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UBC Theses and Dissertations

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 o f M a n i t o b a , 1973 M . S c , U n i v e r s i t y o f M a n i t o b a , 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 t h e Department of Mathematics  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA September , 1978 (c) N e i l F. F a l k n e r , 1978  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  by h i s of  at make  that  written  thesis  it  may  is  financial  of  University  of  British  September 18, 1978  of  Columbia,  British  by  for  gain  Columbia  shall  the  requirements I  agree  r e f e r e n c e and copying  of  that  not  copying  or  for  that  study.  this  thesis  t h e Head o f my D e p a r t m e n t  understood  Mathematics  2075 Wesbrook Place Vancouver, Canada V6T 1W5  of  for extensive  be g r a n t e d  It  fulfilment  available  permission.  Department  Date  freely  permission  purposes  for  in p a r t i a l  the U n i v e r s i t y  representatives.  this  The  thesis  or  publication  be a l l o w e d w i t h o u t  my  Supervisor:  Dr. R. V. Chacon  A b 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 , and l e t (B )  be Brownian  n  motion w i t h i n i t i a l  distribution  y .  (For t h o s e u n f a m i l i a r w i t h Brownian m o t i o n we i n s e r t a b r i e f h e u r i s t i c explanation.  C o n s i d e r a drunk on a pub c r a w l i n ]R . n  Imagine t h a t 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 d e s c r i b e d by  the p r o b a b i l i t y l a w y . ]R  Imagine t h a t t h e r e i s a pub a t each p o i n t i n  , and t h a t t h e drunk wanders from pub t o pub i n a t o t a l l y random  n  f a s h i o n , as i s t o be expected from h i s i n e b r i a t e d s t a t e .  Imagine t h a t  the drunk t a k e s o n l y an i n f i n i t e s i m a l d r i n k a t each pub, so t h a t he keeps moving c o n s t a n t l y . through  ]R  n  We can d e s c r i b e t h e drunk's random p r o g r e s s  by c o n s i d e r i n g t h e s e t o f a l l p o s s i b l e p a t h s 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 .  I n more p r e c i s e  m a t h e m a t i c a l t e r m s , we d e s c r i b e t h e m o t i o n o f t h e drunk by means o f a c e r t a i n p r o b a b i l i t y measure from  [ 0 , °°)  into  3R  n  The p r o b a b i l i t y space  .  P  Then  on t h e space  C  B ( t o ) = co(t)  f o r co e C , t e [ 0 , °°) .  t  of c o n t i n u o u s maps  (C,P) , t o g e t h e r w i t h t h e f a m i l y  (B ) o f fc  random v a r i a b l e s d e f i n e d on i t , i s c a l l e d Brownian m o t i o n , a f t e r t h e b o t a n i s t Brown who, i n 1827, observed t h e random m o t i o n o f m i c r o s c o p i c p a r t i c l e s suspended i n w a t e r . ) For each random t i m e variable  B^ .  T  l e t y^  be t h e d i s t r i b u t i o n o f t h e random  ( T h i s random v a r i a b l e i s t h e one d e f i n e d by  B (to) = B - .(to) .) I t i s n a t u r a l t o a s k w h i c h measures 1  i(to)  of t h e form  y  T  where  T  i s a stopping  time.  v  (A s t o p p i n g  on ]R  n  are  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 s e t i n H  i s a stopping  n  time.)  Skorohod [1] f i r s t c o n s i d e r e d t h i s s o r t of q u e s t i o n . if  n = 1  and  m easure on 2 a  E.  that  v = (We  i s the p o i n t mass a t  w i t h c e n t r e of mass a t  x dv(x)  =  v .  u  0 , and 0  if  v  He showed t h a t  i s a probability  and w i t h f i n i t e v a r i a n c e  , then t h e r e i s 'a "randomized" s t o p p i n g time and the e x p e c t a t i o n of  remark t h a t i f  u  T  T  such  i s e q u a l to the v a r i a n c e of  i s the p o i n t mass a t  0  and  T  is a  s t o p p i n g t i m e , 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 define  v  t o be  , then  the e x p e c t a t i o n o f  T .  " i f and o n l y i f " form.)  v  has mean  0  and v a r i a n c e e q u a l t o  Thus Skorohod's r e s u l t can be s t a t e d i n an Dubins [1] and Root [ 1 ] , by d i f f e r e n t methods,  showed t h a t Skorohod's r e s u l t can be improved i n t h a t a not randomized can be o b t a i n e d . s t o p p i n g time  T  whenever  v  mean i s d e f i n e d and e q u a l to variance.  which i s  i s a p r o b a b i l i t y measure on 0 , even when  v  ]R  whose  does not have f i n i t e  The meaning of " n a t u r a l " h e r e i s e n l a r g e d upon i n t h e paper  d i f f e r e n t but e q u i v a l e n t d e f i n i t i o n of s t a n d a r d and a l s o another  p r o b a b i l i t y measures on = y .  stopping times.  TR  n = 1  and  u  and  We  give a  s t o p p i n g times i n  c h a r a c t e r i z a t i o n of them i n 8.13.  [3]) has p o i n t e d out t h a t i f  u  T  The method of Dubins y i e l d s a " n a t u r a l "  of Chacon [ 1 ] , where s t a n d a r d s t o p p i n g times a r e d e f i n e d .  that  i f we  8.2,  Doob (see Meyer v  a r e any  then t h e r e i s a s t o p p i n g time  T  two such  T h i s i s the r e a s o n f o r the s p e c i a l i n t e r e s t i n s t a n d a r d But we d i g r e s s .  To get back t o our s t o r y , Chacon and  Walsh [ 1 ] , u s i n g p o t e n t i a l t h e o r y , gave a v e r y t r a n s p a r e n t proof the r e s u l t of Dubins and Root.  A l s o u s i n g p o t e n t i a l t h e o r y , Rost  of [1]  - i i i -  g e n e r a l i z e d Skorohod's r e s u l t t o Markov p r o c e s s e s w i t h proper k e r n e l s ( e g . , Brownian m o t i o n i n dimension  potential  g r e a t e r than or e q u a l t o 3 ) .  R o s t ' s method, however, produces a randomized s t o p p i n g time even when a non-randomized one e x i s t s . parameters,  Now a p h y s i c i s t has s a i d , "Give me f i v e  and I w i l l f i t an e l e p h a n t ; g i v e me s i x parameters, and  w i l l make him w i g g l e h i s t r u n k ! "  I n o t h e r words, i t i s n a t u r a l t o a s k  when one can g e t a non-randomized s t o p p i n g t i m e , f o r n >_ 2 . B a x t e r and Chacon [1] have g i v e n 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 t o be s o , but t h e i r hypotheses a r e r a t h e r s t r o n g — i n 8.21.  see t h e d i s c u s s i o n  I n 7.11 and 8.20, we have succeeded i n p r o v i n g r e s u l t s  a l o n g these l i n e s w h i c h appear much c l o s e r t o b e i n g b e s t p o s s i b l e . They a r e not a c t u a l l y b e s t p o s s i b l e , however, as t h e example 7.13 shows.  I t i s much e a s i e r t o get b e s t p o s s i b l e r e s u l t s i f one a l l o w s  o n e s e l f t o work w i t h randomized s t o p p i n g t i m e s .  I n 11.12, t h e  2 p r o b a b i l i t y measures  v  on R  such t h a t  s t a n d a r d randomized s t o p p i n g time and ]R^  such t h a t  u  v = v  , where  T  i sa  i s a p r o b a b i l i t y measure on  i s a p o t e n t i a l (see 1.4), a r e c h a r a c t e r i z e d .  r e s u l t appears t o be new. t h e r e s u l t of Rost  This  A l s o , i n 10.5, we prove a p a r t i c u l a r case of  [ 1 ] , by a d i f f e r e n t method.  