whenever t € tO, 0 0) , k e U , a ^ . . . , ^ e [0,°°) and \Ji e C([0,°°)) such t h a t \\>. = 0 on [ t , 0 0 ) f o r i = 0,...,k , then the map x(co)(dh) II i(j.(h(a.)) i s F -measurable mod Q . From t h i s i=0 1 1 * we can con c l u d e t h a t i f t e [0,°°) , A e H , and E > 0 then x(«)(A) i s F^ +^-measurable mod Q . Then u s i n g the r i g h t - c o n t i n u i t y o f (F ) 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 . b) => a) By a monotone c l a s s argument, one f i n d s t h a t i f B e F^ ® H then co I—• x (w) (B (co)) i s F^-measurable mod Q . U s i n g the r i g h t - c o n t i n u i t y o f (^ ) one f i n d s t h a t t h i s c o n c l u s i o n i s s t i l l t r u e i f B i s o n l y i n G . But f o r any B e G , X ( 0 ( B ( . ) ) = E ( 1 B U , F ) Q - a.e. 9.27. Now he r e i s a v e r s i o n o f a r e s u l t due to Bax t e r and Chacon [3] Theorem. L e t ( A »F , F t ,Q ) be a f i l t e r e d measure space, where ) i s r i g h t - c o n t i n u o u s . L e t H, H, H be as i n 9.24, and l e t M, G, G^ be as i n 9.26. L e t R be the s e t of measures R on G such t h a t (M ,G,G t»R , i|0 i s an o p t i o n a l enlargement o f (A ,F,F ,Q) . L e t R be g i v e n the weak t o p o l o g y i n d u c e d by the maps of the form - 122 -f ( u ) ( h ) R ( d _ , d h ) where f e L X(A,F,Q;C(H)) . Then R i s compact and s e q u e n t i a l l y compact. P r o o f : L e t RRV denote the compact, s e q u e n t i a l l y compact, t o p o l o g i c a l space RRV(A,F,Q;H) , as d i s c u s s e d i n 9.21. We can do t h i s , s i n c e H i s compact and m e t r i z a b l e . L e t S be the s e t of measures S on G whose p r o j e c t i o n on A i s Q , and l e t S be t o p o l o g i z e d a n a l o g o u s l y to R . L e t d> : RRV -»- S be d e f i n e d by *(X)(G) = Q(d_)x(o>) (G(oj)) Then d> i s a c o n t i n u o u s map o f RRV onto S . I f x> x ' e RRV then <Kx) = M x ' ) i f f X = x ' Q - a.e. C l e a r l y S i s H a u s d o r f f . Thus d> maps c l o s e d s u b s e t s o f RRV onto c l o s e d s u b s e t s o f S . Now R £ S and, s i n c e <|> i s onto , R = <f>[d> X [ R ] ] . Thus t o complete the p r o o f , i t s u f f i c e s to show t h a t d) X [ R ] i s c l o s e d i n RRV. But by (a<=>b) of 9.26, 4'~ 1[R] = " X e RRV : f o r a l l t e [0,°°) and a l l A" e H . x ( * ) ( A ) i s e q u a l Q - a.e. to an F -measurable f u n c t i o n , t and t h i s s e t i s c l o s e d i n RRV, by (b<=>c) o f 9.26. (The f u n c t i o n s g of the form c o n s i d e r e d i n 9.26(c) a r e c o n t i n u o u s on H , as p o i n t e d out i n 9.24.) - 123 -Remark: In view of (a<=>b) of 9.25, and the observation following the definition in 9.24, R may be regarded as the set of "randomized right-continuous (F^)-time changes" (where we identify pairs of time changes which are equal Q - a.e.). Thus from any sequence of right-continuous (F^)-time changes, we may extract a subsequence converging (in the sense explained above) to a randomized right-continuous (F^)-time change. - 124 -10. EMBEDDING MEASURES IN BROWNIAN MOTION IN A GREEN REGION, USING RANDOMIZED STOPPING TIMES. 10.1. Throughout t h i s s e c t i o n , D w i l l be a Green r e g i o n i n R n w i t h Green f u n c t i o n G , and R w i l l be i n f { t >_ 0 : I D} . I f y i s a measure i n D and T i s a randomized ( 8 t ) - s t o p p i n g time, then y_ w i l l denote the measure on B o r e l D d e f i n e d by V_(A) = P P(du.)x(„) ({t e [0,R(o))) : B (ID) e A}) Note t h a t i f T a r i s e s from a genuine ( B ^ - s t o p p i n g time T then P T = V- where y_(A) = P M ( B _ e A, T < R) . More g e n e r a l l y , i f (X,A,m) i s a measure space w i t h m(X) = 1 , and i f T a r i s e s as i n 9.18(a) from a s t o p p i n g time T over the p r o d u c t enlargement of ( f t , B , B t , P y ) by (X,A,m) , then f o r A £ B o r e l D , V T(A) = m(dx)P y(B , s £ A , T(',x) < R) m ( d x ) y _ ( . ) X ) ( A ) X 10.2. N o t a t i o n . I f ( A , F ) i s a measurable space and a, x a r e r r v ' s i n [0, 0 0] over ( A , F ) , we w r i t e a <_ x to mean a(„) ( ( t , 0 0 ] ) <_ x(_)((t,»]) f o r a l l u> £ A and a l l t e [0,«) • 10.3. Lemma. L e t p be a measure i n D such t h a t Gy i s a p o t e n t i a l . - 125 -Let a, T be randomized (B t)-stopping times such that a <_ x . Then Gy > Gy > Gy . (It follows that y_ and y are fi n i t e on — 0 — T O T compact subsets of D .) Proof: Let A = Borel (0,1) and let S, T be the ((8 ® A ) + ) -stopping times associated with a, T by 9.18(b). '1 Then S < T . Also y = — o 0 du Vg^. uy a n c* similarly for y^ Moreover, S(-,u) and T(-,u) are (8 t)-stopping times for 0 < u < 1 . Now app ly 7.4. • 10.4. Lemma. Let y be a measure in D such that Gy is a potential. Suppose (T ) is a net converging to x in RST(fi,B,Bt,Py) . Let v = y T , y ± = yx_ Then: a) For any compactly supported continuous function <f> in D , 4>Gv b) For any compactly supported continuous function <j> in D , <(>dy , 4>dv Proof: a) Let <J> be a non-negative compactly supported continuous function in D , and let (Zj)n be the non-negative decreasing t 0<t<°° process defined by Z t = <t>(Bs)ds t A R - 126 -Then E (ZQ) = <()Gy by 6.9. S i n c e Gy i s a p o t e n t i a l , i t f o l l o w s t h a t E Y ( Z Q ) i s f i n i t e . Now t Z^(IXI) i s co n t i n u o u s on [0,°°] f o r any co f o r which ZQ(CO) i s f i n i t e . Thus <T , Z> + <T, Z>. (See 9.23.) But i f c i s any randomized ( B ^ ) - s t o p p i n g time then <o, Z> P y(dco) a(co)(dt)Z t(co) P y(dco) du Z . , v (co) S(co,u) du E V [ S ( - , u ) A R * ( B g ) d s ] du * G y S ( - u) ^ b y 6 ' 9 3 n d t h e s t r o n g M a r k o v p r o p e r t y ) 4>Gyo , where S i s the ( ( 8 ^ ® B o r e l ( 0 , 1 ) ) + ) - s t o p p i n g time a s s o c i a t e d to a by 9.18(b) . b) From a ) , i t f o l l o w s t h a t the c o n c l u s i o n o f b) h o l d s i f <f) 2 i s a compactly s u p p o r t e d C f u n c t i o n i n D . But Gy^ <_ Gy f o r a l l i , by 10.3. Thus, by 7.8, sup y. (K) < 0 0 f o r any compact i 1 s u b s e t K o f D . We can t h e r e f o r e complete the p r o o f o f b) by an a p p r o x i m a t i o n argument. The n e x t r e s u l t i s a p a r t i c u l a r case of a r e s u l t due to Rost [1] Our method o f p r o o f , which was sugg e s t e d to me by R. V. Chacon, i s - 127 -q u i t e d i f f e r e n t from R o s t ' s . 10.5. Theorem. L e t y, v be measures i n D , and suppose Gy i s a p o t e n t i a l . Then the f o l l o w i n g a r e e q u i v a l e n t : a) Gy _> Gv . b) There e x i s t s a randomized ( 8 t ) - s t o p p i n g time T such t h a t y = v . x P r o o f : b) => a ) . See 10.3. a) => b ) . F o r each n a t u r a l number i , l e t v. be the measure — l i n D such t h a t Gv. = iAGv . l Then, by 7.11, t h e r e are (8^)-stopping times T_^  such t h a t v. = y T f o r each i . By 9.23, th e r e i s a randomized ( 8 t ) - s t o p p i n g 1 i time T such t h a t some subsequence o f (T^) converges to T i n RST(fi,8,8 t,P V) . Now Gv ± + Gv . Hence, by 10.4, y^ = v . ^ 10.6. C o r o l l a r y : L e t y,v be measures i n D such t h a t Gy i s a p o t e n t i a l and Gv <_ Gy . Then v(Z) <_ y(Z) f o r ev e r y B o r e l p o l a r s u b s e t Z of D . P r o o f : By 10.5, v = y^ f o r some randomized ( 8 t ) - s t o p p i n g time T . L e t T be the ( (B f c ® B o r e l ( 0 , 1 ) ) + ) - s t o p p i n g time a s s o c i a t e d to T by 9.18(b). (-1 Then v = du y , x . Now i f Z i s any B o r e l p o l a r s u b s e t J Q T(-,u) of D then y , , (Z) <_y(Z) f o r each u i n (0,1) , by 7.3(b); - 128 -hence v(Z) <_ y(Z) . 10.7. The p r o o f o f 10.6 t h a t we have j u s t g i v e n depends on a r a t h e r l a r g e f r a c t i o n of what has gone b e f o r e . T h e r e f o r e , l e t us i n d i c a t e how t h i s r e s u l t can be proved more d i r e c t l y , u s i n g o n l y c l a s s i c a l p o t e n t i a l t h e o r y . F i r s t of a l l t h e r e i s a B o r e l s e t A £ D such t h a t v _> u on B o r e l s u b s e t s o f A and v u on B o r e l s u b s e t s o f D\A . Now by 2.2, we have o n l y t o prove t h a t v(Z) <_ y(Z) f o r every B o r e l s e t Z £ {Gv = °°} . L e t H = A n {Gv = °°} . Then t h e r e a r e unique measures a, B i n D such t h a t a + u = u , 3 + u = v . (Note n rl t h a t u < v .) rl Now v < u on B o r e l s u b s e t s o f {Gv = <*>}\H , and u <_ u , so — n i t c e r t a i n l y s u f f i c e s to show t h a t 8(H) = 0 . Observe t h a t GB <. Ga and a(H) = 0 . Now a ( f r i n g e ( H , D ) ) = 0 . (H i s a p o l a r s e t , so f r i n g e ( H , D ) Hence, by a theorem of c l a s s i c a l p o t e n t i a l t h e o r y , bal(Ga,H,D) i s e q u a l to the lower r e g u l a r i z a t i o n of in f { b a l ( G a , U , D ) : H £ U open £ D} . But H i s p o l a r . Thus t h i s infimum i s e q u a l to zero except on a p o l a r s e t . Now i f H £ U open £ D then U £ base(U,D) so 3 l i v e s on n base(U.D) . Hence bal(Ga,U,D) >_ G3 U by the d o m i n a t i o n p r i n c i p l e H 4.1. T a k i n g the infimum over a l l such U , we f i n d t h a t {^ 8^  > ^ i s a p o l a r s e t , whence i t must be empty. Thus 3(H) = 0 . - 129 -10.8. Now we e s t a b l i s h the f o l l o w i n g improvement of 10.6. Note t h a t i t completes the p r o o f s o f (b => a) of 4.2 and (a => b) of 4.8. P r o p o s i t i o n : L e t W be an open subset o f H n and l e t u, v be superharmonic f u n c t i o n s i n W w i t h R i e s z measures y, v r e s p e c t i v e l y , L e t E = b a s e ( { u J> v}) n W . Then v(Z) <_ y(Z) f o r every B o r e l p o l a r s e t Z £ E . P r o o f : Case 1. Assume W i s a Green r e g i o n and u, v are p o t e n t i a l s i n W . L e t F = {u >_ v} , and l e t a = bal(y,F,W) , g = bal(v,F,W) . y(dx)bal(<5_,F,W) , and s i m i l a r l y f o r g , by 2.15. Now a = x I f x e E then bal(<!>x,F,W) = 6^ by the d o m i n a t i o n p r i n c i p l e 4.1. On the o t h e r hand, i f x e W\E then G__6 i s f i n i t e on E w x so b a l ( 6 x > F , W ) does not charge p o l a r s u b s e t s of E by 2.2. Thus ot(Z) = y(Z) and g(Z) = v(Z) f o r e v e r y B o r e l p o l a r s e t Z £ E . But u •> v on F so bal(u,F,W) >_ bal(v,F,W) . Hence a(Z) _> g(Z) f o r e v e r y B o r e l p o l a r s e t Z c W . Case 2. L e t W, u, v be as i n the statement of the p r o p o s i t i o n . L e t Z be a B o r e l p o l a r s u b s e t of E . Then Z i s a c o u n t a b l e u n i o n of r e l a t i v e l y compact B o r e l s u b s e t s o f W , so i t s u f f i c e s t o c o n s i d e r the case i n which Z i t s e l f i s r e l a t i v e l y compact i n W . Then t h e r e a r e open s e t s U and V such t h a t Z £ u , U i s r e l a t i v e l y compact i n V , and V i s r e l a t i v e l y compact i n W . Then u, v a r e bounded below i n V so t h e r e i s a r e a l number c such t h a t u+c and v+c a r e n o n - n e g a t i v e i n V . - 130 -L e t f = b a l ( u + c , U, V) g = b a l ( v + c , U, V) . Then f and g a r e p o t e n t i a l s i n V whose R i e s z measures c o i n c i d e w i t h y and v r e s p e c t i v e l y on B o r e l s u b s e t s o f U ; i n d e e d f = u+c i n U and g = v+c i n U . Now {f _> g} n U = {u _> v} n U . Thus, by 3.9 and 3.10, i f x e U then {f >_ g} i s t h i n a t x r e l a t i v e t o V i f f {f >_ g} n U i s t h i n a t x r e l a t i v e to U i f f {u >_ v} n U i s t h i n a t x r e l a t i v e to U i f f {u >_ v} i s t h i n a t x . Hence Z c b a s e ( { f _> g}, V) . T h e r e f o r e y(Z) _> v(Z) by case 1. 10.9. Remark. L e t T be the s e t of r r v ' s T i n [0,°°] over (ft,B) of the form T ( _ ) = f ( _ ) 6 Q + ( 1 - f ( w ) ) « T ( a ) ) . where f ranges over B^-measurable [ 0 , 1 ] - v a l u e d f u n c t i o n s on „ and T ranges over non-randomized ( B ^ ) - s t o p p i n g t i m e s . The element of T a r e randomized ( B t ) - s t o p p i n g times which we might say a r e "randomized o n l y a t time 0" . We c l a i m t h a t the f o l l o w i n g a r e e q u i v a l e n t : a) The c o n j e c t u r e 7.12 i s v a l i d f o r the Green r e g i o n D . b) Whenever u, v a r e measures i n D such t h a t Gy i s a p o t e n t i a l and Gy >_ Gv , t h e r e e x i s t s T e T such t h a t - 131 -P r o o f : a) => b) : By 10.6, v <_ y on B o r e l s u b s e t s of E = {Gv = °°} L e t 1)1 be a [ 0 , l ] - v a l u e d B o r e l f u n c t i o n on E such t h a t v(Z) = <f> dy f o r every B o r e l s e t Z £ E . Extend <|> to be 0 on H^\E , and l e t f = <}>(B ) . Then f i s B -measurable; i n d e e d , f o 0 u i s B^-measurable. Now t h e r e a r e unique measures a, 8 i n D such t h a t a + v = y , 6 + v = v . We have Ga _> Gg , and g({Gg = « = } ) = 0 . Hence, by E 2.2, g does not charge p o l a r s e t s . But then a c c o r d i n g to the c o n j e c t u r e 7.12, t h e r e i s a ( 8 t ) - s t o p p i n g time T such t h a t T L e t T(U>) = f(u>)6 n + ( 1 - f ( u ) ) 6 _ / , , (to e Q) . Then T e T 0 1 (.to; I f A i s any B o r e l s u b s e t o f D then y x ( A ) = P y(dto)x(to)({t e [0,R(to)) : B t ( u ) e A}) P y(dto) [f ( t o ) l A ( B 0 ( t o ) ) l { ( ) < R } ( t o ) + ( l - f ( t o ) ) l A ( B T ( t o ) ) l { T < R } ( t o ) ] y(dx)<j>(x)l A(x) + P a ( d a J ) l A ( B T ( t o ) ) l { T < R } ( t o ) = v £ ( A ) + a T ( A ) = v(A) Thus y = v T - 132 -b) => a ) : L e t y, v be measures i n D such t h a t Gy i s a p o t e n t i a l , Gy _> Gv , and t h e r e e x i s t s a B o r e l s e t A £ D such t h a t v(Z) = y(ZnA) f o r a l l B o r e l p o l a r s e t s Z £ D . L e t T e T such t h a t y_ = v . ( A c t u a l l y , any randomized ( B ^ - s t o p p i n g time T such t h a t y_ = v w i l l do here.) L e t E = { G y = ° ° } , y ' = y_^_ , y" = y_ , v' = y^ , and l e t v" = y" . Then Gv' _< Gy' and Gv" <_ Gy" . A l s o y' does not charge p o l a r s e t s . Thus, by 7.11, t h e r e i s a ( B ^ - s t o p p i n g time T' such t h a t y_j,, = v' . Now f o r any B o r e l p o l a r s e t Z £ D , v"(Z) = v(Z) = y(ZnA) = y(ZnAnE) = y"(ZnA) . Thus, a c c o r d i n g to the h y p o t h e s i s o f b ) , t h e r e e x i s t s T" e T such t h a t y",, = v" . By the d e f i n i t i o n o f T , t h e r e i s a 8 ..-measurable T U [ 0 , l ] - v a l u e d f u n c t i o n f and a ( B t ) - s t o p p i n g time T" such t h a t T ( _ ) = f ( u)6Q + ( 1 - f ( _ ) ) 6 _ „ ^ ^ f o r a l l cu i n . C l e a r l y we may assume t h a t f = 1 on {T = 0} . L e t g = _ A ° B Q . Then f o r any B o r e l s e t H £ D , g d P y " = y"(HnA) {BQ e H} = y(HnAnE) = v(H E) = v"(HnE) ( s i n c e v' does not charge p o l a r s e t s ) = y"(HnE) T P V " ( d „ ) [ f ( _ ) l { B o £ H n E j Q < R } ( _ ) + < l - f ( o > » l { B _ e HnE, T < R} (03) ] - 133 -f dP u {B Q e H} I t f o l l o w s t h a t g dP1" f dP M f o r a l l F i n Br S i n c e g and f a r e b o t h B 0-measurable, g = f P - a.e. L e t T = < T' on { B Q I E} 0 on {BQ e E n A} T" on {BQ £ E \ A} Then T i s a ( B t ) - s t o p p i n g time, and y T = v . 10.10. Here i s a c o r o l l a r y o f 10.5 which s h o u l d make i t p o s s i b l e t o reduce to the case where u i s a p o i n t mass i n a t t e m p t i n g to prove the c o n j e c t u r e 7.12. P r o p o s i t i o n : L e t y, v be measures i n D such t h a t Gy i s a p o t e n t i a l and Gy _> Gv . Then t h e r e i s a f a m i l y (v ) „ o f measures i n D such t h a t : x xeD a) F o r a l l x e D , Gv < G6 x — x b) F o r a l l A e B o r e l D , x H- ^ x ( A ) i s a B o r e l f u n c t i o n i n y ( d x ) v x ( A ) D and v(A) = P r o o f : By 10.5, t h e r e i s a randomized (B^)-stopping time x such - 134 -t h a t v = v . By 9.15, we can modify T on a s e t of P y-measure T 0 so t h a t i t becomes a randomized ( B t + ) - s t o p p i n g time. Now l e t v = (5 ) f o r each x i n D . X X T g 10.11. Here i s a n o t h e r a p p l i c a t i o n o f 10.5. P r o p o s i t i o n : L e t D^, be Green r e g i o n s i n ]R n w i t h Green f u n c t i o n s G^, r e s p e c t i v e l y . Suppose - 2^ * L e t u, v be measures i n such t h a t u ( D 2 \ D ^ ) = 0 = vCD^XD^) , G 2 U i s a p o t e n t i a l i n T)^ , and G^u >^  G^v . Then G^V 2. ^ 2V ' P r o o f : L e t R ± = i n f { t >_ 0 : B t I D ±} ( i = 1, 2) . Now G^ <_ G 2 on x ; hence G^u £ G^P i n .' In p a r t i c u l a r , G^u i s a p o t e n t i a l i n . Thus by 10.5, t h e r e i s a randomized (8 t)-s t o p p i n g time x such t h a t f o r a l l A i n B o r e l , v(A) = P U(dco)x(co)({t e [0, R-^to)) : Bt(u>) £ A}) L e t a ( u ) ( E ) = x (co) (En [ 0 ^ (co) ) ) + x (co) ( (R^co) , »])6oo(E) f o r co e n , E e B o r e l [0,°°] . Then f o r any t e [0,°°] and any co £ ft , a(co)([0,t]) = x(co)(H(co)) where H = (ft * [0,t]) n [0,R) £ 8 ® B o r e l [0,t] . Thus a i s a randomized (8^)-stopping time . I f A £ B o r e l , then v(A) = v(AnD x) P y(dco)a(co) ({t £ [0,R 1(w)) : Bt(u>) e A n D ^ ) - 135 -P y(d_ ) a ( _ )({t e [0,R (_)) : _ t(_) e A}) P y(d_)a(_)({t e [0,R 2(u)) : B (_) e A}) Thus G 2v £ G 2u , by 10.3. - 136 -2 11. EMBEDDING MEASURES IN BROWNIAN MOTION IN R , USING RANDOMIZED STOPPING TIMES 11.1. I n 10.5, we saw t h a t i f y, v a r e measures i n a Green r e g i o n D i n n n such t h a t Gy i s a p o t e n t i a l , t h e n Gy >^  Gv i f f t h e r e e x i s t s a randomized (B._)-stopping time T such t h a t y_ = v , where y i s as d e f i n e d i n 10.1. A l s o , i n 8.20, we saw t h a t i f T y v y, v a r e measures on R such t h a t U and U a r e p o t e n t i a l s , then t h e r e e x i s t s a y - s t a n d a r d ( 8 t ) - s t o p p i n g time T such t h a t y_ = v i f f U y _> U V and y(R.) = v(lR) . I n t h i s s e c t i o n , we a r e g o i n g to prove the analogue o f the 2 l a t t e r r e s u l t f o r measures on E . In t h i s c a s e , the example g i v e n a t the b e g i n n i n g o f s e c t i o n 9 shows t h a t we must use randomized s t o p p i n g t i m e s . 11.2. Throughout t h i s s e c t i o n , i f T i s a randomized ( B t ) - s t o p p i n g time and y i s a measure on E n , then y_ w i l l denote the measure on B o r e l E n d e f i n e d by P T(A) = P y(dco)x(u.) ( { t e [0,°°) : B^w) e A}) Of c o u r s e i f T a r i s e s from a genuine ( B t ) - s t o p p i n g time T then y_. = y_ where y_ i s as d e f i n e d i n 8.1. More g e n e r a l l y , i f (X,A,m) i s a measurable space and T a r i s e s as i n 9.18(a) from a s t o p p i n g time T over the p r o d u c t enlargement o f (n,B,8t,Py) by (X,A,m) then xm ( d x ) y T ( . , x ) ' - 137 -11.3. D e f i n i t i o n : L e t y be a measure on ]R n such t h a t U y i s a p o t e n t i a l . L e t T be a randomized ( 8 ^ )-stopping time. We s h a l l say T i s y - s t a n d a r d i f f whenever p, 0 a r e randomized (B ..-stopping times such t h a t p <_ a <_ T (see 10.2 f o r the meaning o f <_ her e ) then U P and U 0 a r e p o t e n t i a l s and U p > U ° . 11.4. D i s c u s s i o n o f 11.3: I f n >_ 3 then any x i s y - s t a n d a r d , by 10.3. Suppose n <_ 2 and x i s y - s t a n d a r d . Then y^0R n) 2. u0R n) by 4.8 ( i f n=2) o r 4.11 ( i f n=l) . Thus "x i s P y - a.s. f i n i t e " ; more e x p l i c i t l y P y ( { _ : X(CJ) ({«=}) 4 0}) = 0 . 11.5. a) L e t m be Lebesgue measure on B o r e l (0,1) . By 9.16, the map f f(m) i s a 1-1 map o f the s e t o f i n c r e a s i n g l e f t - c o n t i n u o u s maps o f (0,1) t o [0,°°] onto the s e t of p r o b a b i l i t y measures on [0,°°] . C l e a r l y t h i s map i s an o r d e r isomorphism, where f o r p r o b a b i l i t y measures a, 6 on [0,°°] , we say a <_ 8 i f f a((t,»]) £ 8((t,°°]) f o r a l l t e [0,°°) . I t f o l l o w s t h a t t h i s o r d e r i n g o f p r o b a b i l i t y measures on [0,°°] i s a l a t t i c e o r d e r i n g . G i v e n any two p r o b a b i l i t y measures a, 6 on [0,°°] , we denote t h e i r l e a s t upper bound i n t h i s o r d e r i n g by a v B , and t h e i r g r e a t e s t lower bound by ot A 8 . b) I f A i s a s e t and a, T a r e maps from A to p r o b a b i l i t y measures on [0,°°] we d e f i n e a A X and a v x i n the o b v i o u s - 138 -p o i n t w i s e f a s h i o n . c) I f (A , F ) i s a measurable space and a , T a r e r r v ' s i n [0,°°] over (A , F ) then so a r e O A T and a v T , by 9.18 (where we take F = F f o r 0 <_ t < 0 0) . d) I f ( A , F , F t ) i s a f i l t e r e d measurable space, where (F ) i s r i g h t - c o n t i n u o u s , and i f a, x a r e randomized ( F t ) - s t o p p i n g times then so a r e a A T and a v x , by 9.18. e) I f a i s a p r o b a b i l i t y measure on [O, 0 0] and t e [O, 0 0] then a A 6 can be d e s c r i b e d q u i t e e x p l i c i t l y ; a A 6 = a r n , t t [ 0 , t ] + a((t,°°])6 t I f f i s any non - n e g a t i v e B o r e l f u n c t i o n on [0,°°] t h e n ( a A 6 t ) ( d s ) f ( s ) = < x ( d s ) f ( s A t ) . 11.6. Lemma: L e t (A ,F,Q) be a a - f i n i t e measure space. L e t (x ) be a net c o n v e r g i n g to x i n RRV = RRV(A , F,Q;[0,«]) L e t T : A ->- [0,°°] be F-measurable. L e t a.(a)) = x.(co) A 5 , a(co) = x(co) A 6r T(u) Then a . ->• a i n RRV l P r o o f : L e t (Z )„ be any c o n t i n u o u s r e a l - v a l u e d p r o c e s s over t 0<_t<°° J (A , F ) s a t i s f y i n g Q(dco) sup IZ (to) I < » 0<t<°° Then (Z m ) i s a l s o such a p r o c e s s , t AT - 139 -Hence Q(doj) T . ( _ ) ( d t ) Z (_) 1 tAT Q(du>) x ( _ ) ( d t ) Z t A _ ( _ ) That i s , by 1 1 . 5 ( e ) , Q(dco) a 1 ( _ ) ( d t ) Z t ( _ ) Q(da») o ( _ ) ( d t ) Z t ( _ ) • 11.7. Lemma: L e t (A,F,Q) be a a - f i n i t e measure space, L e t ( x ± ) be a n e t c o n v e r g i n g to x i n RRV = RRV(A,F,Q;[0,»]) Suppose l i m sup t-*» i Q(d_)x (_)((t,»]) = 0 . Then: a) Q(d_)x.