Now Skorohod [1] a l s o showed t h a t i f (X^) i s a s q u a r e - i n t e g r a b l e m a r t i n g a l e w i t h t i m e s e t I = { 0 , 1, 2,...} , s a t i s f y i n g and i f (^ ) t  i - Brownian m o t i o n , s  i n one d i m e n s i o n ,  then t h e r e i s an i n c r e a s i n g sequence times w i t h f i n i t e  (T )  E(X^) = 0 ,  s t a r t i n g from  0 ,  o f randomized s t o p p i n g  e x p e c t a t i o n s , such t h a t t h e p r o c e s s e s  (B^ ) and i (X.) have t h e same j o i n t d i s t r i b u t i o n . (Note t h a t i f X. = X,. f o r l l 0 all i , t h i s reduces t o t h e r e s u l t f o r measures w h i c h we d e s c r i b e d J  - iv -  first.)  Dubins [1] remarks t h a t h i s method can be a p p l i e d here t o  y i e l d non-randomized s t o p p i n g t i m e s , and t h a t t h e c o n d i t i o n t h a t 2 E(X_^) < «>  can be dropped, though of c o u r s e the  T^'s  need not have  f i n i t e e x p e c t a t i o n then (though one can show t h a t they w i l l be i f c o n s t r u c t e d as Dubins d e s c r i b e s ) . of t h i s to Brownian m o t i o n i n randomized s t o p p i n g t i m e s . )  n  I n 12.7  dimensions.  we prove a g e n e r a l i z a t i o n (For  n >_ 2 , we  We a l s o g i v e a d e t a i l e d p r o o f of  a s s e r t i o n c o n c e r n i n g the one d i m e n s i o n a l case. that a right-continuous martingale  ^t^0<t<°°  standard  use Dubins'  Monroe [1] has shown C a n  ^  e  e m  k d d e d -*e  n  a n  "enlargement" of one d i m e n s i o n a l Brownian m o t i o n by means of a r i g h t continuous times.  increasing family  (T ) fc  of m i n i m a l  (= s t a n d a r d )  stopping  I n 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  dimensions.  F i n a l l y , i n 12.18  n  we d e s c r i b e an example, d i s c o v e r e d by  R. V. Chacon, w h i c h shows t h a t enlargement r e a l l y i s n e c e s s a r y i n the theorem of Monroe. Now  f o r a few words about t h e s e c t i o n s of t h i s t h e s i s from w h i c h  r e s u l t s have not yet been mentioned. devoted  t o e s t a b l i s h i n g n o t a t i o n and  S e c t i o n s 1, 5, and 6 a r e m a i n l y t e r m i n o l o g y , and t o 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 r e f e r e n c e . a r e devoted  S e c t i o n s 2, 3, and  t o e s t a b l i s h i n g those a s p e c t s of the f i n e t h e o r y of  balayage w h i c h a r e needed throughout  t h e r e s t of t h e t h e s i s .  the p r o b a b l e e x c e p t i o n of the s t r o n g form of the d o m i n a t i o n 2 for  t h e l o g a r i t h m i c p o t e n t i a l i n ]R  s e c t i o n s a r e not new. novel.  4  , t h e r e s u l t s of these  With principle  three  I b e l i e v e , however, t h a t some of the p r o o f s a r e  F o r example, t h e r e a d e r might f i n d i t amusing t o compare our  proof of 2.1 w i t h the p r o o f of 8.43  of Helms [ 1 ] , or t o compare our  - v proof of 3.2 w i t h the d i s c u s s i o n Brelot  [1].  Section  i n sections  9 and 10 of t h e paper of  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 s p a c e s , 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 r e v i e w o f , and enlargement on, p a r t s of t h e papers of B a x t e r and Chacon [2 and 3 ] .  The theorem 9.13 however,  though a s i m p l e 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 e n l a r g e m e n t s " defined  i n B a x t e r and Chacon [ 3 ] .  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 One l a s t t h i n g :  ( D i s t r i b u t i o n a l enlargements a r e enlargements.)  towards t h e end of s e c t i o n 10, we d i s c u s s  a  c o u p l e 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.  - vi -  T a b l e of C o n t e n t s Page 1.  Potential Theoretic Preliminaries  1  2.  The I n t e g r a l R e p r e s e n t a t i o n of Balayage  8  3.  Thinness  23  4.  The S t r o n g Form of the Domination P r i n c i p l e  31  5.  Brownian M o t i o n 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 M o t i o n and P o t e n t i a l Theory  57  Embedding Measures i n Brownian M o t i o n i n a Green R e g i o n , U s i n g Non-Randomized S t o p p i n g Times  61  7.  1 8. 9. 10.  2  Embedding Measures i n Brownian M o t i o n i n H U s i n g Non-Randomized S t o p p i n g Times  or  1  , 76  Randomized S t o p p i n g Times, and Enlargements P r o b a b i l i t y Spaces  of 98  Embedding Measures i n Brownian M o t i o n i n a Green R e g i o n , U s i n g Randomized S t o p p i n g Times  124  2 11.  Embedding Measures i n Brownian M o t i o n i n H Randomized S t o p p i n g Times  , Using 136  12.  Embedding P r o c e s s e s i n Brownian M o t i o n  147  13.  Appendix  184  of M i s c e l l a n e o u s N o t a t i o n and Terminology  References  189  Index of S e l e c t e d N o t a t i o n and Terminology  192  - v i iAc knowled g ement s I w i s h t o thank my s u p e r v i s o r , R a f a e l Chacon, whose g u i d a n c e and encouragement have b e e n i n v a l u a b l e ' t o me. about how t o do mathematics.  He has t a u g h t me much  I would a l s o l i k e t o thank M a u r i c e S i o n ,  whose l u c i d t e a c h i n g made i t i n f i n i t e l y e a s i e r f o r me t o l e a r n t h e "heavy d u t y " measure t h e o r y needed i n t h e s t u d y of p r o c e s s e s , and John Walsh, who was always ready t o answer my q u e s t i o n s about Brownian m o t i o n and P o t e n t i a l Theory.  I w i s h t o e x p r e s s my g r a t i t u d e  t o t h e N a t i o n a l R e s e a r c h C o u n c i l of Canada, and t h e I z a a k Walton K i l l a m M e m o r i a l Fund of t h e 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 s u p p o r t .  F i n a l l y , I would l i k e t o 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 T h i s s e c t i o n i s m a i n l y 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 1.1.  of p o t e n t i a l theory  For each  x eH  ,  6 x  t h a t we s h a l l use.  denotes t h e D i r a c measure a t  6 (A) = l . ( x ) X  If  x = 0  1.2.  A  x:  for A £ Borel H  A.  we s h a l l j u s t w r i t e  6  for  denotes the L a p l a c i a n on  E. ;  6  .2  n i=l  9x:  We s h a l l f r e q u e n t l y use t h e 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  open s e t  D  in H  then  n  i s a superharmonic f u n c t i o n i n an  -Au  i s c a l l e d the R i e s z measure of distribution in D then  T  such t h a t  i s a ( p o s i t i v e ) measure i n D  which  u .  On the o t h e r hand, i f T  is a  i s a ( p o s i t i v e ) measure i n  D  -AT  a r i s e s from a unique superharmonic f u n c t i o n i n D .  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 t h e f u n c t i o n on  2 $(x)  =  2TT  n  d e f i n e d by  if n = 1  x  1_  ]R  log  (n-2)a  n  x  x ,n-2  if n = 2 if  n > 3  (We  - 2 -  where  a n  i s the n - 1  d i m e n s i o n a l Lebesgue measure o f t h e s u r f a c e  of t h e u n i t sphere i n ]R . n  (we  Then  take  $(0) = + °°  i f  n >^ 2 .)  <3> has t h e f o l l o w i n g t h r e e  a)  $  b)  A$ =  c)  $  properties:  i s superharmonic. -6 .  