(_)({»}) = 0 . b) I f (Z ) i s any c o n t i n u o u s r e a l - v a l u e d p r o c e s s w i t h time s e t [0,°°) (not [0,°°]) such t h a t Q(dui) sup |Z ( u ) | < 0<t<~ then Q(du) x. (uO(dt)Z (ui) l t Q(dco) T ( _ ) ( d t ) Z t ( _ ) P r o o f : a) L e t f be a s t r i c t l y p o s i t i v e F-measurable f u n c t i o n on A - 140 -such that f dQ For each t in [0,°°) let gfc be a continuous [0,1]-valued function on [0,°°] such that g = 0 on [0,t] and gfc = 1 on [t+1, 0 0 ] . Then Q(du) T i ( u ) ) ( d s ) f ( o ) ) g t ( s ) + Q(do)) T ( u ) ( d s ) f ( a > ) g t ( s ) for each t . But lim sup t-*» i ' Q(dco) T_^ ((JJ) (ds)f (co)gt(s) = 0 . Hence lim Q(doj) x(to)(ds)f(to)g (s) = 0 . That i s , Q(dto)f (to) T (to) ({<»}) = 0 . As f is s t r i c t l y positive, a) is proved. For each s e [0,°°) let h be a continuous [0,l]-valued s function on [0,°°] such that h = 1 on [0,s] and h = 0 on [s+1, <*>] . Then s Q(dto) x. ( to)(dt)h (t)Z. (to) X S t T(t0)(dt)h g(t)Z t(t0) for each s . Letting s go to <*> , and using the "tightness condition" on (x ) , we obtain the desired result. - 141 -11.8. Lemma: L e t T be a s e t of randomized ( 8 ^ ) - s t o p p i n g times such t h a t i f a, x a r e randomized ( B t ) - s t o p p i n g times w i t h a <_ x and xeT then a e T . L e t y be a f i n i t e measure on ]R n . Suppose t h a t l i m sup y ({x e ]R n : | |x| | r } ) = 0 T T r-x» Tel Then l i m sup t-**> xeT P y ( d u ) ) x ( _ ) ( ( t , °°]) = 0 P r o o f : Emulate the p r o o f o f 8.10, n 11.9. Lemma: L e t y be a measure on IR (where n = 1 o r 2) such t h a t U y i s a p o t e n t i a l . L e t (ft' , B ' ,8J_,Q,if>) be the p r o d u c t enlargement o f ( f t , 8 , B t , P y ) by (X,A,m) , where X = (0,1) , A = B o r e l X , and m = Lebesgue measure on A . L e t x be a randomized ( B ^ - s t o p p i n g time, and l e t T be the ( B p - s t o p p i n g time a s s o c i a t e d to x by 9.18(b). Suppose T dQ < 0 0 . Then x i s y - s t a n d a r d . P r o o f : F o r each t , l e t B^ = B t ° v) . Then (ft' ,8' ,8|.,Bj.,Q) i s a g e n e r a l i z e d Brownian motion p r o c e s s w i t h i n i t i a l d i s t r i b u t i o n y Note t h a t i f a i s any randomized (B ) - s t o p p i n g time and i f S i s the ( B ^ ) - s t o p p i n g time a s s o c i a t e d t o a by 9.18(b), then y (E) = Q(B e E , S < °°) f o r each B o r e l s u b s e t E o f ]R n . A l s o , i f p, a a r e randomized ( B ^ - s t o p p i n g times such t h a t p <_ a , and i f R, S a r e the ( B | _ )-stopping times a s s o c i a t e d to - 142 -p, a by 9.18(b), then R _< S . Now emulate the p r o o f of 8.5, wo r k i n g w i t h the p r o c e s s (ft' ,8' ,BJ_,Bj_,Q) i n s t e a d of ( f t , 8 , B t , B t , P U ) 11.10. C o r o l l a r y : L e t u be a measure on ]R n such t h a t U P i s a p o t e n t i a l . L e t T be a randomized ( B ^ J - s t o p p i n g time such t h a t P y({u) : x ( t o ) ( ( t , «>]) 4 0}) = 0 f o r some t i n [0, °°) . Then x i s y - s t a n d a r d . P r o o f : C l e a r . • 11.11. Theorem: L e t y be a measure on 1R (where n = 1 o r 2) such t h a t U P i s a p o t e n t i a l . L e t T be a s e t of good measures on ]R n such t h a t whenever a, B a r e measures on ]R n such t h a t U a, U B a r e p o t e n t i a l s and U dy > U Bdy f o r a l l y e T , then U a >_ U 6 Suppose (x^) i s a net c o n v e r g i n g to x i n RST(ft,B,B t,P ) , and each x_^  i s y - s t a n d a r d . C o n s i d e r the f o l l o w i n g s t a t e m e n t s : a) There i s a measure a on ]R n such t h a t U a i s a p o t e n t i a l , a C R n ) = yQR n) , and U 1 >_ U 0 1 f o r a l l i . b) l i m sup t-x» i P M ( d a i ) x . ( c o ) ( ( t , - ] ) = 0 . c) f dy + x. I f dy^ f o r every bounded c o n t i n u o u s f u n c t i o n on T d) U i s a p o t e n t i a l and y < T i U dy -> U dy f o r a l l - 143 -e) x i s y - s t a n d a r d . Then a) => b) => c ) , and a) => d) => e ) . P r o o f : a) => b ) . L e t S be the s e t of randomized ( B t ) - s t o p p i n g times a such t h a t a <_ x_^ f o r some i . Then a e S i m p l i e s y v t h a t U i s a p o t e n t i a l and U _. U , s i n c e the T i ' s a r e y - s t a n d a r d . Hence l i m sup y ({x e l R n : | |x| | >_ r } ) = 0 , by r-x» aeS 8.9. The statement b) then f o l l o w s by 11.8. b) => c ) . Apply 11.7, w i t h (Z ) = (f(B..)) . (Note t h a t P y ( f t ) = y ( R n ) < 0 0 as U y i s a p o t e n t i a l and n <_ 2 .) a) => d ) . S i n c e a) => c ) , we have t h a t y -»• y weakly. — x . x l Then, r e a s o n i n g as i n the p r o o f of (a => b) of 8.11, we can deduce the statement d ) . d) => e ) . For each randomized ( B ^ ) - s t o p p i n g time a and each s i n [0, °°) , l e t oAS denote the randomized ( B t ) - s t o p p i n g time d e f i n e d by ( O A S ) ( _ J ) = O ( O ) ) A 6 . Then f o r each s i n [0, °°) , s x ^ s -> X A S i n RST , by 11.6, and y x . A S y u 1 > u s by 11.10. Thus, a p p l y i n g (a => d) w i t h ("^^ r e p l a c e d by ( x ^ A s ) ^ , x r e p l a c e d by x A s , and a e q u a l to y g , we f i n d t h a t y. X . A s U 1 dy -»-y T T X A s , U dy f o r a l l measures y e Y . Now f o r each i , X . A S X . u 1 > u 1 - 144 -s i n c e T i s y - s t a n d a r d . Thus we f i n d t h a t 1 U A U u T A s > u T f o r a l l s i n [0, °°) . From t h i s , w i t h s = 0 , i t f o l l o w s t h a t t h e t o t a l mass of y i s > t h a t of y , whence t h e s e two measures have T — the same t o t a l mass, and so P y ( { u e Q : x(w)({°°}) 4 0}) = 0 . Now suppose p and a are randomized ( 8 t ) - s t o p p i n g times w i t h p <_ a <_ T . Then f o r any s i n [0, 0 0) , U V V u u p A s > u a A s > u T A s > u T . L e t t i n g s °° and a p p l y i n g 8.7 and 8.8 (we can t a k e a = y^ i n 8.8 y y here) we c o n c l u d e t h a t U D and U a r e p o t e n t i a l s and y y u 0 > u 0 . (As s ->• 0 0 , y -*• u weakly and y y weakly, by p A s p O A s a (b => c ) . T h i s i m p l i c a t i o n a p p l i e s because p , a <_ x and » p y ( T = ») » = o .) 11.12. C o r o l l a r y : L e t y be a measure on 3R (where n = 1 or 2) PC time. Then the f o l l o w i n g a r e e q u i v a l e n t : such t h a t U P i s a p o t e n t i a l . L e t x be a randomized ( B ) - s t o p p i n g a) x i s y - s t a n d a r d . b) U T i s a p o t e n t i a l and U T A t > U T f o r a l l t i n [0,™) ( x A t i s as d e f i n e d i n the p r o o f o f 11.11(d).) P r o o f : a) => b ) . C l e a r . - 145 -u. b) => a) . We always have y TOR n) £ yOR n) • Now s i n c e U P >^  U T and n <_ 2 , we have y T Q R n ) >_ yQR n) , by 4.8 ( i f n = 2) o r 4.11 ( i f n = 1) . Now T A t i s y - s t a n d a r d f o r each t i n [0,°°) , by 11.10, and T A t T i n RST(ft,B,8 t,P y) as t -> °° . To con c l u d e the p r o o f , a p p l y 11.11 w i t h a = y^ and ^ x ^ ) ^ = ( T A t ) t • 11.13. C o r o l l a r y : L e t y be a measure on ]R n (where n = 1 or 2) such t h a t U P i s a p o t e n t i a l . L e t (ft* ,8' ,8j.,Q,<Jj) be an o p t i o n a l enlargement o f (ft,B,B t,P y) and l e t B' = B t ° I|J f o r each t . For each (8J_)-stopping time S l e t y g be the measure on H n d e f i n e d by y g ( E ) = Q ( B g e E, S < °°) L e t T be a (8 j_)-stopping time, and l e t T be (a v e r s i o n o f ) the randomized ( 8 t ) - s t o p p i n g a r i s i n g from T as d e s c r i b e d i n 9.8 and a) => b) o f 9.13. Then the f o l l o w i n g a r e e q u i v a l e n t : a) Whenever R and S a r e (8J . )-stopping time and R <_ S <_ T y R y S P R y S then U and U a r e p o t e n t i a l s , and U >_ U b) T i s y - s t a n d a r d . P r o o f : a) => b) . F o r each t i n [0,°°) l e t ( T A t ) (co) = T(to )A<5 t (co e Q) . T A t a r i s e s from T A t i n the same way t h a t T a r i s e s y from T . A l s o , y = y m and y = y m . Thus U i s a T T T A t T A t P T A t PT p o t e n t i a l , and U >^  U f o r a l l t i n [0,°°) . Thus T i s y - s t a n d a r d , by 11.12. b) => a ) . L e t p, o be ( v e r s i o n s o f the) randomized (B ) -s t o p p i n g times a r i s i n g from R, S r e s p e c t i v e l y , as d e s c r i b e d i n 9.8 - 146 -and a) => b) o f 9.13. Then p <_ a <_ T Q - a.e. L e t a' = O A X , y a y p p ' = p A a ' . Then y . = y and y , = y . Thus U and U a r e a a p p y y 0 p o t e n t i a l s , and U >_ U . But y = y and y = y . p R o o |-j 11.14. Theorem: L e t y, v be measures on H 2 such t h a t U y, U V a r e p o t e n t i a l s . Then the f o l l o w i n g a r e e q u i v a l e n t : 0 0 a) U P > U V and p ( l ) = v ( l ) . b) There i s a y - s t a n d a r d randomized (B^_)-stopping time x such t h a t y = v . x P r o o f : b) => a ) . C l e a r . a) => b) . For 0 < r < 0 0 , l e t A be the u n i f o r m u n i t 2 d i s t r i b u t i o n on the open b a l l o f r a d i u s r c e n t r e d a t 0 i n 1R , and l e t v = v*A , where * denotes c o n v o l u t i o n . Then r r v v r v r U = U *\ . Thus U i s f i n i t e (and c o n t i n u o u s ) . A l s o , by 4.18 v r v of Helms [ 1 ] , U tU as r+0 . F o r each r , t h e r e e x i s t s a y - s t a n d a r d (B ) - s t o p p i n g time T_ such t h a t y_ = v_ , by 8.20. r By 9.23, t h e r e i s a d e c r e a s i n g sequence ( r ( i ) ) i n (0,°°) and a randomized ( 8 ^ - s t o p p i n g time x such t h a t r ( i ) -»- 0 and T r ( i ) T i n RST ( n , B , B_,P y) . Then y = v and x i s y - s t a n d a r d , by 11.11 t x (we can take a = v i n o r d e r t o a p p l y 11.11 . ) r ( 0 ) ^ g - 147 -12. EMBEDDING PROCESSES IN BROWNIAN MOTION In t h i s s e c t i o n we d e f i n e a c l a s s o f p r o c e s s e s which, f o l l o w i n g Chacon [ 1 ] , we c a l l p o t e n t i a l p r o c e s s e s . E s s e n t i a l l y , t h e se t u r n out to be the p r o c e s s e s which can be embedded i n an o p t i o n a l enlargement o f Brownian motion by means of an i n c r e a s i n g f a m i l y o f s t a n d a r d s t o p p i n g times - see 12.7, 12.16, and 12.17. 12.1. N o t a t i o n : I f y i s a measure on B o r e l ]R^ then we s h a l l o w r i t e U P f o r U V where v i s the r e s t r i c t i o n of y to B o r e l H n . 12.2. D e f i n i t i o n : L e t I £ [-«=, °°] , and l e t n be a p o s i t i v e i n t e g e r . An n - d i m e n s i o n a l p o t e n t i a l p r o c e s s w i t h time s e t I i s a system (A,F,F^,X^,P) where: a) (A,F,P) i s a p r o b a b i l i t y space. b) ^ j ^ i e i "*"s a n i n c r e a s i n g f a m i l y of sub - a - f i e l d s of F . c) ( X . ) . T i s a f a m i l y o f ]R n-valued random v a r i a b l e s over l l e i J 3 (A,F,P) . d) For each i e I , i ) X. i s F.-measurable, l l . .. T Ilaw(X-j) . . n ) U -1- xs a p o t e n t i a l . i i i ) I f n <_ 2 , P C X ^ S ) = 0 . e) Whenever S and T a r e ( F ^ ) - s t o p p i n g times t a k i n g on o n l y f i n i t e l y many v a l u e s , and S <_ T , we have law(X ) law(X ) U b > U 1 . - 148 -law(X-) (We remark t h a t i t f o l l o w s from d) t h a t U and law(X_) U a r e p o t e n t i a l s . A l s o , an ( F ^ ) - s t o p p i n g time i s assumed to take v a l u e s i n I .) 12.3. P r o p o s i t i o n : L e t I _ [-°°, 0 0] , l e t n be a p o s i t i v e i n t e g e r , and l e t ( A ,F , F^.X^P) be a system s a t i s f y i n g a) through d) of 12.2. Then a) and b) below are e q u i v a l e n t : a) ( A , F , F ,X ,P) i s an n - d i m e n s i o n a l p o t e n t i a l p r o c e s s . b) F o r each y e ] R n , ( $ ( X i 9 y ) l ^ x ^ 9}^ i s a s u P e r m a r t i n g a l e i i n the extended sense over (A , F , F^,P) . Moreover, i f n = 1 then a t h i r d e q u i v a l e n t c o n d i t i o n i s : c) (X_^) i s a m a r t i n g a l e over (A , F , F^,P) . P r o o f : I f S i s an ( F _ ^ )-stopping time t a k i n g on o n l y f i n i t e l y many v a l u e s and y e ] R n then law(X ) U (y) = E ( * ( X - , y ) l ^ 3 } ) . Thus (b => a) i s a consequence of the o p t i o n a l s a m p l i n g theorem. a) => b) . L e t i , j e I w i t h i <_ j and l e t F e F_^  . We w i s h to show t h a t Z Y dP > F 1 Z. dP f o r each y e ]R , where F 3 ( X ^ j y ) ! ^ . ^ ^ 3 j f o r each k e I and each y e l R n . L e t S = i on F j on A\F - 149 -and l e t T = j on a l l of A . Then S and T a r e (F ) - s t o p p i n g times t a k i n g on o n l y f i n i t e l y many v a l u e s and S £ T so law(X_) law(X ) U > U That i s , zl dP > Z y dP f o r a l l y e R n From t h i s we can c o n c l u d e t h a t law(X_) Z Y . dP > ZY dP f o r each y e TR such t h a t U " (y) < 0 0 • Thus i f we l e t y, v be the measures on TR d e f i n e d by y(A) = P ( { X ± e A} n F) v(A) = e A} n F) law(X_) then U y > U V except p o s s i b l y on the p o l a r s e t {U = 00} Hence U y > U V on a l l o f TRn . That i s , Z y dP > Z Y . dP f o r a l l y e H n , as d e s i r e d . Now suppose n = 1 . Then P ( X ± = 9) = 0 f o r each i e I c) => b) . S i n c e n = 1, 3>(x,y) = - -|-|x-y| . Thus we need o n l y a p p l y Jensen's i n e q u a l i t y f o r c o n d i t i o n a l e x p e c t a t i o n s . a) => c ) . L e t S, T be (F ) - s t o p p i n g times assuming o n l y f i n i t e l y many v a l u e s , such t h a t S <_ T . law(X ) law(X ) Then U _. U , so law(X.) and lawCX.,) have the same c e n t r e o f mass, by 4.11. Hence E ( X g ) = E(X_) . I t f o l l o w s t h a t (X^) i s a m a r t i n g a l e over (A,F,F_^,P) . - 150 12.4. D e f i n i t i o n : L e t y be a measure on H n such t h a t U y i s a p o t e n t i a l , and l e t (ft' , B ' , B ',Q,ij;) be an o p t i o n a l enlargement o f ( f t , B , 8 t , P y ) . L e t B' = B f c ° i|; f o r 0 <_ t < » . A ( B p - s t o p p i n g time T w i l l be c a l l e d s t a n d a r d ( r e l a t i v e to Q) i f f whenever R, S a r e ( B p - s t o p p i n g times and R <_ S <_ T then l a w ( B R ) l a w ( B g ) l a w ( B R ) law (Bp U and U a r e p o t e n t i a l s and U ^ U 12.5. D i s c u s s i o n o f 12.4: a) Note t h a t the d e f i n i t i o n 12.4 i s a g e n e r a l i z a t i o n o f the d e f i n i t i o n 8.2. b) By 11.13, T i s s t a n d a r d r e l a t i v e to Q i f f " t h e " randomized ( B p - s t o p p i n g time a r i s i n g from T i s y - s t a n d a r d . c) One can show t h a t T i s s t a n d a r d r e l a t i v e to Q i f f f o r eve r y compact s e t K £ ]R n , T(w) Q(du>) l K ( B ' ( o 3 ) ) d t law (Bp i s f i n i t e and e q u a l to I (U - U ) . T h i s i s a si m p l e 'K g e n e r a l i z a t i o n o f 8.13, and we omit the p r o o f . d) I f n >_ 3 , e v e r y ( B p - s t o p p i n g time i s s t a n d a r d r e l a t i v e to Q • e) I f n <_ 2 and y ( { 8 } ) = 0 and T i s s t a n d a r d r e l a t i v e to Q then T i s f i n i t e Q - a . s . 12.6. P r o p o s i t i o n : L e t y be a p r o b a b i l i t y measure on ]R n such t h a t U y i s a p o t e n t i a l , and l e t (ft' , B ' ,8pQ,iJ>) be an o p t i o n a l enlargement - 151 of ( f t , B , B t , P y ) . L e t = B ° 1J1 f o r 0 <_ t < 00 L e t I c [ - o o o o ] and l e t ( T . ) . _ be an i n c r e a s i n g f a m i l y o f - l i e l ( B p - s t o p p i n g times which a r e s t a n d a r d r e l a t i v e to Q . Then (ft',8',8.j, jB^-Q) i s an n - d i m e n s i o n a l p o t e n t i a l p r o c e s s w i t h time i i s e t I . P r o o f : C l e a r l y the i n d i c a t e d system s a t i s f i e s a) through d) of the d e f i n i t i o n 12.2. To f i n i s h the p r o o f we s h a l l show t h a t i t s a t i s f i e s b) of p r o p o s i t i o n 12.3. L e t i , j e I w i t h i <_ j and l e t F e B_ . i L e t S = \ T. on F l T. on S7'\F J and l e t T = T. . Note t h a t S i s a J law(B^) law(B^) ( B p - s t o p p i n g time. As T i s s t a n d a r d , U > B . For law (Bp y e {U < °°} , we can conclude from t h i s t h a t * ( Y > y ) 1 { B T ^ } d Q -r 1 1 . X _ $ ( B T . ' y ) 1 { B _ * _} d Q ' ? 1 • Then, by the method used i n the p r o o f of (a => b) of 12.3, we can co n c l u d e t h a t t h i s i n e q u a l i t y a c t u a l l y h o l d s f o r a l l y e ]R n . • 12.7. Theorem: L e t I = {0,...,k} where k e U , o r l e t I = K . L e t n be a p o s i t i v e i n t e g e r , and l e t (A,F,F^,X^,P) be an n - d i m e n s i o n a l p o t e n t i a l p r o c e s s . L e t u = law(X^) . Then: a) There i s a p r o d u c t enlargement (ft' ,8' ,B|. ,Q,^) of ( f t , B , B t , P y ) - 152 -and an i n c r e a s i n g f a m i l y ^ ± ^ ± e j _ °^ ( B j . )-stopping times which a r e s t a n d a r d r e l a t i v e to Q , such t h a t (B*, ) . and i ( X . ) . _ have the same j o i n t d i s t r i b u t i o n , where o f cou r s e 1 1 e l B£ = B ° IJJ f o r 0 <_ t <_ <=° . ' b) I f n = 1 , no enlargement i s n e c e s s a r y i n a ) . That i s , t h e r e i s an i n c r e a s i n g f a m i l y ^ ± ^ ± e j °^ u - s t a n d a r d (8 f c)-stopping times such t h a t ^ B x . ^ i e l 3 n d ^ X i ^ i e l n a v e the same j o i n t d i s t r i b u t i o n . P r o o f : a) L e t (ft 1,8' ,B^,Q,IJJ) be the p r o d u c t enlargement of (f t , B , 8 t , P M ) by ( L , L , A ) , where L = ( 0 , 1 ) , L = B o r e l L, and A = Lebesgue measure on L . L e t B^ = B ° f o r 0 <_ t <_ °° . E v i d e n t l y we can l e t T Q = 0 . Suppose j e I such t h a t j+1 e I , and suppose s t a n d a r d (8*.)-stopping times T n < ... < T. have been chosen to t h a t (Y.,...,Y.) 0 — — j 0 j = (B' ,...,B' ) and X ,...,X. have the same j o i n t d i s t r i b u t i o n . We 0 j 2 s h a l l show t h a t one can choose a s t a n d a r d ( B j.)-stopping time T.,, > T. so t h a t Y_,...,Y., B' and X.,...,X., X... have the J+1 ~ J 0' j T j + 1 0' j ' j+1 same j o i n t d i s t r i b u t i o n . L e t D = ]R n and l e t E = E." . L e t a be the j o i n t d i s t r i b u t i o n o f X_,...,X. and l e t 3 be the j o i n t d i s t r i b u t i o n o f X_.,...,X. , X.,., 0 3 0 3 j+1 Then a i s a p r o b a b i l i t y measure on E ^ + ^ , t p r o b a b i l i t y measure on E2+^ , and the p r o j e c t i o n map ( XQ 5 • • • > X j ' X j +1^ ' ^ ^ ^TJ ' * * ' ' ^ j ^  i s a - 153 -c a r r i e s onto a Hence by 9.4, t h e r e i s an a - e s s e n t i a l l y unique B o r e l measurable f a m i l y (8 ) . . of p r o b a b i l i t y measures on E such t h a t f o r a l l X x e E 3 l A e B o r e l ( E j + 2 ) , 8(A) a(dx)B ( A ( x ) ) . S i n c e X.,...,X., X.,. i s a p o t e n t i a l p r o c e s s , 0' J j+1 F x E $ ( x j , y ) l D ( x j ) d 8 ( x 0 , . . . , x j + 1 ) > FxE l ( x . + 1 , y ) l D ( x . + 1 ) d 8 ( x 0 , . . . , x . + 1 ) f o r a l l F e B o r e l ( E ^ + ± ) and a l l y e D . T h i s may be deduced from (a => b) of 12.3, by change o f v a r i a b l e s . Now the f i r s t i n t e g r a l i n t h i s i n e q u a l i t y may a l s o be w r i t t e n as •KXj , y ) l _ ( x ) d a ( x Q , . .. ,x..) w h i l e the second may be w r i t t e n as U J ( y ) d a ( x Q , . . . , X j ) . From t h i s one may deduce, by a v e r a g i n g w i t h r e s p e c t to y over each element of a c o u n t a b l e s e t o f open b a l l s which forms a base f o r the t o p o l o g y o f D , t h a t f o r a-almost a l l ( X Q , . . . , X _ . ) e E ^ + ± we have X Q , • • • , x. J 1 • ) l - ( X j ) on D Of c o u r s e we s h o u l d v e r i f y t h a t U i s a p o t e n t i a l f o r a - a.a. x e - 154 -e As the $ 's a r e f i n i t e measures, the U 's a r e OK. We need o n l y X B check t h a t f o r a - a.a. x , U i s f i n i t e - v a l u e d . F i x any y e D Then U x ( y ) d a ( x ) . + 2 ^ " ( x j + 1 , y ) 1 D ( x j + 1 ) d B ( x Q , . . . , x . , x . + 1 ) $ _ (Vl' y ) 1{X. + 1 * o) law(X ) = U_ J (y) < » . B x i+1 Thus, i n p a r t i c u l a r , U_ (y) < °° f o r a - a.a. x e E J , f o r t h i s v p a r t i c u l a r y . But i f y i s a f i n i t e measure on E and U_ i s f i n i t e a t one p o i n t t h e n i t i s f i n i t e a t a l l p o i n t s ' — see 1.5 f o r 8 the case n = 1 and 1.6 f o r the case n = 2 ; i f n > _ 3 , U_ = 0 f o r a l l x , as § >_ 0 i n t h i s c a s e . Thus f o r a - a.a. x = (x.,...,x.) e E^ x D , we can choose a 0 j 6 - s t a n d a r d randomized (B ) - s t o p p i n g time T such t h a t X. t X J B x ( A ) -x. P J(dco)x (u))({t e [0, » ] : B (CJ) e A}) f o r a l l A e B o r e l E (more b r i e f l y , B = (6 ) ) . x x. T J x (For the p r o o f of t h i s see 10.5 f o r the case n >^  3 and 11.4 n = 2 . The' method used to prove 11.14 a l s o works f o r n = 1 , though i n t h i s c ase we can g e t non-randomized s t o p p i n g times; see 8.20.) Now i f n < 2 then t h i s takes c a r e of a - a.a. x e E^"*" s i n c e then - 155 -B P(X = 3) = 0 . On the o t h e r hand i f n >_ 3 then we have U X <_ 0 f o r a - a . a . x e E ^ x { 3 } ; f o r any such x the p r o b a b i l i t y measure B must be e q u a l to 6„ so we can take T = "0" . x 3 x Thus we o b t a i n a f a m i l y (x ) . ,.. o f randomized (8 )-x -J+1 t x e E J s t o p p i n g times such t h a t f o r a - a.a. x = (x-.,. . . ,x^) e E"^ + ± , x i s 6 - s t a n d a r d and 8 = (6 ) x x . x x. x 3 3 x Now i t w i l l become ap p a r e n t , from ja s e r i e s o f lemmas which f o l l o w , t h a t the x 's can be chosen so t h a t f o r each t e (0,°°) the map x ( X , U > ) «• T x ( _ ) ( [ 0 , t ) _ i s ( B o r e l E ^ + X ) ® B^-measurable. F o r x e E ^ + 1 l e t T be the ((8 ® A ) ) - s t o p p i n g time a s s o c i a t e d to x^ by 9.18(b), and f o r _' = (_,u) e ft1 l e t e ' c o ' = ( 9 _) ( f o r 0 < t < °°) t t - — F i n a l l y l e t T... = T. + S . J + l 3 F i r s t of a l l we c l a i m t h a t S i s a (8^ + t ) - s t o p p i n g time. 3 W e l l , f o r any t e (O, 0 0) , {S < t} = u {_'-(_,u) e ft' : u < x f n M ( 6 - , , ^ ) ( [ 0 , r ] ) } r e [ 0 , t ) Y 0 ( U ) ' - - " Y j ( w > T j ( w } r r a t i o n a l (See p r o o f of 9.18(b).) A l s o , f o r any r e [0,t) , the map 156 -= ( _ , _ ) • * ((Y 0(_'),...,Y j(a»')), 6-i s (B^ + r > ( B o r e l E"^ + ±) ® B ^ )-measurable and the map j (( X f.,...,x.),w) •> x ( w ) ( [ 0 , r ] ) u j XQ , .. . 5 x j i s ( B o r e l E ) ® B^-measurable. I t f o l l o w s t h a t {S < t} e B_j, + t f o r 0 < t < 0 0 . Thus, as j i s r i g h t - c o n t i n u o u s , S i s a (B ,^ + t _ ) - s t o p p i n g time, as j j c l a i m e d . But then, u s i n g the r i g h t - c o n t i n u i t y o f ( B p , one e a s i l y checks t h a t T_ + S i s a ( B p - s t o p p i n g time. That i s , T j + i i s a (8|_)-stopping time. Now l e t Y = B' and suppose f f a r e non-n e g a t i v e B o r e l f u n c t i o n s on E . Then f 0 ( x 0 ) . . . f . ( x . ) f . + 1 ( x . + 1 ) d P f 0 ( x Q ) . . . f . ( x . ) f j + 1 ( x j + 1 ) d B ( x 0 , . . . , x . , x . + 1 ) fo (V"- fj ( x. ) [j f - _ i ( x - . i ) d 3 v (x ) ] d a ( x , j+1 j+1 x Q,...,x j+1 0 (*) Vi ( u , ) [ n f . ( Y . ( u . ' ) ) ] E i=0 1 1 [ n f . ( Y , (_'))]_,_,__ ( B l [ fj + l ( B T . )]dQ( i=0 j + lx T Y 0 ( u ' ) , . . . , Y (a ) ' ) (6' „'))dQ (a> ' ) Y. ( a ) ' ) , . . . , Y ( i _ ' ) j - 157 -f 0 ( Y 0 ) . . . f . ( Y . ) f . + 1 ( Y . + 1 ) d Q Thus Y„,...,Y. , Y.,., and X„,...,X. , X.,. have the same j o i n t 0' j j+1 0' j j+1 d i s t r i b u t i o n . (The s t e p (*) above, and a l s o the s t e p (**) below, f o l l o w from a "souped up" v e r s i o n o f the s t r o n g Markov p r o p e r t y d e s c r i b e d i n Meyer [ 2 ] . F o r the r e a d e r ' s convenience we s t a t e and prove t h i s r e s u l t i m mediately f o l l o w i n g the p r e s e n t p r o o f . ) Now l e t us show t h a t T.,, i s s t a n d a r d . We s h a l l prove t h i s J+1 by a p p l y i n g 1 2 . 5 ( c ) . L e t K be any compact s u b s e t o f D . Then T. r J l K ( B ' ) d t dQ law(B^ ) [ U y - U j ] = K law(X.) [UM - U ] K w h i l e Lj+1 T'. J l K ( B ' ) d t dQ (**){ T v ( I N T , , * (6 ' io') Y Q(oo ) , . . . , Y (co ) 1\ J V Bt ( 9T a ) ' ) ) d t A W ) j Y.