i s i n v a r i a n t under r o t a t i o n s about t h e o r i g i n .  In f a c t , these p r o p e r t i e s determine constant.  Of c o u r s e p r o p e r t y  We s h a l l a l s o use t h e l e t t e r  $ . $  f i r s t , t h e f u n c t i o n on ]R x ]R n  The $  t o w i t h i n an a d d i t i v e  b) has t h e s i m p l e form t h a t i t does  because o f t h e way we n o r m a l i z e d  second, t h e f u n c t i o n on  $  t o denote two o t h e r defined  n  [0,°°)  by  satisfying  $(x,y) = $ ( x - y ) ; 4>(x) = $ ( | |x| |) .  c o n t e x t w i l l always make c l e a r w h i c h f u n c t i o n we mean. depends on  n  b u t t h i s dependence i s not made e x p l i c i t .  p a r t i c u l a r l y i m p o r t a n t t o keep t h i s i n mind when a f u n c t i o n on 1.4.  If y  It is  i s r e g a r d e d as  00  i s a measure on t h e B o r e l s u b s e t s o f H  =  $ (x,y)dy(y)  uV)=  $ (x,y)dy(y)  tr>)  we d e f i n e  $  Note t h a t  [O, ) .  f i n i t e on compact s e t s ) then we d e f i n e  and  functions:  U^,  U^, U^: H  n  n  ( n o t assumed  —>• [0,°°] by  +  on t h e s e t where  and  are not both  infinite,  - 3-  by  We say  i s a potential i f f U  i s everywhere d e f i n e d and s u p e r -  M  harmonic on ]R • n  T h i s happens i f f  i s f i n i t e a t a l l p o i n t s of E.  and  n  i s f i n i t e a t a t l e a s t one p o i n t of 3R . n  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  f i n i t e on compact s e t s (so d i s t r i b u t i o n s on ]R ) n  r e c o v e r e d from  and  and  y  AU^ = -y;  can be r e g a r d e d as Schwartz i n particular,  y  can be  .  The f o l l o w i n g t h r e e r e s u l t s g i v e more e x p l i c i t of  y is  those measures  y  f o r which  characterizations  i s a potential.  We s t a t e these  r e s u l t s w i t h o u t p r o o f , but t h e statements a r e so d e t a i l e d t h a t they almost prove 1.5.  themselves.  Proposition: a) b)  = 0 If U  y  Suppose  and  U  y  n = 1 .  = -U  y  i s f i n i t e a t two d i s t i n c t p o i n t s then  i s f i n i t e a t a l l p o i n t s and U  y  c) 1.6.  i s L i p s c h i t z and  i s a p o t e n t i a l i f f yOR) <  Proposition: a)  yOR)  Suppose  I f U^(x) for  some  2 i n K.  y  i s finite,  i s c o n t i n u o u s ; indeed  i s a L i p s c h i t z constant f o r  00  and  x d y ( x ) < °°  n = 2 .  i s f i n i t e and x  U  y  y ( { y £ K. : | |y-x| | <_ r } )  and some  r  in  (l, ) 0 0  then  i s finite y is  - 4-  finite,  U.^  i s f i n i t e a t a l l p o i n t s and 1  indeed  i s L i p s c h i t z and  constant f o r b)  If u  i s continuous;  2  —  u0R )  i s a Lipschitz  .  i s f i n i t e on compact s e t s then  i s locally  Lebesgue  integrable. c)  U  i s a potential i f f u  P  i s f i n i t e and  log ||x||dp(x) < » . +  1.7.  Proposition: a) b)  = 0 If U  P  and  U  P  n >_ 3 .  =  .  i s f i n i t e a t one p o i n t  s e t s and c)  Suppose  U  i s locally  P  then  u  i s f i n i t e on compact  Lebesgue i n t e g r a b l e .  i s a potential i f f  lA$du <  00  •  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.  Proposition:  U  P  i s a potential i f f  u  i s f i n i t e on compact  •  s e t s and  I $(x) I du (x) <  00  .  J||x||>l 1.9.  C o n s i d e r an open s e t D I f f o r some p o i n t  minorant i n D in  D,  i n ]R . n  X Q in D  then f o r a l l x  and indeed has a g r e a t e s t  denote by  the function i n D,  $(x,*)  $(XQ,«)  has a harmonic  has a harmonic  one which we s h a l l ,  f o r t h e moment,  h(x,•) .  I n t h i s case we say D G: D x D —*• [0,°°]  d e f i n e d by  minorant  i s a Green r e g i o n and t h e f u n c t i o n  - 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 x e D,  For each  harmonic f u n c t i o n  D .  G(x,*)  u  in  D  i s the s m a l l e s t n o n - n e g a t i v e s u p e r satisfying  -6  Au =  x  J  G(x,') D  s h o u l d 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  t h a t would a r i s e from a u n i t p o i n t charge a t  made of e l e c t r i c a l l y c o n d u c t i n g One  can show t h a t  G(x,y) = G(y,x) Clearly i f function  G  of  G  m a t e r i a l , provided  e D .  n >_ 3  E  then  3D  3D  i s connected.  were  D x D  i s a Green r e g i o n and  n  i s g i v e n by  n  if  i s j o i n t l y c o n t i n u o u s on  f o r a l l x,y  lR  x  and  the Green  G(x,y) = $(x,y) .  I t i s a l s o c l e a r t h a t any open s u b s e t of a Green r e g i o n i s a Green r e g i o n .  I n p a r t i c u l a r , any open s u b s e t of  E ,  where  n >_ 3,  in  ]R  is a  then  D  is a  n  i s a Green r e g i o n . A moment's thought w i l l show t h a t an open s e t Green r e g i o n i f f D ^ E  D  .  2 One  can show t h a t i f  2  Green r e g i o n i f f E \D see 8.33 of Helms [1].  D  i s an open s e t i n  i s not a p o l a r s e t .  (A p o l a r s e t i s a s e t c o n t a i n e d  E  T h i s i s Myrberg's theorem -  i n the s e t 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 .  Polar sets are  the  " s m a l l " s e t s of p o t e n t i a l t h e o r y . )  2 One  can a l s o show t h a t i f  matter, i n  K.) n  then  D  D  i s an open s e t i n  E  (or f o r t h a t  i s a Green r e g i o n i f f t h e r e i s a n o n - c o n s t a n t  -  6  -  n o n - n e g a t i v e superharmonic f u n c t i o n i n  D .  T h i s i s p a r t of 8.33  of  Helms [1], but i t ' s not hard t o g i v e a p r o o f u s i n g o n l y the m a t e r i a l i n the f i r s t s i x c h a p t e r s o f Helms; i e . , the m a t e r i a l up t o and i n c l u d i n g the c h a p t e r on Green 1.10.  Let If  D  y  define  be a Green r e g i o n i n ]R  Gy  Gy  w i t h Green f u n c t i o n  i s superharmonic i n  as a measure i n y  we  i s f i n i t e a t a t l e a s t one p o i n t of each component of  In t h i s case we say  If  D,  G(x,y)dy(y)  D,  y  D  Gy  i s the p o t e n t i a l of  such t h a t  i s a measure i n  D  Gy  D)  and  y,  AGy = -y .  and we  describe  i s a potential.  such t h a t  the g r e a t e s t harmonic m i n o r a n t of  D  i s f i n i t e on compact s e t s (and  so can be r e g a r d e d as a Schwartz d i s t r i b u t i o n i n  y  G .  [0,°°]. by  Gy(x) =  then  n  i s a measure on the o - f i e l d of B o r e l s e t s of  Gy: D —»-  If  potentials.  Gy  in  Gy D  i s a p o t e n t i a l then i s zero.  the s m a l l e s t n o n - n e g a t i v e superharmonic f u n c t i o n i n  D  Gy  i s thus  having  y  as  R i e s z measure. If  u  i s a superharmonic f u n c t i o n i n  m i n o r a n t , and i f p o t e n t i a l and of  u .  1.11.  i s the R i e s z measure of  u = Gy + h  where  h  w h i c h has a u,  then  Gy  subharmonic is a  i s the g r e a t e s t harmonic m i n o r a n t  T h i s i s known as the R i e s z d e c o m p o s i t i o n theorem.  Let If  y  D  u  D  be an open s e t i n H  n  .  i s a n o n - n e g a t i v e superharmonic f u n c t i o n i n  D  and  E  is  - 7 -  a subset of  D  then the r£duite of  (which we s h a l l denote by  u  over  red(u,E,D))  E  r e l a t i v e to  i s d e f i n e d t o be the infimum  of the s e t of n o n - n e g a t i v e superharmonic f u n c t i o n s i n majorize  u  on  E .  