(co') Y 0 ( c o ' ) , . . . , Y . ( W ' ) 1 (B')dt]dQ(co*) IN. L 6 law(Bj, X . x_» • • • > x. U J - u 0 2 ;P J ® A) K ] d a ( x Q , . . . , X j ) K X X j • • • j x U j ( y ) - U °' j ( y ) d a ( x 0 , . . . , x j d y - 158 -law(X.) law(X ) U 2 (y) - U J (y)dy K Thus 1 ( B ' ) d t dQ IN. L law(B^ ) [u y - U 2 + 1 ] K l a w ( X . ^ ) [UP - U 2 + 1 ] < - . K I t f o l l o w s t h a t T.,, i s s t a n d a r d . J+1 (b) The p r o o f of t h i s p a r t i s q u i t e s i m i l a r to t h a t of p a r t a ) , except t h a t i n p l a c e of the f a m i l y (T ) . of randomized (8 ) -X xeE-5 s t o p p i n g times, we a p p e a l to 8.20 to get a f a m i l y (T ) . - of X x e E 3 + i genuine ( B p - s t o p p i n g t i m e s . I t w i l l become a p p a r e n t , from the s e r i e s of lemmas which f o l l o w , t h a t the T 's can be chosen so t h a t x f o r each t e (O, 0 0) the map (x,a>) + 1 { T < t j ( u ) i s ( B o r e l E2+^~) ® B ^ m e a s u r a b l e , • Now he r e i s the v e r s i o n o f the s t r o n g Markov p r o p e r t y mentioned i n the p r o o f of 12.7. 12.8. P r o p o s i t i o n . L e t ( W , M , M t > Z t , 9 t > P ) be a s t r o n g Markov p r o c e s s - 159 -w i t h t r a n s l a t i o n o p e r a t o r s , w i t h s t a t e space (E,E) L e t T be an ( M ^ - s t o p p i n g time, and l e t f : W x W [O, 0 0] be M T ® M-measurable. L e t u be any p r o b a b i l i t y measure on E . Z T ( w ) u , u Then E (f(w,«)) = E M ( f ( w , » ) ° 6 |M )(w) f o r P M - a.a. w £ W More p r e c i s e l y : Z T(w) a) The map w E (f(w,«)) i s M - m e a s u r a b l e . b) F o r any M-measurable f u n c t i o n g : W -> [0, 0 0 ] , we have Z (w) g(w)E ( f ( w , . ) ) d P P ( w ) g(w)f(w,6 Tw)dP^(w) P r o o f : I t s u f f i c e s to c o n s i d e r f of the form a ® b , where a and b a r e n o n - n e g a t i v e f u n c t i o n s on W which a r e M-measurable and Z T(w) T z T(w) M-measurable r e s p e c t i v e l y . Then E (f(w,«)) = a(w)E (b) , which i s M^,-measurable, and the f i r s t i n t e g r a l i n b) i s T y g a E x ( b ) d P y g a E y ( b o 0x|MT) d P y g(w)a(w)b(e Tw)dP H(w) , where the second s t e p f o l l o w s from the f a c t t h a t g a i s M - m e a s u r a b l e . • - 160 -Before concluding, let us emphasize that, in a slight departure from the usual formulation of the translation operator version of the strong Markov property, we are not assuming here that M = a(Z t : 0 _< t <_ oo) mod P y . We must work in a more general setting than this, since we want to apply this proposition to the process (ft* ,B' ,8^,Bj.,eppX » A) considered i n the proof of 12.7(a). Now we take up the proof of the measurability results used in the proof of 12.7. 12.9. Lemma. Let X and Y be sets, let R and S be a-rings of subsets of X and Y respectively, and l e t y be a measure on 5 such that each set belonging to S is of o-finite y-measure. Also. let T = R ® 5 , and for each 5-measurable function f : Y H , let [f] = {g : Y -*-]R| g is S-measurable and y({f 4 g}) = 0} . Then the following are equivalent, for F : X ->• L 1 = L±(Y,S,y) . a) range(F) is a separable subset of L X , and and x F(x)dy is R-measurable for each S e S S b) range(F) is a separable subset of L X , and F X[U] n {F 4 0} e R for every norm-open set U £ L X . c) There is a T-measurable function f : X x Y TR such that F(x) = [f(x,«)l for a l l x e X . Proof: a) => b). One f i r s t verifies that x ^ f(x) g dy is R-measurable for each bounded S-measurable function g on Y . - 161 -Next, as range (F) i s s e p a r a b l e , t h e r e i s a s e t S e S o f a - f i n i t e y-measure such t h a t r a n g e ( F ) c {f e L 1 : f = 0 y - a.e. on Y\S} . Thus f o r any bounded l i n e a r f u n c t i o n a l A on L"^ , t h e r e i s a bounded S-measurable f u n c t i o n g on Y such t h a t A ( F ( x ) ) F ( x ) g dy x f o r a l l x e X ; thus A ° F i s R-measurable. The c o n c l u s i o n b) now f o l l o w s from the g e n e r a l t h e o r y of m e a s u r a b i l i t y f o r Banach space v a l u e d f u n c t i o n s ; see P e t t i s [ 1 ] . b) => c ) . U s i n g the s e p a r a b i l i t y o f range(F) , and the f a c t t h a t F 1 [ B ] n { F ^ 0 } e R f o r each open b a l l B i n L 1 , we can c o n s t r u c t a sequence (F_^) of c o u n t a b l e - v a l u e d R-measurable f u n c t i o n s from X i n t o such t h a t f o r each i , and f o r each i n X , ||F(x) - F.(x) || ± < 2"1 . L Now f o r each i , t h e r e i s a T-measurable r e a l - v a l u e d f u n c t i o n f . on X x Y such t h a t F.(x) = [ f . ( x , • ) ] f o r a l l x e X . Moreover, f o r a l l x e X , f j [(x,») F(x) y - a.e., because of the r a t e o f L^-convergence o f (F^.(x)) to F(x) . - 162 -L e t E = {(x,y) e X x Y : ( f \ ( x , y ) ) does n o t converge i n E } and d e f i n e f : X x Y -> E by f i l m f . ( x , y ) i f (x,y) I E f ( x , y ) = \ 0 i f (x,y) e E Then E e T and f has the d e s i r e d p r o p e r t i e s . c) => a ) . The m e a s u r a b i l i t y p a r t f o l l o w s from F u b i n i ' s theorem. L e t us e s t a b l i s h the s e p a r a b i l i t y p a r t . W e l l , t h e r e a r e c o u n t a b l e s e t s RQ £ R and S^ £ S such t h a t f i s measurable w i t h r e s p e c t to the o - r i n g g e n e r a t e d by the s e t s o f the form RxS where R e R„ and S e Sn . L e t S., be the o - r i n g g e n e r a t e d by , 0 0 and l e t Z = u S . Then Z e S , so Z i s of a - f i n i t e u-measure, and f o r each x e X , f(x,«) i s S^-measurable and v a n i s h e s o u t s i d e Z . Thus range(F) £ " L ^ Z , S ^ y ) " , which i s s e p a r a b l e . • Remark: The assumption t h a t each s e t i n S i s of a - f i n i t e y-measure r e a l l y i s needed f o r the p r o o f o f (c => a) of the above lemma. T h i s i s shown by the f o l l o w i n g example. L e t X = Y = [0,1] , R = S = B o r e l X , y = c o u n t i n g measure on S , and l e t f ( x , y ) = { 1 i f x = y 0 i f x 4 y Then f i s R ® S-measurable, but {[f(x,»)] : x e X} i s n o t a - 163 -s e p a r a b l e s u b s e t o f L"*" . 12.10. Lemma: L e t ( A , F , F t > P ) be a f i l t e r e d measure space, where P i s f i n i t e . L e t RST = RST(A,F , F ,P) . Suppose F i s c o u n t a b l y g e n e r a t e d mod P . Then t h e r e i s a map T •+ x o f RST i n t o i t s e l f s uch t h a t a) F o r e v e r y x , x = x P - a . e . b) The map (x,to) H- x ( t o ) ( [ 0 , t ) ) i s ( B o r e l RST) ® F - m e a s u r a b l e f o r each t e (0,°°) . P r o o f : F o r each p o s i t i v e r a t i o n a l number r , l e t f : RST x A [0,1] be d e f i n e d by f (x,co) = x ( w ) ( [ 0 , r ) ) . Then f o r each x , f (x,«) i s F^_-measurable; a l s o , i f F e F , then f (x,-)dP = F r P(dco) x ( a 3 ) ( d t ) l [ 0 ) r ) ( t ) l F ( t o ) . Thus f o r F e F , the map x f (x,«)dP i s B o r e l ( R S T ) - m e a s u r a b l e . F r Indeed i t i s the p o i n t w i s e l i m i t o f a sequence of c o n t i n u o u s f u n c t i o n s on RST. A l s o note t h a t L ^ ( A , F ,P | F r ) i s s e p a r a b l e s i n c e F i s c o u n t a b l y g e n e r a t e d mod P . Thus, by 12.9, f o r each r we can s e l e c t a map : RST x A [0,1] such t h a t i ) g i s ( B o r e l RST) ® F -measurable r r i i ) F o r every x , g (x,«) = f r ( x , « ) P - a.e. Fo r each t e (0,») , l e t h = sup g r . Then each h t i s a 0<r<t - 164 -( B o r e l RST) ® F - m e a s u r a b l e map of RST x A i n t o [0, 1] , and f o r each (x,co) e RST * A , the map t H- h t(x , c o ) i s i n c r e a s i n g and l e f t - c o n t i n u o u s . A l s o , f o r each T e (0,°°) , h (x,») = x(»)([0,t)) P - a.e. Now f o r each x , l e t T be d e f i n e d by x(co) = the unique p r o b a b i l i t y measure u on [0, 0 0] such t h a t y ( [ 0 , t ) ) = h (x,co) f o r 0 < t < » . Then c l e a r l y the map x •-»- x has the r e q u i r e d p r o p e r t i e s . 12.11. Lemma: L e t (A , F,P) be a f i n i t e measure space, and l e t RRV = RRV(A ,F,P;[0, «]) . L e t (E , E ) be a measurable space, and l e t (X V be a measurable p r o c e s s over (A , F ) w i t h s t a t e space t 0<t<°° (E , E ) . F o r each x i n RRV , l e t y(x) be the measure on E d e f i n e d by y(x ) (A) = P(dco)x(co) ( { t e [0, » ] : X (u>) e A}) Then the map x y(x) i s measurable, i n the sense t h a t f o r each A e E , u(»)(A) i s a Borel(RRV)-measurable f u n c t i o n on RRV . P r o o f : L e t A e E , and l e t H = {(co,t) e A x [0, o o ] : x (u>) e A} . Then H e F ® ( B o r e l [ 0 , «=]) } and H(co) = {t e [0, o o ] : X (w) e A} f o r each co e A . Note t h a t f o r each x e RRV , the map co x(co)(H(co)) i s F-measurable, so the d e f i n i t i o n of y(x) i s a l r i g h t . - 165 -L e t y be the s e t o f bounded measurable r e a l - v a l u e d p r o c e s s e s (Y )_ over (A,F) such t h a t the map t 0<t«=° T P(dco) x(o))(dt)Y t((_) i s B o r e l ( R R V ) - m e a s u r a b l e . I f (Y ) has c o n t i n u o u s sample paths then t h i s map i s a c t u a l l y c o n t i n u o u s by the d e f i n i t i o n o f the t o p o l o g y o f RRV (see 9.21), so i n t h i s case 0^) e V . By a monotone c l a s s argument, i t f o l l o w s t h a t V c o n s i s t s o f a l l bounded measurable r e a l - v a l u e d p r o c e s s e s over (A,F) . In p a r t i c u l a r 1 e V . That i s , the map H X H - P(dw)x(a.) (H(u.)) i s B o r e l ( R R V ) - m e a s u r a b l e . 12.12. Lemma: L e t (A,F,F ,P) be a f i l t e r e d measure space, where P i s f i n i t e , ) i s r i g h t - c o n t i n u o u s , and F i s c o u n t a b l y generated mod P . L e t RST = RST(A,F,F ,P) , and l e t T be a B o r e l s u b s e t of RST . L e t (E,E) be any c o u n t a b l y generated measurable space and l e t (X__)„ be a measurable p r o c e s s over (A,F) w i t h t 0<t<°° s t a t e space (E , E ) . F o r each x e RST , l e t u(x) be as i n 12.11. L e t (Z,A , a ) be a a - f i n i t e measure space and suppose z H - V ( Z ) i s a measurable map from Z to f i n i t e measures on E . L e t Z ^ = { z e Z : v ( z ) = u ( x ) f o r some x e T } . Then: a) Z^ i s measurable w i t h r e s p e c t to the c o m p l e t i o n of a . b) I f a(Z\Z.) = 0 , then t h e r e i s a f a m i l y (x ) „ of I z ze_ randomized ( F ^ - s t o p p i n g times such t h a t : - 166 -i ) x e T and u(x ) = v ( z ) f o r a - a.a. z e Z . z z i i ) F o r each t e (0,°°) , the map (z,w) H- T z ( a ) ) ( [ 0 , t ) ) i s A ® F^-measurable. P r o o f : L e t M be the s e t of f i n i t e measures on E , and l e t M be the s m a l l e s t a - f i e l d o f s u b s e t s o f M which makes the maps of the form m H- m(A) (A e E) measurable. Then (M,M) i s a s e p a r a t e d c o u n t a b l y g e n e r a t e d measurable space. The maps x >-> u(x) and z H- v ( z ) a r e ( B o r e l RST, M)-measurable and (A,M)-measurable r e s p e c t i v e l y . Now RST i s a compact p s e u d o m e t r i z a b l e space, by 9.21 and 9.23(a). Hence (RST, B o r e l RST) i s a B l a c k w e l l space i n the sense of Meyer [ 1 ] . Hence M^ E {u(x) : x e T} b e l o n g s to S o u s l i n M . T h e r e f o r e Z^ = {z e Z : v ( z ) e M^} b e l o n g s to S o u s l i n A ; p a r t a) f o l l o w s from t h i s . Now assume a(Z\Z^) = 0 . Now by the theorem on p. 251 o f D e l l a c h e r i e and Meyer [ 1 ] , t h e r e i s a f u n c t i o n ¥ : M^ -> T such t h a t : ¥ i s ( B o r e l f i e l d (Souslin(M|M )) , B o r e l RST)-measurable and y(H'(m)) = m f o r a l l m e M^ . L e t p z = ¥(v(z)) i f z e Z "0" i f z e Z\Z, Then p e T and u(p ) = v ( z ) f o r a - a.a. z e Z , and the map z z z H- p i s ( B o r e l f i e l d ( S o u s l i n A) , B o r e l RST)-measurable. In z p a r t i c u l a r , z H- p i s (A, B o r e l RST)-measurable, where X i s - 167 the c o m p l e t i o n o f A w i t h r e s p e c t t o a . Now B o r e l RST i s c o u n t a b l y g e n e r a t e d , so f o r some e A w i t h a ( Z \ Z ^ ) = 0 , we have t h a t z »-»• p i s A-measurable on Z . L e t a = { z p i f z e Z A z U '0" i f z e Z\Z 0 Then a = p f o r a - a.a. z € Z , and z a i s (A, B o r e l RST)-z z z measurable. F i n a l l y , l e t x ^ x be as i n 12.10 and l e t x^ = 5 z f o r a l l z e Z . Then c l e a r l y the f a m i l y (x ) „ has the d e s i r e d z zeZ p r o p e r t i e s , • 12.13. Lemma: L e t y be a measure on H n such t h a t U y i s a p o t e n t i a l . L e t RST = R S T ^ B . B ^ P ^ ) and l e t T = {x e RST : x i s y-standard} . Then T e B o r e l RST . P r o o f : I f n >_ 3 then T = RST , and we a r e done. Suppose n = 1 or 2 . Then y i s f i n i t e . F o r each x e RST , l e t y_ be as i n 11.2. u y A . y X X A i T Now x i s y - s t a n d a r d i f f U i s a p o t e n t i a l and U >_ U f o r each n a t u r a l number i . T h i s i s e s s e n t i a l l y 11.12. (See 11.5 f o r the meaning o f x A i .) Now U i s a p o t e n t i a l i f f |$(x)|dy (x) i s f i n i t e , x >1 y . y A l s o , U >_ U T i f f u T A 1 > U whenever V i s an open b a l l - 168 -whose r a d i u s i s r a t i o n a l and whose c e n t r e has r a t i o n a l c o o r d i n a t e s . Thus the c o n d i t i o n t h a t x be u - s t a n d a r d can be e x p r e s s e d i n terms o f c o u n t a b l y many measurable c o n d i t i o n s . 12.14. Lemma: L e t (A,F,P) be a a - f i n i t e measure space, where F i s c o u n t a b l y generated mod P . L e t H be a compact m e t r i z a b l e sp ace, and l e t RRV = RRV(A,F,P;H) . L e t RV = {Y e RRV X = P - a.e. f o r some F-measurable f u n c t i o n f : A -»• H} Then: a) F o r x e RRV > we have x e RV i f f whenever x'» x" e RRV and x = \ X 1 + \ x" p " a- e- t h e n X = x' = x" P - a.e. b). RV i s a G - s e t i n RRV o P r o o f : a) (=>) i s c l e a r . (<=) Suppose x e RRV\RV . L e t U be a c o u n t a b l e open base f o r H . L e t A = {oo e A : x(w) i s n ot a p o i n t mass} . Then A = {to e A : f o r some U e U , 0 < x(w)(U) < 1} . Thus A e F . As x I RV , P(A) > 0 . But then f o r some U e U , P(B) > 0 , where B = {u> e A : 0 < x(^) (U) < 1} . D e f i n e a : A [0, 1] by a(w) = x(^)(U) • D e f i n e a, x e RRV by a (GO) (E) = y(u>) (E\U) l-a(cj) x(oo)(E) = y(oj)(EnU) a(oo) a(u>) = x(u) = x(w) } (UJ e B , E e B o r e l H) (_ e A\B) - 169 -Then {a 4 x} = B , so a and x d i f f e r on a s e t of p o s i t i v e P-measure. A l s o x ( w ) = [1 _ a(co)]a(co) + a(co)x(co) f o r a l l co e A . Now d e f i n e x '» x" E R R V as f o l l o w s : i f a (to) L \ > l e t X' (co) = o(co) X"(ai) = [1 - 2a(co)]o(co) + 2a(co)x(co) ; i f a(co) >_ , l e t X'(co) = [1 - 2(l-a(co))]x(co) + 2(1 - a(oi))a(ai) x"(co) = x(co) . Then (x* 4 x " l = B , so x ' and x " d i f f e r on a s e t o f p o s i t i v e P-measure. A l s o , x = ^ x ' + "J x" • b) By 9.21, RRV i s a compact p s e u d o - m e t r i z a b l e space; l e t K be i t s m e t r i c i d e n t i f i c a t i o n , and l e t ij; be the c a n o n i c a l map o f RRV onto K . F o r a , x e RRV , \p(o) = \J/(x) i f a = x P - a.e. L e t D be the d i a g o n a l i n K x K . Then (K><K)\D i s a-compact. D e f i n e iji : K x K -»• K by <()(x,y) =" y x + -| y " . Then ()> i s c o n t i n u o u s , and by b) , <|>[(KxK)\D] = K \ ^ [ R V ] . Thus K\\JJ[RV] i s a-compact i n K . Hence RV = \p "*"[^[RV]] i s a G^-set i n RRV 12.15. At l a s t we can prove the m e a s u r a b i l i t y a s s e r t i o n s made i n the p r o o f of 12.7. Lemma: L e t (Z,A,a) be a a - f i n i t e measure space. - 170 -L e t D = 3Rn , E = . Suppose IT i s a measurable map o f Z o i n t o D and g i s measurable map from Z to p r o b a b i l i t y measures on E such t h a t f o r a - a.a. z e Z , U ^ Z ^ i s a p o t e n t i a l and U 6 T r ( z ) > U 6 ( Z ) . Then: a) There i s a f a m i l y (x ) _ o f randomized (8 ) - s t o p p i n g z zeZ t times such t h a t : i ) f o r a - a.a. z e Z , x i s 6 . . - s t a n d a r d and z T T ( Z ) (« /_0_ = S(z) . TT (. Z ) X Z i i ) f o r each t e (0,°°) , the map (z,u>) ^ x (w)([0, t ) ) z i s A ® B -measurable, t b) I f n = 1 , t h e r e i s a f a m i l y (T ) of genuine (B ) -2 Z £ _ J t s t o p p i n g times such t h a t : i ) f o r a - a.a. z e Z , T i s 6 , ^ - s t a n d a r d and z T T ( Z ) TT (.z; T z i i ) f o r each t e (0,°°) , the map (z,o.) 1^_ t ^ i s A ® 8^-measurable. P r o o f : F i r s t l e t us e x p l a i n some o f the n o t a t i o n used i n the statement o f the lemma. I f u i s a measure on B o r e l E , x i s a randomized ( B p - s t o p p i n g time, and T i s a genuine ( B p - s t o p p i n g time, then u_ denotes the measure on B o r e l E d e f i n e d by - 171 -V T ( A ) = P y (dto)x(co)({t e [0, »] : B (a>) £ A}) w h i l e y^, denotes the measure on B o r e l E d e f i n e d by y T ( A ) = P y ( B T £ A) . Now f o r each z e Z , l e t v ( z ) be B(z) t r a n s l a t e d by - T T ( Z ) , so t h a t u v ( z ) < u 6 f o r a - a.a. z . I f n >_ 2 , l e t T be the s e t o f 6 - s t a n d a r d randomized ( B p -s t o p p i n g t i m e s . I f n = 1 , l e t T be the s e t o f 6 - s t a n d a r d randomized (8 ) - s t o p p i n g times x such t h a t x(co) = 6 , . f o r t t (co; P - a.a. co e ft , f o r some B-measurable f u n c t i o n f : ft •+ [0, °°] . In e i t h e r c a s e , T i s a B o r e l s e t i n RST(ft,8,8 t,P ) . (Apply 12.13 i f n >_ 2 ; a p p l y 12.13 and 12.14 i f n = 1 .) Now f o r a - a.a. z e Z , t h e r e e x i s t s x e T such t h a t <5T = v ( z ) . (Apply 10.5 i f n >_ 3 , 11.14 i f n = 2 , o r 8.20 i f n = 1 .) Thus by 12.12, t h e r e i s a f a m i l y (a ) such t h a t : z Z £ Z i ) f o r a - a.a. z £ Z , a e T and 6 = v(z) . z a z i i ) f o r each t £ (O, 0 0) , the map (z,co) o (co)([0,c°)) i s z A ® B -measurable, t F o r each z , l e t x be "the t r a n s l a t i o n o f a by + T T ( Z ) " . The z z f a m i l y ( x ) then has the p r o p e r t i e s a s s e r t e d i n a ) . Now suppose z n = 1 . F o r each z , l e t T be the (B ) - s t o p p i n g time d e f i n e d by z t - 172 -T (GJ) = i n f { t : T ( u ) ) ( [ 0 , t ) ) > 0} . 2 2 Then x (u ) = 6_ , f o r P Z - a.a. co e ft , by the c h o i c e o f T . z T (_) z A l s o {T < t} = {x (»)([0,t)) > 0} . C l e a r l y then, the f a m i l y z z (T_) has the p r o p e r t i e s a s s e r t e d i n b) . ~ l 12.16. Theorem: L e t n be a p o s i t i v e i n t e g e r , and l e t ( A , F , F t > X t , P ) be an n - d i m e n s i o n a l p o t e n t i a l p r o c e s s , w i t h time s e t [0,°°) , and w i t h r i g h t - c o n t i n u o u s sample p a t h s . L e t u = law(X^) . Then t h e r e i s an o p t i o n a l enlargement (ft' ,8' ,8|.,Q,i>) of ( f t , 8 , 8 t , P y ) and an i n c r e a s i n g f a m i l y ( T t ) p < t < _ o f ( B ^ ) - s t o p p i n g times which a r e st a n d a r d r e l a t i v e to Q , such t h a t : a) F o r each w e f t ' , t H- T t (w) i s r i g h t - c o n t i n u o u s on [0,-) . b) The p r o c e s s e s ( x p and (B_j_ ) have the same f i n i t e d i m e n s i o n a l j o i n t d i s t r i b u t i o n s , where B' = B ° ii f o r J s s 0 <_ s <_ »• . P r o o f : L e t (ft",8",8J],R,d)) be the p r o d u c t enlargement o f (f t , 8 , 8 t , P P ) by (L, L ,A) , where L = (0,1) , L = B o r e l L , and A = Lebesgue measure on L . A l s o , l e t B'^  = B ° <j) f o r 0 <_ t <_ 00 . Then by 12 . 7 ( a ) , f o r each n a t u r a l number k , t h e r e i s an i n c r e a s i n g k f a m i l y (S ) ^ o f ( B ' p - s t o p p i n g times which a r e s t a n d a r d r e l a t i v e k to R such t h a t the p r o c e s s e s (X ) and (B", ) have the ' t e l . c K k S t t e l k -k same j o i n t d i s t r i b u t i o n , where 1^ = {j2 : j = 0 , 1, 2,...} . Now 173 -d e f i n e X k , T k f o r a l l t e [0 , ») by X? = X j 2 -k T = T j 2 -k } f o r ( j - l ) 2 ~ k < t < j 2 k , j = 1, 2, 3, Then (X ) and (B ' \ )„ have the same f i n i t e d i m e n s i o n a l t 0<t<°° 0<t<°° j o i n t d i s t r i b u t i o n s , and X -»• X^ as k -»• °° , f o r each t . t t L e t H be the space of r i g h t - c o n t i n u o u s i n c r e a s i n g maps o f [O, 0 0) i n t o [0, 0 0] , t o p o l o g i z e d as i n 9.24. A l s o , l e t H and (H ) „ be as i n 9.24. t 0<t<?° L e t ft' = ft x H B' = 8 9 H ( B p = ( ( B t 9 H f c ) + ) ijj = p r o j e c t i o n of ft' on ft B ' = B o ib (0<t<°°) . t t Y  For each k , l e t T be the map of ft" i n t o ft' d e f i n e d by r, (co) = ((f) (co) , T . ( c o ) ) . Then r i s ( B " , B ' ) - m e a s u r a b l e , and a l s o ( B " , B ' ) - m e a s u r a b l e f o r k t t each t . (co T^(co) i s (B'p H p -measurable f o r each t , and (B") i s r i g h t - c o n t i n u o u s . ) Next, f o r each k , l e t Q, be the t K - 174 -measure on 8' d e f i n e d by Q F C(A) = R O ^ C A ) ) . Suppose f i s a no n - n e g a t i v e 8'-measurable f u n c t i o n on ft' . Then f ° r i s B'^-measurable, and for any A e 8 , f dQ ,-1 f ° r , dR - 1 - 1 k ip [A] r k " [ A ] ] f ° r , dR . -1 k <j> X [ A ] Thus E, ( f | ^ , 8) = E ( f o r |<f>, 8) , which i s 8 -measurable mod P V . Thus f o r each k , (ft' ,8' , B^,Q k , i j j ) i s an o p t i o n a l enlargement o f (f t , B , 8 t , P y ) . F o r each t e [0, ») , d e f i n e T f c on ft' by T t ( u , h ) = h ( t ) . Then each T f c i s a ( B p - s t o p p i n g time s a t i s f y i n g T t o = T f c . F o r each k , the p r o c e s s (B^ ) has the same f i n i t e d i m e n s i o n a l j o i n t d i s t r i b u t i o n s , r e l a t i v e t o Q K , as (X^) has ( r e l a t i v e t o P ) . Each T i s s t a n d a r d r e l a t i v e to Q . A l s o , f o r each (co,h) e ft' , t H- T t(_),h) i s i n c r e a s i n g and r i g h t - c o n t i n u o u s on [0, °°) . By 9.27, t h e r e i s a subsequence ( \ ( ^ ) ) o f (\) a n d a p r o b a b i l i t y measure Q on 8' such t h a t (ft',8',8^,Q,IJJ) i s an o p t i o n a l f ( u ) ) ( h ) Q k ( £ ) ( d o j , d h ) enlargement o f (ft,B,B t,P K) and f o r a l l f e L 1(ft,B,P y;C(H)) . We c l a i m t h a t the s t o p p i n g times T f c a r e s t a n d a r d r e l a t i v e to Q and t h a t the p r o c e s s (B_ ) Q < t < a o has the same f i n i t e d i m e n s i o n a l j o i n t d i s t r i b u t i o n s r e l a t i v e t o Q as (X ) . has ( r e l a t i v e to P ) . T h i s i s somewhat e a s i e r to t 0<t<°° f (u) (h)Q(doo,dh) - 175 -prove when n 21 3 , so l e t us c o n s i d e r t h i s case f i r s t . I n t h i s c a s e, e v e r y s t o p p i n g time i s s t a n d a r d , and we need o n l y check t h a t the j o i n t d i s t r i b u t i o n s a r e r i g h t . L e t D = ]R n , E = , l e t 0 <_ t < ... < t . <°° , and l e t 0 1 J g be a c o n t i n u o u s f u n c t i o n on E 3 which v a n i s h e s o u t s i d e a compact s u b s e t o f D 3 . Then g(B' ,...,B' )dQ fcl fcj (*) l i m e+0 1 e l i m l i m e4-0 I-** re g ( B l 0 t±+s 1 e re ,B; )ds dQ t.+s 3 g ( B l ,...,B'^ Jds d Q k W j 0 t±+s l i m l i m e+0 £->~ 1 e g(B" k(£) ' t 1 + s . . B « ( i ) ) d . dR t.