The b a l a y a g e of  (which we s h a l l denote by r e g u l a r i z a t i o n of  bal(u,E,D)  of the boundary of  over  which  r e l a t i v e to  D  i s d e f i n e d t o be the lower  bal(u,E,D) d i f f e r s from  i s superharmonic i n red(u,E,D)  D  and the  i s a p o l a r subset  E .  We n o t e t h a t the more u s u a l n o t a t i o n s f o r bal(u,E,D)  E  D  u .  One can show t h a t s e t where  u  bal(u,E,D))  D  are  and  Of c o u r s e , i f D  red(u,E,D)  and  respectively.  i s not a Green r e g i o n then re"duite and b a l a y a g e  a r e t r i v i a l n o t i o n s , s i n c e then e v e r y n o n - n e g a t i v e superharmonic function i n 1.12.  D  i s constant.  The main r e f e r e n c e we recommend f o r b a s i c p o t e n t i a l t h e o r y i s  the book of Helms [ 1 ] . Du P l e s s i s  Other u s e f u l r e f e r e n c e s a r e the books of  [ 1 ] and B r e l o t  [1].  I n 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 t h e o r y , whereas Helms avoids  it.  - 8 -  2.  THE  INTEGRAL REPRESENTATION OF BALAYAGE  Throughout t h i s s e c t i o n , Green r e g i o n i n TR , U  If  E  and  n  G  i s a subset of  i s a positive integer,  i s the Green f u n c t i o n o f  ]R  n  and  Gy  i s a p o t e n t i a l then bal(Gy,E,D)  in  D  w h i c h we s h a l l denote by  bal(Gy,E,D)  y  D  is a  D .  i s a measure i n  D  such t h a t  i s the p o t e n t i a l of a measure  bal(y,E,D)  .  Thus  = G bal(y,E,D) .  I n 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 r e p r e s e n t a t i o n formula f o r balayage: If f o r every  u  i s any n o n - n e g a t i v e superharmonic x  in  function i n  D  then  D, u(y)bal(6  bal(u,E,D)(x) = D  ,E,D)(dy) X  T h i s was proved by B r e l o t [ 1 ] . Our method of p r o o f d i f f e r s from B r e l o t ' s i n t h a t i t r e q u i r e s no a p p e a l t o the t h e o r y of the D i r i c h l e t problem; i t uses o n l y the c l a s s i c a l p o t e n t i a l t h e o r y c o n t a i n e d i n the f i r s t seven c h a p t e r s of the book of Helms [ 1 ] . We b e g i n w i t h a weak v e r s i o n of the " d o m i n a t i o n p r i n c i p l e " .  2.1.  Proposition:  a measure i n  D  Let  E  such t h a t  be a c l o s e d subset of Gy  D  and l e t  y  be  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 p o l a r s u b s e t s of the boundary of E. Suppose  v  i s a n o n - n e g a t i v e superharmonic  function i n  D  such  - 9 -  that  v >_ Gy Then  Proof: v  on  E\Z  v >_ Gy  vAGy  where  Z  throughout  i s some p o l a r s e t . D .  i s a potential i n D  so  vAGy = Gv  in D .  D .  Now  Gv >_ Gy  on E\Z  Let  x  For  0 < r < distance(x^R \D)  be i n  and we w i s h t o show  g  r  Now  Gv >_ Gy  throughout  D . l e t a^  n  d i s t r i b u t i o n on t h e c l o s e d b a l l of r a d i u s let  f o r some measure  be the u n i f o r m u n i t r  centred  at  x,  and  = b a l ( a ,E,D) . r B r  l i v e s on  E  and  Ga  p o l a r subset of the boundary of s e t s , by 2.2 below.  r  = Gg  E .  r  Also  on B  E\P r  where  P  i s some  does not charge  polar  Hence Gvda  r  =  Ga dv r GB dv r f  Gv dB Gy dB  GB Ga  r  r  dy dy  Gy da  Letting  r  go t o  0,  we  obtain  Gv(x) > Gy(x) •  - 10 -  2.2. in  Lemma:  H ,  Let u  and l e t y  n  be a superharmonic f u n c t i o n i n an open s e t V be t h e R i e s z measure of  charge p o l a r s u b s e t s of t h e s e t where Proof:  u  set  y  does n o t  i s finite.  By a s t a n d a r d argument we c a n reduce t o t h e case where  a Green r e g i o n (even an open b a l l ) and Then  u . Then  u = G^u . of  u  i s a potential i n V .  I t s u f f i c e s t o show t h a t i f K  {u < } then 00  y  i s any compact sub-  does n o t charge p o l a r s e t s .  Now  G y  K  V  i s f i n i t e so by 6.21 of Helms [ 1 ] , g i v e n se t  C £ K  such t h a t  If  i s a polar set i n V  P  y(K\C) < e  compact s u p p o r t i n V Now  V  d y  =  y  c  and  dy < °°,  so P  K  e > 0  was a r b i t r a r y ,  G y  G^y =  Hence t h e y - o u t e r measure of As  e > 0  y  00  o  n  K  t h e r e i s a compact  i s c o n t i n u o u s on  then t h e r e i s a measure  such t h a t G y  c  V is  y  V  with  P n K .  y ( { G y = «>})= 0 c  v  i s l e s s than  e .  does n o t charge p o l a r  sets. •  K  2.3.  Lemma:  Let E  potential i n D the  be a c l o s e d subset of  Let  u  E .  be a Let h  be  i n t h e open s e t D\E .  h  in  D\E  u  on  E  v =  v = reM(u,E,D) .  Proof: D\E  and l e t u  which i s f i n i t e on t h e boundary o f  g r e a t e s t harmonic m i n o r a n t o f  Then  D  so  L e t w = r£d(u,E,D) . w <_ v . L e t  Then  w £ u  and w  i s harmonic i n  be a sequence o f open b a l l s such t h a t  - 11 -  D\E = u B . i and V i ,  { j : B. = B.}  Let u  i s infinite. P I ( u . ; B .) u. l  Then each  u. l  Hence D  in  i  D\B. l  i s superharmonic i n D,  v,  and  t h e lower r e g u l a r i z a t i o n o f  u. 4- v . l v,  boundary o f Now  E .  w >_ v  and  Remark:  which l i v e s on t h e  and (by 2.2) does not charge p o l a r s u b s e t s of t h e  on  E  and  {w < w}  d o m i n a t i o n p r i n c i p l e 2.1, w >_ v w = w  i s superharmonic i n v  and i s , i n f a c t , t h e p o t e n t i a l o f a measure  closed set E  and l e t  in B.  1 1  u i+1  = u  v = v .  If u  A l s o , on  E,  i s a polar set.  throughout  D .  w = u = v .  E  of t h e above lemma may f a i l .  F o r i n s t a n c e suppose  u = G(x,*)  f o r some  and  least  Then  2.4.  2 .  Corollary:  x e E,  v = u  bal(u  D\E,  v = w .  •  the conclusion E = {x} and  (the dimension of l R ) i s a t n  but red(u,E,D) =  Let E  be p o t e n t i a l s i n D  n  Now i n  Thus  i s n o t f i n i t e on t h e boundary o f  Thus by the  °°lr ;. v  be a c l o s e d subset of  D  and l e t u ,u  w h i c h a r e f i n i t e on t h e boundary o f  E .  Then  + u  = baKu^E.D) + bal(u ,E,D) . 2  Proof:  The g r e a t e s t harmonic m i n o r a n t o f a sum of two superharmonic  - 12 -  functions  (each o f which has a subharmonic m i n o r a n t ) i s t h e sum o f  t h e i r g r e a t e s t harmonic m i n o r a n t s - see 5.22 o f Helms [ 1 ] .  Hence  r e d ( u ^ + U £ , E, D) = r e d ( u E , D ) + r£d(u ,E,D) . 1 }  It follows that  bal(u +u ,E,D) 1  2  d i f f e r a t most on a p o l a r s e t .  2  and  b a l ( u ,E,D) + b a l ( u , E , D ) 2  But two superharmonic  functions  which a r e e q u a l e x c e p t on a p o l a r s e t a r e e q u a l everywhere. • The  following result  i s proved i n c h a p t e r 8 o f Helms [1] u s i n g  the D i r i c h l e t problem t h e o r y developed i n t h a t c h a p t e r . s t a n t i a t e our c l a i m t h a t t h e r e s u l t s  To  sub-  o f t h i s s e c t i o n can be e s t a b l i s h e d  w i t h o u t t h i s p a r t o f t h e t h e o r y , we i n c l u d e a p r o o f . 2.5.  Lemma:  Let Z  be a p o l a r subset o f  Then t h e r e i s a f i n i t e measure but  in D  and l e t  such t h a t  x e D\Z .  Gu =  00  on  Z  Gy(x) < °° .  