+s 3 l i m l i m e+0 ; 0 £ i + s l i m e+0 g(X ,...,X )ds dP 0 V S 3 g(X ,...,X )dP 1 3 T h i s s u f f i c e s t o show t h a t (B^, ) Q < t < O 0 h a s t n e r i g h t f i n i t e - 176 -d i m e n s i o n a l j o i n t d i s t r i b u t i o n s . However, the s t e p (*) r e q u i r e s f u r t h e r e x p l a n a t i o n . (We remark t h a t a l l the o t h e r s t e p s i n the above c a l c u l a t i o n h o l d f o r any n , but the method o f v e r i f i c a t i o n of the s t e p (*) depends on n , and i s s i m p l e s t when n _> 3 .) W e l l , g(B' ( o ) , h ) , ...,B' (_,h))ds 0 l i + s l 2 + S re Q g ( B h ( t 1 + s ) ( w ) ' - - " B h ( t . + s ) ( a , ) ) d s Now i f h -»• h i n H then h (t.+s) •+ h(t.+s) f o r i = l , . . . , j , f o r m m i l a l l b u t c o u n t a b l y many s (see 9.24). Now ( B ^ depends c o n t i n u o u s l y on t , except a t t = 0 0 ; but g i s s u p p o r t e d by a compact s u b s e t o f D J and, as n _ _ 3 , | | B t | | ->- °° P P - a .s. as t -»- 0 0 . Thus f o r P y - a.a. w e f t , the map re Q 8 ( B h ( t 1 + s ) ( w ) ' - - " B h ( t j + s ) ( w ) ) d s i s c o n t i n u o u s on H . T h i s j u s t i f i e s the s t e p (*) i n the case n >_ 3 Now suppose n = 1 o r 2 . L e t D, E, t ^ , . . . , t j , and g be as b e f o r e . F i x e e [0, °°) . D e f i n e f on ft x H by re f ( o i , h ) = Q g ( B h ( t 1 + s ) ( t ° ) ' " - ' B h ( t . + s ) ( w ) ) d s and f o r each a e [0, °°) d e f i n e f on ft x H by cL re f ( 0),h) 0 8 ( B h ( t 1 + s ) A a ( w ) ' - - - ' B h ( t j + s ) A a ( w ) ) d S - 177 -Then f , f (0<a<°°) a r e u n i f o r m l y bounded, each f i s c o n t i n u o u s i n i t s second v a r i a b l e , and f = f on ft x H , where a a H = {h e H : h < a on [0, t . + e)} . Now u s i n g 8.9 and a s l i g h t a — J g e n e r a l i z a t i o n o f 8.10, one f i n d s t h a t l i m sup Q (f2'\(fixH )) = 0 . a-*>° k But i f h h i n H and each h i s bounded by a on [0, t . + e) then h ( t ) <_ a f o r each c o n t i n u i t y p o i n t t o f h i n [0, t ^ + e) , whence h < a on [0, t . + e) . Thus each H i s c l o s e d i n H . - 3 a Hence Q(ftxH ) _> l i m sup Q . (fixH ) . I t f o l l o w s t h a t a i + oo k W a l i m Q(S7*\ftxH ) = 0 . a But f dQ - f dQ k(£) f - f dQ a + f dQ -a f a d\(i) + f a " f d C W ) Now the f i r s t term on the r i g h t hand s i d e h e r e can be made s m a l l by t a k i n g a l a r g e , the t h i r d term can be made s m a l l u n i f o r m l y i n I by t a k i n g a l a r g e , w h i l e the second term goes to 0 as I •+ 0 0 , f o r each a . Thus f dQ . T h i s s u p p l i e s the j u s t i f i c a t i o n o f the st e p (*) above i n the case n <_ 2 , so we now - 178 -know t h a t (B^ ) has the r i g h t f i n i t e d i m e n s i o n a l j o i n t d i s t r i b u t i o n s t r e l a t i v e to Q i n a l l c a s e s . I t remains to show t h a t the s t o p p i n g times T a r e s t a n d a r d r e l a t i v e to Q when n = 1 o r 2 . L e t t t e [0, 0 0) . L e t o = law(X . Then a i s a p r o b a b i l i t y t+1 measure on E and a(D) = 1 . Now re law(B' ;Q) = l i m -Lt e4-0 law(B' ;Q)ds 0 t+s = l i m l i m — e+0 £-*=° £ o l a " ( B i t + 6 ; Q M ^ ) ) d s • where the second s t e p f o l l o w s from the f a c t t h a t law(B' ;Q , %) law(B' ;Q) f o r each I , as was shown above, T t + s k W T t + s (The l i m i t s a r e w i t h r e s p e c t to the vague t o p o l o g y . ) Now c o n s i d e r any a e [0, °°) . Then l a w ( B l ;Q) = l i m -T t a e+0 e re law(B' A a ; Q ) d s 0 t+s l i m l i m e+0 l-**> l a w ( B T t + s A a ; Q k ( £ ) ) d S ' where i n t h i s c a s e , the second s t e p f o l l o w s from the f a c t t h a t i f f i s a bounded c o n t i n u o u s f u n c t i o n on E then the map (o),h) f(B, , , s ( u ) ) d s h(t+s)Aa i s c o n t i n u o u s i n h Now f o r 0 <_ s <_ — and k(£) >_ 1 , - 179 -U law(B t+s ;Q. ) > u a A p p l y i n g 8.7 and 8.8, one f i n d s t h a t law(Bj, A a;Q) law(B^, ;Q) U > U Then, by 11.12, T i s s t a n d a r d r e l a t i v e t o Q • I n the case n = 1 , the above theorem j u s t says t h a t any r i g h t -c o n t i n u o u s m a r t i n g a l e can be embedded i n an o p t i o n a l enlargement of Brownian motion by means of a r i g h t - c o n t i n u o u s i n c r e a s i n g f a m i l y o f s t a n d a r d s t o p p i n g times — a r e s u l t due to Monroe [ 1 ] . (We remark t h a t Monroe works w i t h minimal s t o p p i n g times, r a t h e r than s t a n d a r d ones. I t i s easy to show t h a t a s t a n d a r d s t o p p i n g time i s m i n i m a l . Monroe i n e f f e c t p r o v e s t h a t the converse i s t r u e when n = 1 , though he does not e x p l i c i t l y d e f i n e s t a n d a r d s t o p p i n g t i m e s . T h i s i s an i n t e r e s t i n g r e s u l t i n i t s own r i g h t , b u t f o r the purpose of embedding, i t i s perhaps e a s i e r t o work w i t h the d e f i n i t i o n o f s t a n d a r d s t o p p i n g times than w i t h the d e f i n i t i o n of minimal s t o p p i n g times.) 12.17. C o r o l l a r y : L e t I = {0, -1, -2,...} and l e t (A,F,F ±,X ,P) be an n - d i m e n s i o n a l p o t e n t i a l p r o c e s s w i t h time s e t I . Then t h e r e i s a p r o b a b i l i t y measure u on ] R n such t h a t U y i s a 0 p o t e n t i a l (and p({9}) = 0 i f n = 1 o r 2 ) , and an o p t i o n a l enlargement - 180 -( f t ' , B ' , B ' , Q , i f O of ( f t , B , B t , P U ) , such t h a t t h e r e a r e BJ.-stopping txmes T 0 ^ - T - l - T - 2 ^ which a r e s t a n d a r d r e l a t i v e to Q , such t h a t the p r o c e s s e s (X^) and (B^ ) have the same j o i n t d i s t r i b u t i o n (where B| = • f i f o r 0 <_ t <_ °° ) . P r o o f : There a r e i n d i c e s i ^ > i ^ > i ^ > .•• i n I such t h a t f ( x ) l a w ( X . ) ( d x ) 1 j f ( x ) y ( d x ) f o r e v e r y compactly s u p p o r t e d c o n t i n u o u s f u n c t i o n f on E = 3Rg , where y i s some measure on E s a t i s f y i n g y(E) <_ 1 . law(X.) law(X ) Now U 1 i u f o r e v e r y i . Thus, by 8.7 and 8.8 ( i f n = 1 o r 2) , U y i s a p o t e n t i a l and U law(X ) 2 dy -> U y dy f o r e v e r y good measure y on D = ]R n . ( I f n _> 3 i t i s t r i v i a l to check t h a t t h i s convergence h o l d s . ) law(X.) law(X ) But U 1 i n c r e a s e s as i d e c r e a s e s . Thus U + U as i -> -°° . (Hence we a l s o have t h a t law(Xj ->• y v a g u e l y as i -»• -°° .) I f n = 1 o r 2 then, by 8.9, l i m sup law(X.) ({x e D : | |x| | >. r}) = 0 sup r-x» i - 181 -In t h i s case we a l s o have law(X^)({9}) = 0 f o r a l l i . Thus i f n = 1 o r 2 then y i s a p r o b a b i l i t y measure and law(X.) y({9}) = 0 . On the o t h e r hand, i f n j> 3 then from U y >_ U 1 we can deduce t h a t u(D) _> law(X i>(D) (see p r o o f of 7.8); a l s o , as 9 i s an i s o l a t e d p o i n t o f E , l a w ( X i ) ( { 9 } ) •*• y({3}) as !->•-«>. Thus i n t h i s case too, y i s a p r o b a b i l i t y measure. Now f o r each y e D , ($(X^,y)) i s a s u p e r m a r t i n g a l e o v e r (A , F , F.,P) , and sup E ( $ ( X . , y ) ) = U P ( y ) . Thus by V, T21 o f Meyer [1] 1 i 1 f o r each y e D such t h a t U y ( y ) < » t $(X^,y) converges almost s u r e l y and i n L X , as !->•--, to an i n t e g r a b l e f u n c t i o n . But from t h i s i t f o l l o w s t h a t t h e r e e x i s t s X : A ->• E such t h a t —GO X i s F -measurable, where F = nF. , and X. -*• X P - a.s. —oo —oo —oo ^ 2. i — 0 0 as 1 -> -« . C l e a r l y law(X ) = y and ( A , F , ( F . ) (x ) P) i s a p o t e n t i a l p r o c e s s , where J = Iu{-°°} • Now l e t G0 = F - o o . Y 0 = X - » and l e t G = F Y = Xn f o r 1 < t < 0 0 t 0 t 0 -G t - 1=_v Y t = X_ x f o r | < t < 1 , and so on. Then ( A , F , G t , Y t » P ) i s a p o t e n t i a l p r o c e s s , w i t h time s e t [0, 0 0) , w i t h r i g h t - c o n t i n u o u s sample p a t h s , and so can be embedded i n an o p t i o n a l enlargement o f Brownian motion w i t h i n i t i a l d i s t r i -b u t i o n law(Yg) = y , by means of a r i g h t - c o n t i n u o u s i n c r e a s i n g f a m i l y o f s t a n d a r d s t o p p i n g times, by 12.16. I t i s now c l e a r how to - 182 -complete the p r o o f . < g I n the above c o r o l l a r y , i t i s n e c e s s a r y t o c o n s i d e r an e n l a r g e -ment o f Brownian motion, even i n the one d i m e n s i o n a l c a s e . T h i s i s shown by the f o l l o w i n g example, which was i n v e n t e d by R. V. Chacon. 12.18. Example: L e t y be the u n i t p o i n t mass a t 0 i n ]R n , and l e t (ft' , B ',8j.,Q,40 be the p r o d u c t enlargement of ( f t , B , B T , P M ) by ( L , L , A ) , where L = {0, 1} , L = power s e t o f L , and A i s o t h e measure on L which a s s i g n s mass to {0} and to {1} . L e t B' = B ° ij; f o r 0 < t < » . For i = 0, 1, 2,... d e f i n e t t — — T! on ft' by T:(OJ,0) = i n f { t > 0 : ||B.(a>)|| = 2 1" 1} x — t T! (W,1) = i n f { t > 0 : | |B. (u>) | | = 2 1 } x — t Then each T^ i s a ( B ^ ) - s t o p p i n g time which i s s t a n d a r d r e l a t i v e t o and _> T 1 _> • • • . Thus ( f t ' , B ' , B ^ , , B ^ , , Q ) i s a p o t e n t i a l p r o c e s s w i t h time s e t i i {0, -1, -2, ...} , by 12.6. Suppose TQ T ^ >_ ... a r e y - s t a n d a r d ( B ^ - s t o p p i n g times such t h a t ( B T ) and (B^,) have the same j o i n t d i s t r i b u t i o n , i i We s h a l l deduce a c o n t r a d i c t i o n . W e l l by 8.13, as T± i s y-y. ( U y - U 1 ) , where y.. = l a w ( B T ) = law(B^,) s t a n d a r d , E^(T.) x x - 183 -But law(B^,,) = j a±-i + \ °± w n e r e f o r e a c h J > °j i s t h e u n i f o r m u n i t d i s t r i b u t i o n on the sphere of r a d i u s 2~* c e n t r e d a t 0 I t f o l l o w s t h a t E M ( T . ) -> 0 as i -»•-<» . Hence T. 4- 0 P P - a.s. ^ l i . . As (8y) i t as i -> -°° . L e t T = l i m T_ . s 8y) i s r i g h t - c o n t i n u o u s , 8y = n8y . Now B y = {E e 8y : E n {T £ t} e B y f o r 0 £ t < »} 1 i s i n c e P y(T=0) = 1 and s i n c e e v e r y s e t of^ P y-measure zero b e l o n g s to 8y . But s i n c e u i s a p o i n t mass, 8y = {0, ft} mod P y . Now f o r each i , l e t E = {||B || = 2 1} . Then P y ( E ± ) = j . i But we a l s o have P y ( E . n E. ,) I x-1 = Q ( | | B l , | | = 2 1 , ||B', || = 2 1 " 1 ) i i - 1 = Q(ftx{l}) = | . Thus, mod P y , E = E = E = ••• . L e t E = n u E. . 0 -1 -2 i = 0 j = i l Then E e n B y = 8y , b u t P y ( E ) = ~ , which c o n t r a d i c t s the i i t r i v i a l i t y of B y . - 184 -13. APPENDIX OF MISCELLANEOUS NOTATION AND TERMINOLOGY 13.1. We take p o s i t i v e to mean s t r i c t l y p o s i t i v e (though by a p o s i t i v e measure we r e a l l y j u s t mean a n o n - n e g a t i v e one; a l s o , by i n c r e a s i n g we j u s t mean n o n - d e c r e a s i n g ) . A , rea d meet, denotes g r e a t e s t lower bound, w h i l e v , rea d j o i n , denotes l e a s t upper bound. I f f i s any [-00,°°]-valued f u n c t i o n then f + denotes fvO and f ~ denotes (-f)vO . 13.2. ]R denotes the s e t of r e a l numbers. ]N denotes the s e t {0,1,2,...} o f n a t u r a l numbers. Q denotes the s e t of r a t i o n a l numbers. 13.3. I f x = ( x l 5 . . . , x ) e ]R n then l l x l l denotes the u s u a l 1 n e u c l i d e a n norm of x : 2 2 1 / 2 ||x|| = ( x j + ...+ x 2 ) . 13.4. We use the symbol A i n two ways; t o denote the L a p l a c i a n : n g2 A = £ — j , and to denote symmetric d i f f e r e n c e o f s e t s : 1=1 3x. l AAB = (A\B)u(B\A) . 13.5. L e t X be a t o p o l o g i c a l space. F o r E c X , i n t ( E ) denotes the i n t e r i o r of E , E denotes the c l o s u r e o f E , and 8E denotes the f r o n t i e r o f E (3E = E \ i n t (E)) . I f f i s a [-°°, °°]-valued f u n c t i o n on X then we s h a l l say f i s c o n t i n u o u s i f f f ''"[U] i s open i n X f o r each U open i n [-°°, 0 0] ; the p o i n t o f t h i s remark i s to emphasize t h a t when we say " f i s c o n t i n u o u s " , we - 185 -are not s u g g e s t i n g t h a t i t assumes o n l y f i n i t e v a l u e s . We s h a l l n o t use the e x p r e s s i o n " c o n t i n u o u s i n the extended s e n s e " . I f Y i s a n o t h e r t o p o l o g i c a l s p a c e , then C(X,Y) denotes the s e t of c o n t i n u o u s f u n c t i o n s from X i n t o Y ; C(X) denotes C(X^R) . B o r e l X denotes the o - f i e l d g e n e r a t e d by the c l o s e d s u b s e t s of X . I f u i s a [-°°, °°]-valued f u n c t i o n on X then the lower  r e g u l a r i z a t i o n of u , denoted by u , i s the l a r g e s t lower-s e m i c o n t i n u o u s f u n c t i o n v on X s a t i s f y i n g v <_ u ; a l a r g e s t such f u n c t i o n always e x i s t s s i n c e the supremum of any c o l l e c t i o n of l o w e r - s e m i c o n t i n u o u s f u n c t i o n s i s a g a i n l o w e r - s e m i c o n t i n u o u s . F o r each x e X , we have u(x) = u(x) i f x i s an i s o l a t e d p o i n t o f X l i m i n f u(y) i f x i s a l i m i t p o i n t of X y x We remark t h a t l i m i n f u(y) denotes sup i n f u(y) , where y -y x Vet/_ yeV\{x} x f i s the c o l l e c t i o n of neighbourhoods of x x 13.6. When we speak o f a measure, w i t h o u t a q u a l i f i e r such as " s i g n e d " or " r e a l - v a l u e d " , we s h a l l always mean a p o s i t i v e measure. An o u t e r measure on a s e t X i s a [0, <=°]-valued f u n c t i o n y on the power s e t of X such t h a t whenever S i s a c o u n t a b l e c o l l e c t i o n o f s u b s e t s of X and R c u S we have y (R) <^  \ y (S) ; (In SeS p a r t i c u l a r , t a k i n g R = 0 and S = 0 , we f i n d t h a t y(0) = 0) . I f y i s an o u t e r measure on X , then M denotes the a - f i e l d y of y-measurable s u b s e t s of X : - 186 -M = {R c X : f o r any S c X , y(S) = y (SnR) + y(S\R} . y I f H i s a c o l l e c t i o n of s u b s e t s o f X then an tf-outer measure on X i s an o u t e r measure y on X such t h a t H c M and ~~ y f o r e v e r y R c X , y(R) = i n f { y ( H ) : R c H e H} . I f X i s a t o p o l o g i c a l space then by a measure on (or i n ) X , we mean a measure y on B o r e l X such t h a t y(K) i s f i n i t e f o r each c l o s e d compact s e t K c x . I f (X ,A) i s a measurable space and y i s a measure on A and A e A then y. denotes the measure on A A d e f i n e d by y.(B) = y(BnA) . 13.7. We have made some use of the th e o r y of S o u s l i n s e t s and of a n a l y t i c s e t s . We r e f e r the r e a d e r to Meyer [ 1 ] , D e l l a c h e r i e and Meyer [ 1 ] , and B r e s s l e r and S i o n [1] f o r e x p o s i t i o n s o f t h i s t h e o r y . 13.8. I f D i s a Green r e g i o n then the symbol G^ , used w i t h o u t e x p l a n a t i o n , denotes the Green f u n c t i o n of D . 13.9. Suppose V i s an open b a l l i n l R n and f i s a [-00 , 0 0 ] - v a l u e d f u n c t i o n on 3V . Then P I ( f ; V ) denotes the f u n c t i o n d e f i n e d by P I ( f ; V ) ( x ) = 1 a r n f 2 2 r ~ I i x " p | 1 f (z) d a ( z ) 3V l | x - z l | n f o r a l l x e V f o r which the i n t e g r a l makes sense, where p i s the c e n t r e o f V , r i s the r a d i u s of V , a i s the " s u r f a c e a r e a " measure on 3V , and a = the " s u r f a c e a r e a " o f the u n i t sphere i n n ]R n . Note PI sta n d s f o r " P o i s s o n i n t e g r a l " . I f f i s i n t e g r a b l e w i t h r e s p e c t t o a over 3V , then P I ( f ; V ) i s harmonic i n V and - 187 -and l i m P I ( f ; V ) ( x ) = f ( z ) f o r each z € 8V a t which f i s x->z c o n t i n u o u s — see 2.4 and 2.7 of Helms [ 1 ] . 13.10. We f o l l o w the c o n v e n t i o n t h a t 0 times 0 0 e q u a l s 0 . In a d d i t i o n , we adopt the c o n v e n t i o n t h a t 0 times u n d e f i n e d e q u a l s 0 ; see 6.1 f o r an example o f t h i s . 13.11. Suppose (A ,F) and (E ,E) a r e measurable s p a c e s , P i s a measure on F , and : A -»• E i s (F ,E)-measurable. Then ^(P) , law(i(j) , and law(^;P) a l l denote the measure u on E d e f i n e d by y (A) = P (ip X [ A ] ) ; we use whic h e v e r n o t a t i o n i s most s u g g e s t i v e and l e a s t ambiguous i n a g i v e n s i t u a t i o n . ' I f g i s a [-°°, °°]-v a l u e d F-measurable f u n c t i o n on A , then E(g|i | ; ,E) denotes any E-measurable f u n c t i o n h on E such t h a t h(x)lawGjj) (dx) = A g dP f o r a l l A e E ; i f law Op) i s a - f i n i t e and g i s e i t h e r n o n - n e g a t i v e or i n t e g r a b l e , such a f u n c t i o n h e x i s t s , and i s unique mod lawGjO . F i n a l l y , a d i s i n t e g r a t i o n of P w i t h r e s p e c t t o , E i s a f a m i l y (P ) of p r o b a b i l i t y measures on F such t h a t f o r each A e E x xeE and each F e F , x B- P (F) is E-measurable, and x P ( F n ^ 1 [ A ] ) = P (F)law(i | 0(dx) . J A X I f lawdlO is a-finite, F is countably generated, and P is - 188 -i n n e r r e g u l a r w i t h r e s p e c t t o a semicompact c l a s s , then a d i s i n t e -g r a t i o n o f P w i t h r e s p e c t t o \\i, E e x i s t s . (See a l s o 9.4.) - 189 -REFERENCES R. B a x t e r and R. V. Chacon 1. P o t e n t i a l s of Stopped D i s t r i b u t i o n s . 111. J . Math. 18 (1974) 649-656. 2. Compactness o f S t o p p i n g Times, Z. W a h r s c h e i n l i c h k e i t s t h e o r i e verw. G e b i e t e , 40 (1977) 169-181. 3. Enlargement of o - a l g e b r a s and Compactness of Time Changes, Can. J . Math., 29_ (1977) 1055-1065. M. Blumenthal and R. K. Getoor 1. Markov P r o c e s s e s and P o t e n t i a l Theory, Academic P r e s s , 1968. B r e l o t 1. M i n o r a n t e s Sous-harmoniques, Extremales e t C a p a c i t e s , J . de Math. Pure e t App., IX, 24. (1945) 1-32. 2. Elements de l a T h e o r i e C l a s s i q u e du P o t e n t i e l , 3 e d i t i o n (1965), C e n t r e du Documentation U n i v e r s i t a i r e , 5 P l a c e de l a Sorbonne, P a r i s . W. B r e s s l e r and M. S i o n 1. C u r r e n t Theory of A n a l y t i c S e t s , Can. J . Math., 16. (1964) 207-230. V. Chacon 1. P o t e n t i a l P r o c e s s e s , T r a n s . Amer. Math. S o c , 226 (1977) 39-58. V. Chacon and J . B. Walsh 1. One D i m e n s i o n a l P o t e n t i a l Embedding, U n i v e r s i t e de S t r a s b o u r g , S e m i n a i r e de P r o b a b i l i t e s X, 19 74/75, S p r i n g e r - V e r l a g , L e c t u r e Notes i n Mathematics, no. 511. - 190 -C. D e l l a c h e r i e and P. A. Meyer 1. P r o b a b i l i t i e s e t p o t e n t i e l , e d i t i o n e n t i e r e m e n t r e f o n d u e , Hermann, 19 75. J . L. Doob 1. Se m i m a r t i n g a l e s and Subharmonic F u n c t i o n s , T r a n s . Amer. Math. S o c , 77 (1954) 86-121. L. E. Dub i n s 1. On a Theorem of Skorohod, Ann. Math. S t a t . 39. (1968) 2094-2097. N. Du P l e s s i s 1. An I n t r o d u c t i o n to P o t e n t i a l Theory, O l i v e r and Boyd, Edi n b u r g h , 1970. L. L. Helms 1. I n t r o d u c t i o n to P o t e n t i a l Theory, John W i l e y and Sons, Inc . , 1969. G. A. Hunt 1. M a r k o f f P r o c e s s e s and P o t e n t i a l s I , ' 44-93. 2. M a r k o f f P r o c e s s e s and P o t e n t i a l s I I , 316-369 • 3. M a r k o f f P r o c e s s e s and P o t e n t i a l s I I I 151-213. A. I o n e s c u F u l c e a and C. Ionescu F u l c e a 1. T o p i c s i n the Theory of L i f t i n g , S p r i n g e r - V e r l a g , 1969. P. A. Meyer 1. P r o b a b i l i t y and P o t e n t i a l s , B l a i s d e l l , 1966. 2. P r o c e s s u s de Markov, S p r i n g e r - V e r l a g , L e c t u r e Notes i n Mathematics, no. 26, 1967. - 191 -3. Sur un A r t i c l e de Dubins, U n i v e r s i t e de S t r a s b o u r g , Seminaire de P r o b a b i l i t e s V, 1969/70, S p r i n g e r - V e r l a g , L e c t u r e Notes i n Mathematics, no. 191. 4. Convergence F a i b l e e t Compacite des Temps d ' A r r e t , d'apres  B a x t e r e t Chacon, U n i v e r s i t e de S t r a s b o u r g , S e m i n a i r e de P r o b a b i l i t e s X I I , 1976/77, S p r i n g e r - V e r l a g , L e c t u r e Notes i n Mathematics, no. 649. I. Monroe 1. On Embedding R i g h t Continuous M a r t i n g a l e s i n Brownian  M o t i o n , Ann. Math. S t a t . , 43 (1972) 1293-1311. J . K. P a c h l 1. D i s i n t e g r a t i o n and Compact Measures, to appear i n Math. Scand. B. J . P e t t i s 1. On I n t e g r a t i o n i n V e c t o r Spaces, T r a n s . Amer. Math. S o c , 44 (1938) 277-304. M. Rao 1. Brownian M o t i o n and C l a s s i c a l P o t e n t i a l Theory, Feb., 1977, Aarhus U n i v e r s i t y L e c t u r e Notes S e r i e s , no. 47. D. H. Root 1. The E x i s t e n c e o f C e r t a i n S t o p p i n g Times of Brownian M o t i o n , Ann. Math. S t a t . 40 (1969) 715-718. H. R o s t 1. D i e S t o p p v e r t e i l u n g e n e i n e s M a r k o f f - P r o z e s s e s m i t  L o k a l e n d l i c h e m P o t e n t i a l , M a n u s c r i p t a Math., 3 (1970) 321-330. 2. The S t o p p i n g D i s t r i b u t i o n s of a Markov P r o c e s s , I n v e n t . Math., 14 (1971) 1-16. A. B. Skorohod 1. S t u d i e s i n the Theory o f Random P r o c e s s e s , Addison-Wesley, 1965. - 192 -INDEX OF SELECTED NOTATION AND TERMINOLOGY a n a l y t i c s e t s 13.7 B 5.4 t 8 ° , B°, B\ B j , 8, B t , °B , J B 5.4 b a l a y a g e , b a l ( , , ) 1.11, 2. base( , ) 3.1 bas e ( ) 3.10 D 5.4 A d i s i n t e g r a t i o n 13.11 E( | , ) 13.11 enlargement 9.7, 9.12 f i l t e r e d measurable space 9.11 f i l t e r e d measure space 9.11 f i n e t o p o l o g y 3.12 f r i n g e ( , ) 3.1 f r i n g e ( ) 3.10 G D 13.8 Gy 1.10 g e n e r a l i z e d Brownian motion p r o c e s s 5.5 good measure 8.6 Green f u n c t i o n 1.9 Green p o t e n t i a l 1.10 Green r e g i o n 1.9 law( ) , law( , ) 13.11 lower r e g u l a r i z a t i o n 13.5 - 193 -measure 13.6 o p t i o n a l enlargement 9.12 o u t e r measure 13.6 p 5.1 P 5.3 t P y , P X 5.4 P I ( ; ) 13.9 p o l a r s e t 1.9 p o t e n t i a l 1.4, 1.10 p o t e n t i a l p r o c e s s 12.2 p r o d u c t enlargement 9.17 randomized random v a r i a b l e , r r v 9.1 randomized s t o p p i n g time, r s t 9.2 r e d u i t e , r e d ( , , ) 1.11 R i e s z measure 1.2 RRV( , , ; ) 9.21 RST( , , , ) 9.23 S o u s l i n s e t s 13.7 s t a n d a r d randomized s t o p p i n g time 11.3 s t a n d a r d s t o p p i n g time 8.2, 12.4 T 5.4 A t h i n n e s s 3.1, 3.10 U y, uj, U_ 1.4 U y 12.1 - 194 -u n i v e r s a l l y measurable space 9.5 6, 6 1.1 t 5.4 u A 13.6 $ 1.3 Q 5.4 ( f t , B , B t , B t , e t , P x ) 5.4 5.2 

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