Proof:  By t h e d e f i n i t i o n o f a p o l a r s e t (Helms [ 1 ] , p. 126) t h e r e i s  an open s e t V <=_ R Z _c V (B_^)  u  D  and  v =  00  and a superharmonic f u n c t i o n  n  on  Z .  Let v  v  in V  be t h e R i e s z measure o f  such t h a t v . Let  be a sequence o f open b a l l s such t h a t (D\{x}) n V = u B . . I  For each  i , l e t v.  be t h e measure on t h e B o r e l s e t s o f  D  defined  I  by  v ^ ( A ) = v(AnB_^) .  Gv. (x) < i  00  .  Each  A l s o , i n B. , l  i s a f i n i t e measure i n D AGv. = -v = Av . i  Hence i n B., l  satisfying Gv. and i  - 13 -  v {Gv  d i f f e r by a harmonic f u n c t i o n . = °°} 1  1  n B. = {v = •»}  In p a r t i c u l a r , Now  n B. . 1  choose a sequence  (a.)  of  p o s i t i v e r e a l numbers such t h a t  Y  a.v. (D) <  . 1  00  11  and y a.Gv.(x) < . 1 1 i The d e s i r e d measure  2.6. in  Corollary: D  and l e t  y  can be t a k e n t o be  Let E  00  u  £ i  •  be a n o n - n e g a t i v e superharmonic  be any subset of  D .  Then  s m a l l e s t n o n - n e g a t i v e superharmonic f u n c t i o n v >_ u  on  Proof:  E  Let  harmonic  w = bal(u,E,D)  function i n  i s a polar set. D  such t h a t  v + eGy ^ u that  on  v ( x ) >_ w(x)  throughout  bal(u,E,D) v  in  D  i s the  such t h a t  except f o r a p o l a r s e t .  D  .  Then  x e D\Z  Let Gy = E, .  on  00  so  w  . Z  i s a polar set.  function i n  D  such t h a t  Let  v  Z = E n  {v<u} u  By 2.5 t h e r e i s a f i n i t e measure but  Gy(x) <  v + eGy >_ w .  Hence  i s a non-negative super-  E n {w<u}  and  be a n o n - n e g a t i v e superharmonic  in  function  v >_ w  on  .  00  Letting D\Z  For any e  As  go to Z  e > 0, 0,  i s polar,  we  find  v >_ w  D . •  2.7.  Corollary:  Let  ( ^)  ^  u  superharmonic f u n c t i o n s i n i n f i n i t e on any component of  D  e  a n  i n c r e a s i n g sequence of n o n - n e g a t i v e  whose supremum  u  i s not  identically  D, and hence i s superharmonic.  Let  - 14 -  (E_^)  be an i n c r e a s i n g sequence of s u b s e t s of  Then  b a l ( u ,E ,D) + b a l ( u , E , D ) .  Proof,:  Let  w = bal(u,E,D) .  a superharmonic f u n c t i o n f o r some  i , x e E^  Clearly  v <_ w .  and  D,  with union  bal(u_^,E^,D)  If  x e E  v ( x ) < u_^(x) .  and  E .  increases to  v ( x ) < u(x)  then  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. in  Lemma: D .  Let  u  be any n o n - n e g a t i v e superharmonic f u n c t i o n  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  potentials i n  D  whose R i e s z measures  Proof: of  D  Let  be  u. = u A i l  u_^  D. . l  u.  Theorem:  and l e t E  Hence  then  i  D  D .  a)  I.  Assume  E  i  Let  u  Also, i f  i s the Riesz  . n  be a n o n - n e g a t i v e superharmonic f u n c t i o n i n  u  D .  Then f o r each  x e D,  u ( y ) b a l ( 6 ,E,D)(dy) .  i s a f i n i t e p o t e n t i a l w i t h R i e s z measure  i s closed i n  =  00  u. l  i  be any s u b s e t of  a  D,  u.(D) = u.(D.) <  x  Assume  i , l e t u. = b a l ( i A u , D . ,D) l l bounded by i , and s a t i s f y i n g  u. i u . l  bal(u,E,D)(x) =  Proof:  F o r each  i s a potential in  in  measure o f  2.9.  sequence of open r e l a t i v e l y compact s u b s e t s  a  w h i c h i n c r e a s e s to  Then each  are f i n i t e .  y  E  D .  B  Let  =  y  D\E  u .  - 15 -  Then  u = Ga + GB  and  Ga  and  G3  a r e 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) . the d o m i n a t i o n p r i n c i p l e 2.1,  F i r s t consider  bal(Ga,E,D) = Ga .  on  bal(G6 ,E,D) = G6  almost everywhere w i t h r e s p e c t to  X  E\Z  where  Z  By  Also  b a l ( G 6 , E , D ) = G8^ x  Ga .  i s p o l a r ; hence a,  by 2.2.  X  Thus Ga(y)bal(6  X  ,E,D)(dy)  G bal(<$ ,E,D) ( z ) a ( d z ) x  G6 (z)a(dz) = x  =  Ga(y)6 (dy) x  Ga(x)  Now c o n s i d e r  GB .  the open s e t  D\E  i)  F i r s t suppose  and l e t G'  x I E  or  be the Green f u n c t i o n of  ( r e s p . k) be t h e g r e a t e s t harmonic m i n o r a n t of V .  As  GB  and  G6  n = 1 .  GB  Let  V  be  V. L e t  ( r e s p . GS^)  a r e f i n i t e on the boundary ( i n D)  of  h in  E,  X  bal(G8,E,D)  in  V  and k = bal(G6 ,E,D) x  by 2.3.  ( I f n = 1,  G'B = G8 - h  in  V .  in  V,  every superharmonic f u n c t i o n i s f i n i t e . ) But  G'B(x) =  G' ( x , z ) g ( d z )  G(x,z) - k ( z ) 3 ( d z ) V = G8(x) -  bal(G6 ,E,D)(z)B(dz) .  Now  - 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 ,E,D)(dy) x  ii)  Now suppose  x e E  and  n _> 2 .  Then  {x}  i s p o l a r so  b a l ( v , E \ { x ) , D ) = bal(v,E,D)  f o r each n o n - n e g a t i v e  function  For each  v  i n D, by 2.6.  b a l l of r a d i u s E. l  centred at  2  i s closed i n D  and  x  i eU  l e t B^  and l e t E  E. + E\{x} . i  superharmonic be t h e open  = E\B^ .  Then each  Hence, by 2.7,  b a l ( v , E ,D) + b a l ( v , E \ { x } , D ) for  each n o n - n e g a t i v e  superharmonic  bal(GB,E for  each  i .  Letting  i  ,D)(x) =  function  v  in D .  Now by i ) ,  bal(G6 ,E ,D)(z)8(dz) x  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 ,E,D)(z)g(dz) x  G8(y)bal(6 ,E,D)(dy) x  Combining our r e s u l t s so f a r we have bal(Gy,E,D)(x) = for  a l l x e D,  b)  Now assume t h a t  of  D .  Let  where  (E^)  E  Gy  Gy(y)bal(6 ,E,D)(dy)  i s f i n i t e and  x  E  i s closed i n D .  i s merely a c o u n t a b l e u n i o n of c l o s e d  subsets  be an i n c r e a s i n g sequence o f c l o s e d s u b s e t s of  D  - 17 -  whose u n i o n i s  E .  Then f o r each i ,  bal(Gy,E  by  a).  Letting  i  Now  let  additional  E  x  i  go t o i n f i n i t y , we o b t a i n  bal(Gy,E,D)(x)  c)  Gy(y)bal(6 ,E ,D)(dy),  ,D)(x) =  Gy(y)bal(6 ,E,D)(dy) .  =  x  be an a r b i t r a r y s u b s e t of  assumption  that  y(D) < °° .  D .  Fix  L e t us impose the  e > 0 .  Let  v = e + bal(Gy,E,D) w = e +  bal(G6 ,E,D) x  S = {v >  Gy}  T = {w > G6  Then  E  i s contained i n  n e g a t i v e superharmonic  S  The  same statement  holds with  S =  Now  f o r each  Thus so i s  S  r,  {v>r}  }  t o w i t h i n a p o l a r s e t so f o r any  function  b a l ( f ,E,D)  x  f  in  D,  <_ b a l ( f ,S,D)  S  u {v>r} reQ  .  r e p l a c e d by  T .  Hence  Also,  n {Gy <_ r }  i s open and  {Gy  <_ r }  i s closed i n  i s a c o u n t a b l e u n i o n of c l o s e d s u b s e t s of T .  non-  D .  D .  Similarly,  - 18 -  bal(Gy,E,D)(x) < bal(Gy,T,D)(x) Gy(y)bal(6 ,T,D)(dy)  (by b)  x  bal(G6 ,T,D)(z)y(dz) x  w dy = ey(D)  bal(G6  +  ,E,D)dy  <_ sy (D) +  bal(G6 ,S,D)dy  = ey(D)  Gy(y)bal(6 ,S,D)(dy)  x  +  x  = ey(D) +  (by b)  bal(Gy,S,D)(x)  < ey(D) + v = ey(D) + e + b a l ( G y , E , D ) ( x ) As  y(D) < «°  and  e > 0  was  arbitrary,  bal(Gy,E,D)(x) Gy ( y ) b a l ( 5 , E , D ) ( d y ) x  < bal(Gy,E,D)(x)  II.  Now  l e t us c o n s i d e r the g e n e r a l case.  i n c r e a s i n g sequence u. + u 1  i  and each 1  u.  ( -^) u  By 2.8,  °^ f i n i t e p o t e n t i a l s i n  has f i n i t e R i e s z measure.  t h e r e i s an D  By I . c ) , f o r each  we have b a l ( u ,E,D)(x) =  u (y)bal(6 1  ,E,D)(dy) X  such t h a t  - 19 -  Letting  i  go t o i n f i n i t y and a p p l y i n g 2.7 on the l e f t and t h e  monotone convergence  theorem  on the r i g h t we o b t a i n a t l a s t u(y)bal(S ,E,D)(dy)  bal(u,E,D)(x) =  x  • 2.10.  Corollary:  measure i n  D  Let  E  such t h a t  be any subset of Gy  D  i s a potential.  and l e t y  be a  Then f o r each- x e D,  bal(Gy,E,D)(x) =  bal(G6 ,E,D)dy  bal(Gy,E,D)(x) =  Gy(y)bal(6 ,E,D)(dy)  x  Proof: x  G bal(6 ,E,D)(z)y(dz) x  bal(G6 ,E,D)dy x  • 2.11.  Corollary:  Let  E  be any subset of  D .  Then f o r a l l  x,y e D, b a l ( G 6 ,E,D)(x) = bal(G<5 y Proof:  2.12.  Take  y = & y  Corollary:  ,E,D)(y) x  i n 2.10.  Let  • E  n o n - n e g a t i v e superharmonic  be any subset of functions i n D .  D  and l e t u,v  be  Then  bal(u+v,E,D) = bal(u,E,D) + b a l ( v , E , D ) . Proof:  T h i s f o l l o w s i m m e d i a t e l y from t h e l i n e a r i t y of t h e i n t e g r a l . •  - 20 -  2.13. let  Corollary: y  Let  E  be any s u b s e t o f  be a measure i n D  D .  u(y)y(dy)  f o r e v e r y n o n - n e g a t i v e superharmonic f u n c t i o n for z e D .  and  such t h a t  bal(u,E,D)(x) =  u = G(z,«)  Let x e D  u  in D  o f t h e form  Then  y = bal(6 ,E,D) . x  Proof:  F o r any  z e D,  G (z) Y  =  G(z,y)Y(dy)  =bal(G(z,-),E,D)(x) G(z,y)bal(6 ,E,D)(dy) x  = G bal(6  That i s , to  Y  a n  d  bal(6 ,E,D) x  ,E,D)(z) .  have t h e same Green p o t e n t i a l  relative  D . •  2.14.  Lemma:  Let  f  be 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 s u p p o r t 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 o f two n o n - n e g a t i v e bounded c o n t i n u o u s superharmonic functions i n D .  Proof:  L e t g = -Af  and l e t u = G g , +  v = Gg  .  Then  a r e f i n i t e c o n t i n u o u s superharmonic f u n c t i o n s i n D, Helms [ 1 ] . Now function  h  A(u-v) = -g = Af  in D  such t h a t  in D .  u  and  v  by 6.22 o f  Hence t h e r e i s a harmonic  u - v = f + h .  Now  u  and  v  are  - 21 -  bounded  on t h e s u p p o r t  domination M = sup h  principle .  a r e bounded  Then  of  g  2.1.  f =  and h e n c e a r e b o u n d e d  Thus  (u+M)  non-negative  h  i s also  - (v+M-h),  continuous  bounded  and  on  D  on  D  u+M,  superharmonic  by t h e .  v +  Let M - h  functions i n  D .  • Remark: in  Using D i r i c h l e t  t h e a b o v e p r o o f must  2.15.  Theorem:  in  D  such  D,  then:  Let  that  problem be  E  Gy  0,  t h e o r y , one b u t we  be a n y s u b s e t  don't  need  of  D  .  If  A  i s a potential.  a)  x>—>• b a l ( 6  b)  bal(y,E,D)(A)  x >  E , D ) (a)  c a n show t h a t this  Let  the  h  here.  y  be a  measure  i s any B o r e l s u b s e t  i s a Borel function  in  of  D  y(dx)bal(6 ,E,D)(A)  =  x  Proof: a)  bal(6  ,E,D)(.D) =  1 bal(6  ,E,D)(dy)  = b a l ( l , E , D ) ( x ) <_ 1  for  each  in  D  with  .  x e D; Suppose  compact  in particular, $  support  i s a twice in  D  .  bal(6  x >  E,D)  i s a finite  continuously differentiable  measure function  Then  <Ky)bal(6 ,E,D)(dy) x  is  a Borel function  class  argument,  function  in  D  in  D  by 2.14  i t follows that  combined w i t h x  f o r any B o r e l subset  2.9.  b a l ( 6 , E , D ) (A) x  A  of  D .  By a m o n o t o n e i s a Borel  - 22 -  b) D  By a) we can d e f i n e a measure  v  on the B o r e l s u b s e t s of  by v(A) =  Then f o r any  u(dx)bal(6 ,E,D)(A) . x  z e D, Gv(z) =  v(dy)G(y,z) U(dx) b a l ( 6 , E , D ) ( d y ) G ( y , z ) x  bal(G6 ,E,D)du z  = bal(Gy,E,D)(z),  where we have a p p l i e d 2.9 and 2.10.  Thus  Gv = G b a l ( u , E , D ) ,  so  v = bal(y,E,D) . •  - 23  3.  -  THINNESS  3.1.  Definition.  and l e t  x e D .  Let We  D  be a Green r e g i o n i n ]R  s h a l l say  i f f there i s a non-negative that  bal(u,E,D)(x)  E  i s t h i n at  x  .  n  Let  E £ D  r e l a t i v e to  superharmonic f u n c t i o n  u  in  D  D  such  < 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 :  f r i n g e ( E , D ) = {x e E: E base(E,D) = {x e D: E  3.2.  Theorem:  i s t h i n at  x  r e l a t i v e to  i s not t h i n a t  x  Let  D  be a Green r e g i o n i n ]R  a bounded p o t e n t i a l  v  in  D  D}  r e l a t i v e to  .  n  such t h a t f o r e v e r y  D} .  Then t h e r e i s E £  D,  base(E,D) = ( b a l ( v , E , D ) = v} . Proof: of  D  Let  be a sequence of open r e l a t i v e l y compact s u b s e t s  such t h a t the range of  each  i , let  v. = b a l ( l , V . , D ) . l l  bounded p o t e n t i a l i n W e ' l l show t h a t  D .  c  Now  {u > c}  x e V  E £ D  u  in  D  i s open i n  £ {u > c} .  Then  bal(u,E,D) > c b a l ( v . , E , D )  non-negative  Now  Then  For  v  is a  x e D\base(E,D) . non-negative  bal(u,E,D)(x) < u(x) .  bal(u,E,D)(x) x .  u >_ c b a l ( l , V ^ , D ) .  D .  Well, there i s a  and c o n t a i n s  and  u(x) .  Thus f o r some j ,  in  D .  Hence  cv.(x) = c > bal(u,E,D)(x),  1  v . ( x ) > b a l ( v . ,E,D) (x) . 3  D  and  such t h a t  be a number s t r i c t l y between  -  v —i v = )2 v. . l  Let  Suppose  open base f o r  a n  bal(v,E,D)(x) < v(x) .  superharmonic f u n c t i o n Let  1 S  so  3  Since  2  _ : l  b a l ( v . ,E,D)  3  superharmonic f u n c t i o n i n  3  D  +  £ 2~\.  is a 1  w h i c h i s g r e a t e r 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.  Corollary:  Let  Then  fringe(E,D)  D  be a Green r e g i o n i n ]Rn  i s a p o l a r s e t and  base(E,D)  Let  E c 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  f r i n g e ( E , D ) = E n {bal(v,E,D) < v } ,  and so i s a p o l a r s e t by 7.40 of Helms [ 1 ] . A l s o D\base(E,D) = {bal(v,E,D) < v } , s u b s e t s of  3.4.  D  as  Corollary:  function  G .  potential.  v  and  Let  D  Let  Let  E  y  and so i s a c o u n t a b l e u n i o n of c l o s e d  bal(v,E,D)  a r e lower  semicontinuous.  be a Green r e g i o n i n H , n  be a measure i n D  be any subset o f  D .  such t h a t Then  •  w i t h Green Gy  isa  bal(y,E,D)  lives  on base(E,D) . Proof: Then  Let  v  be as i n 3.2.  bal(w,E,D) = w  by 2.6.  C o n s i d e r any  x e D .  Thus  w ( y ) b a l ( S ,E,D)(dy) = w(x) =  by 2.9.  Also,  w(x)  v ( y ) b a l ( 6 ,E,D)(dy),  i s f i n i t e and  w <_ v .  b a l ( 6 ,E,D)({w < v } ) = 0 .  Thus  Let  w = bal(v,E,D).  - 25 -  As t h i s i s t r u e f o r a l l x e D,  we have  b a l ( y , E , D ) ( { w < v } ) = 0,  by 2.15.  But {w < v} = D\base(E,D> . •  3.5.  Corollary:  and l e t  u,v  Let  D  be a Green r e g i o n i n ]R ,  let  E £  functions i n  D .  n  be n o n - n e g a t i v e superharmonic  D, 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)  u <_ v  on  E\Z  b)  u <_ v  on  base(E,D) .  Proof:  a) = >  Then  f <_ g  b) .  and  b) = > by 3.3, and  Let  by 2.6.  on base(E,D)  f o r some p o l a r s e t  f = bal(u,E,D)  Let  u <_ v  and l e t  But by t h e d e f i n i t i o n of  g = v  a) .  Z  on  on  base(E,D) .  Z = fringe(E,D) E\Z  g = bal(v,E,D) .  base(E,D),  Hence .  u <_ v  Then  Z  f = u  on  base(E,D) .  i s a polar set  . •  Now in  ]R ,  to  D  3.6.  n  we a r e g o i n g to show t h a t i f D E £H , n  and  i f f E n D' Proposition:  x e D n D'  i s t h i n at Let  D  a)  If  x 6 D\E  then  b)  If  n = 1,  x e D,  then  x  and E n D  r e l a t i v e to  D'  a r e Green r e g i o n s  i s t h i n at  i s thin at and  E  relative  D' .  be a Green r e g i o n i n JR E  x  n  and l e t  E £ D .  x .  i s t h i n at  x  then  x i E .  - 26 -  Proof:  a)  Let G  be t h e Green f u n c t i o n o f  and l e t v = bal(u,E,D) such t h a t u  W n E = 0  i s not.  in  W  Thus  .  and  u - v  Let W  D,  l e t u = G(x,*)»  be a connected open s u b s e t o f  x e W .  Then  v  i s harmonic i n W  at  x .  b)  Let u  W .  In particular,  one.  Thus  v ( x ) < u(x) .  W  and hence a t  Thus  E  i s thin  be a n o n - n e g a t i v e superharmonic f u n c t i o n i n D .  red(u,E,D) = bal(u,E,D)  on  E .  harmonic f u n c t i o n s a r e c o n t i n u o u s . E  Then  s i n c e o n l y t h e empty s e t i s p o l a r i n d i m e n s i o n  u = bal(u,E,D)  on the c l o s u r e o f  but  i s a n o n - n e g a t i v e superharmonic f u n c t i o n  w h i c h i s s t r i c t l y p o s i t i v e a t some p o i n t o f  a l l p o i n t s of  D  But i n one d i m e n s i o n , a l l s u p e r Hence  u  and  bal(u,E,D)  agree  in D . •  3.7. W  Lemma:  Let D  be a Green r e g i o n i n ] R , where  be an open subset of  n  D,  and l e t x e W .  c o n t i n u o u s n o n - n e g a t i v e superharmonic f u n c t i o n w <_ w(x)  n >_ 2 . L e t  Then t h e r e i s a f i n i t e w  in D  such t h a t  in 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  unbounded component o f Proof: (If  D\W  x  b e l o n g s t o an  D .)  L e t m = sup G(x,*) D\W  where  i s empty, l e t m = 0 .)  G  A  i s t h e Green f u n c t i o n o f  Then  Helms [ 1 ] . L e t w = G(x,«) (m+l) . sup w < m + 1 . D\W  and  Then  m  D .  i s f i n i t e , by 5.8 o f w <_m + 1 = w ( x ) , and  •  - 27 -  3.8. Let  Theorem.  Let  D  be a Green r e g i o n i n ]R , n  E £ D  and l e t x e D .  a)  E  i s thin at  b)  There i s an open s u b s e t  Proof: a) = >  x  harmonic f u n c t i o n  g  x e W,  and  Let b)  W  D .  of  in D  D,  a non-negative super-  and a c o n s t a n t  g >_ a  on  a  such t h a t  W n (E\{x}) .  F = E\{x} .  Let  such t h a t  n >_ 2 .  Then t h e f o l l o w i n g a r e e q u i v a l e n t :  r e l a t i v e to  g ( x ) < a,  where  u  be a n o n - n e g a t i v e superharmonic f u n c t i o n i n D  bal(u,E,D)(x) < u(x) .  Now  Z = F n ( b a l ( u , E , D ) < u}  a p o l a r s e t so by 2.5, t h e r e i s a n o n - n e g a t i v e superharmonic v  in D  such t h a t  g = v + bal(u,E,D) and  u(x)  a)  function  .  Let  a  Z  g^_u  on  F,  but  g(x) < u(x)  Then  g>a,  on  x e W  and  in D  g(x)  g(x) < a .  WnF.  By 3.7, t h e r e i s a f i n i t e n o n - n e g a t i v e w  function  where  be a number s t r i c t l y between  and l e t W = {u > a} .  Also, since b) = >  v = °° on  superharmonic  such t h a t w <_ w(x)  in  D  and w(x) > sup w . D\W We'll  show t h a t  b a l ( w , E , D ) ( x ) < w(x) . L e t m = sup w . D\W  (If  D\W  i s empty, l e t m = 0 .)  p o s i t i v e r e a l number  b  such t h a t  Since  w(x) > m,  there i s a  w(x) - ba >_ m . L e t  is  - 28 -  v = w(x) + b(g-a) .  Then  v >_ w(x) - ba >_ m, v >_ w(x) as  on  WnF  w(x) >_ w  n >_ 2 .  so  i s superharmonic i n D .  v >_ 0  as  in D .  Thus  v  in D  g - a >_ 0 Thus  v >_ w  there.  v >_ w  v >_bal(w,E,D)  and  on  Hence  F .  i n D,  on  Now  by 2.6.  Now  D\W  v >_ w {x}  .  Also  on  W n F  i s p o l a r , as  But  v ( x ) = w(x) + b ( g ( x ) - a) < w(x) . • 3.9.  Theorem:  Green r e g i o n s at  x  Let i n E.  r e l a t i v e to  E  be any s u b s e t of IR  .  Suppose  n  D  and l e t D,D'  n  x £ D n D' .  i f f E n D'  i s thin at  Then x  E n D  i s thin  r e l a t i v e to  Proof:  C l e a r l y we need o n l y prove t h e f o r w a r d i m p l i c a t i o n .  n = 1,  t h e r e s u l t f o l l o w s from 3.6.  E n D  i s thin at  open subset  W  of  in  D,  on  W n (E\{x}) .  x  u(U) <  .  Let  00  D .  such t h a t  e > 0  If and  Then by 3.8, t h e r e i s an  Let  x e W,  G (resp. G ) 1  g ( x ) < a,  and  g  g >_ a  be t h e Green f u n c t i o n o f  D  be an open r e l a t i v e l y compact neighbourhood Let  u  be t h e R i e s z measure of  f o r A e B o r e l D' .  g' = g + h  n >_ 2  D' .  a n o n - n e g a t i v e superharmonic f u n c t i o n a  U  Hence assume  so we can d e f i n e a f i n i t e measure  Ag' = -v = Ag .  Choose  D,  i n W n D' .  v(A) = y(AnU)  that  r e l a t i v e to  and a number  ( r e s p . D') of  x  be  Let  v  i n D'  g' = G'v .  Thus t h e r e i s a harmonic f u n c t i o n in U .  such t h a t  open neighbourhood of  x  Now  Then  by  Now i n h  in U  U, such  g'(x) = g ( x ) + h ( x ) < a + h ( x ) .  g'(x) < a + h ( x ) - e . in U  g .  such t h a t  Let  W  h >_ h ( x ) - e  be an in W  .  - 29 -  Let  a' = a + h(x) - e .  on  W  n (E\{x}) .  by  3.8.  Then  Thus  x e W,  E n D'  g' (x) < a' ,  i s thin at  x  and  g' >_ a'  r e l a t i v e to  D',  • Note that the above theorem implies that thinness i s a l o c a l property.  3.10. E  Definition.  i s thin at  x e D  and  x  E  Let  E £ ]R  and l e t  n  x e TR  i f f there i s a Green region  i s thin at  x  r e l a t i v e to  U  .  D  D .  We  s h a l l say  i n TR  such that  n  Also, we s h a l l use  the following notations:  f r i n g e ( E ) = {x e E : E base(E) = {x e ]R : E  Corollary: a)  If D  Let  x e ]R  n  in  H  n  E £ ]R  n  and with  E  x}  i s not thin at  n  3.11.  i s thin at  x} .  .  i s thin at  x e D,  x  E n D  then for any Green region  i s thin at  x  r e l a t i v e to  D • b)  For any Green region  D  in H , n  fringe(EnD,D) = D n f r i n g e ( E )  and base(EnD,D) = D n base(E) . c)  fringe(E)  i s a polar set and  base(E)  intersection of open subsets of d) Proof: c)  If  n = 1,  fringe(E)  n >_ 3,  2,  ]R  Now  apply  n  3R  n  n  countable  .  i s empty and base(E) = E .  a) and b) follow immediately from If  E.  is a  3.9.  i t s e l f i s a Green region.  If  n = 1  or  can be written as the union of two Green regions. 3.3.  - 30 -  d)  apply  3.6. •  3.12.  L e t 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 t o p o l o g y i n what f o l l o w s ,  but i t p r o v i d e s a h e l p f u l p e r s p e c t i v e . Let D  D  be an open subset of  i s the weakest t o p o l o g y i n  functions i n  D  D  continuous.  Then the f i n e t o p o l o g y on  I t i s s t r o n g e r than the u s u a l t o p o l o g y  D, w i t h e q u a l i t y i f f n = 1..  on  D  One  can show t h a t the f i n e t o p o l o g y  i s a l s o generated by the f u n c t i o n s of the form  ranges over measures on  3R  D  of the f i n e t o p o l o g y on If  D  where  y If  H  D  D  superharmonic  functions i n  D  ]R  n  the s e t s of the form  W n {u < c}  E .  i s thin at  Also i f V £ H  open i f f ]R \V n  n  i s any c o l l e c t i o n of  u,v e S,  (W open £ D, u e S ,  ceH)  c o n j u n c t i o n w i t h 3.8, one f i n d s t h a t f o r E  Gy,  D .  for a l l  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 on  n >_ 2 ) ,  S  and  and i f u + v e S  D,  of  then the f i n e  which i s l a r g e enough to g e n e r a t e  the f i n e t o p o l o g y on  (where  G  i s a l s o generated by the f u n c t i o n s of the form  i s any open subset of  D .  .  n  ranges over measures w i t h compact support i n D  where  y  i s e q u a l to the r e s t r i c t i o n  i s a Green r e g i o n w i t h Green f u n c t i o n  t o p o l o g y on  U |D,  w i t h compact support c o n t a i n e d i n  n  I t f o l l o w s t h a t the f i n e t o p o l o g y on to  .  n  which makes a l l superharmonic  of  y  ]R  (where  x  iff x n  D . E £ H  n  Using t h i s i n and  x e ]R  n  i s not a f i n e l i m i t p o i n t  i s arbitrary),  i s t h i n a t each p o i n t of  then  V .  V  i s finely  - 31 -  4.  THE STRONG FORM OF THE DOMINATION PRINCIPLE In t h i s s e c t i o n we a r e g o i n g t o prove t h e s o - c a l l e d s t r o n g  form  of t h e d o m i n a t i o n 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 t h e l o g a r i t h m i c p o t e n t i a l ) .  F o r t h e case of a Green 2  r e g i o n , t h i s r e s u l t i s due t o B r e l o t [ 2 ] . F o r E , We a l s o g i v e a p r o o f o f t h e d o m i n a t i o n p r i n c i p l e i n E  i t may be new. 1  .  Of c o u r s e  t h i s i s q u i t e e a s y , and t h e a d j e c t i v e " s t r o n g " i s s u p e r f l u o u s  i n this  c a s e , owing t o t h e f a c t t h a t i n d i m e n s i o n one a l l superharmonic f u n c t i o n s a r e f i n i t e and c o n t i n u o u s . 4.1.  Theorem ( B r e l o t ' s s t r o n g d o m i n a t i o n p r i n c i p l e ) Let  y  D  be a Green r e g i o n i n E  be a measure i n D  subset of  D .  such t h a t  y(D\base(E,D)) = 0 .  b)  Whenever  c)  v  such t h a t  then  Gy  i sa potential.  G . Let  Let E  be any  Then t h e f o l l o w i n g a r e e q u i v a l e n t :  a)  D  w i t h Green f u n c t i o n  n  v >_ Gy  i s a n o n - n e g a t i v e superharmonic f u n c t i o n i n v >_ Gy  throughout  Whenever  v  E,  v >_ Gy  then  on  E\Z  where  Z  i s a polar set,  D .  i s a potential i n D throughout  such t h a t  v >_ Gy  on  D .  Proof: a) — > b) function  I f x e base(E,D) u  i n D,  then f o r any n o n - n e g a t i v e superharmonic  - 32 -  u(y)bal(6  ,E,D)(dy)  x  bal(u,E,D)(x)  so  bal(6  ,E,D) = 6  b) = >  c)  c)  c) = >  a)  Let  Let  = u(x)  u(y)6 ( d y ) , x  by 2.13.  Hence  x  bal(Gy.E,D) = Gy .  Hence  f = bal(Gy,E,D)  v(D\base(E,D)) = 0 so  v >_ Gy  throughout  Then  by 3.4.  Gy 4 Gv .  As  .  Z  Gv <_ Gy  x e D\Z,  Gv(x) < Gy(x) .  we have  such t h a t  v = Gv + g . but  g = °° on  Then  v  Then  Suppose  {Gv < Gy}  D  D  by 2.15.  Thus  by 2.6.  f = Gv  where  i s a polar set.  t h i s implies that  in  bal(y,E,D) = y  i s j u s t a s p e c i a l case of b) .  Z = { f < G y } n E .  y 4 v,  (by 2.9)  v = bal(y,E,D)  Now  y(D\base(E,D)) 4 0 . and  Gv, Gy  Then  a r e superharmonic,  i s not a p o l a r s e t . Hence f o r some  Z  but  By 2.5, t h e r e i s a p o t e n t i a l  g  Gv(x) + g ( x ) < Gy(x) . L e t  i s a potential i n D  and  v >_ u  on  E,  v(x) < u(x). •  4.2.  Theorem.  G .  Let  let  v  measure a)  y  Let  be a Green r e g i o n i n R  be a measure i n D  such t h a t  n  Gy  w i t h Green f u n c t i o n i s a p o t e n t i a l , and  be a n o n - n e g a t i v e superharmonic f u n c t i o n i n D v .  with  Then t h e f o l l o w i n g a r e e q u i v a l e n t :  v >_ Gy  a l m o s t everywhere w i t h r e s p e c t t o  y(Z) £ v(Z) b)  D  v >^ Gy  f o r every B o r e l p o l a r set  everywhere  in D .  y,  and  Z £ D .  Riesz  - 33 -  Proof: a) = >  b)  Let  y(D\E) = 0 .  E = {v ^ G u }  Also  P  Let  a = M \p  and v  y  in  v = GB + Gy + h in  D .  Let  y(Z)  6 = y D  where  {GB  a(D\base(E,D)) = 0 . throughout  P = fringe(E) .  = ) 00  <_ v(Z) .  p  h  .  Thus  Then  i s B o r e l by 3.3,  since  f o r every B o r e l  Then  such t h a t  6 <_ v,  set  so t h e r e i s a  8 + y = v .  Now  Gy = Ga +  G8  i s the g r e a t e s t harmonic m i n o r a n t of  w = Gy + h .  on the p o l a r s e t  v >_ Gy  so  >  n  (unique) measure  let  i s a p o l a r s e t (and  f r i n g e ( E ) = E\base(E,D)) Z £ P .  and  Then Also,  w >^ Ga  w >_ Ga  on  E,  except p o s s i b l y  a(D\(E\P)) = 0 throughout  D,  so by 4.1.  Hence  D . •  b) = >  a)  Obviously  v >_ Gy  y - a.e.  charges p o l a r s e t s l e s s than  v,  For the p r o o f t h a t  l o o k ahead t o  y  10.8. •  2 We 4.3.  now  t a k e up the p r o o f of the s t r o n g d o m i n a t i o n p r i n c i p l e i n Lemma:  potential.  Let  y  be a measure on  such t h a t  3R  is a  Then:  a)  l i m (U^(x) - yCR )1>"(x)) = 0 . | | x | |-*»  b)  If  2  y  has  compact s u p p o r t , or i f  y(dx) = f ( x ) d x  where  2 f and  i s n o n - n e g a t i v e and  l o c a l l y Lebesgue i n t e g r a b l e on  tends t o zero a t i n f i n i t y , then lim (U (x) - y(R )$(x)) = 0 . | |x| |-*» M  2  H  -  -  34  Proof: a)  U ( x ) - yOR )$ (x) y  2  (x-y) - $ ( x ) d u ( y )  Now  if  x  > 1  and  | |x-y| | >_ 1  (x-y) - $ (x) = Thus  $ (x-y) - $ (x) —*• 0  l o  as  then  1 x-y H  §  | |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 t h a t of  U  i s a p o t e n t i a l i s e q u i v a l e n t t o the f i n i t e n e s s  y  log ||x||dy(x),  Claim:  +  by 1.6).  For a l l x,y £ H I $ (x-y) -  2  ,  () I± Y n ( g x  F i r s t n o t e t h a t f o r any  l o  2  +  l o  g l |y| +  r >_ 0,  l o g ( l + r ) <_ l o g 2 + l o g r . +  ( C o n s i d e r the two cases | | x | | •> 1  and  0 <_ r <_ 1  || x-y | | >_ 1 .  $~(x-y) - $ (x) = j- l o g Thus i f  | |x-y| | >_ | |x| |  then  and  r >^ 1 .)  Then  1 x-y 1 |x|  First  suppose  - 35 -  (x-y) - $ (x)  2TT  27  1  1 x-y 1  log  |x |  1 1  log  < ^ l o g d  y  _l_ T  X  + ||y||)  < |^ (log 2 + l o g | | y | | ) +  while i f  I I x-y I I <_ ||x||  then  |$~(x-y) - $ ( x ) | = ^  log  1*1 1 x-y  (x-y) + y  1  o l ° °g 2TT  =  l o  -  l o g ( 1 +  ||x-y|| £ 1 .  |y|  1 +  i27  §  1 x-y  I' l y  (log 2 + log ||y||)  < ^  Now suppose  —V\ x-y —F\  -  27  |  +  Then  |$"(x-y) - <2>~(x)| = $~(x) l o g | |x| | <_j^ l o g ( | |x-y| | + | |y| +  - 17 Finally  suppose  x  +  +  I' l  < 1 .  Then  l o g ( 1  ( l o g 2 + l o g | |y  <_  y  +  |$ (x-y) - $ ( x ) | = <3> ( x - y ) = ^log ||x-y|| +  l^log (||x||  < i ^ l o g d + ||y||) < ^  +  + ||y||)  (log 2 + log ||y| +  - 36  The  claim  b)  If  i s now  u  has  y(dx)  =  f(x)  clearly $ =  $  -  y  4.4.  dx  ,  and  a potential.  open b a l l U  as  satisfies  =  0  for  =  y  1  | |x| | —*- »  +  U (x) .  On  the .  0  the  stated  Now  U  other  = U .  - U  y  x  e  1^  hand, i f  conditions  y  | |x| | >_ 1  for a l l  then and  y  Hence  as  •  Let  y  be  a m e a s u r e on  such  that  U  is  y  Then  Lemma:  =  f  then  inf  (U (x) y  )*(x)) > 0 .  - yQR  is clear.  potential.  y'  least  $ (x)  lim  4.5.  i s at  )$(x) —• 0  yQR  proof  K  K  0  Corollary:  The  support  where —>-  y  - $  U (x)  from  U (x)  +  established.  compact  whose d i s t a n c e  -  Let For  of  be  a measure  each p o s i t i v e r e a l  radius •  7  K\B  y  r  centred  on  ]R  such  2  number  at  0  r  let  2 ~R  in  that  and  U  B  is a  y  denote  let  u  the » r  =  Then:  r 2  a)  For  a l l  x  e H  u^OO  and  > - ^  a l l positive real  r  a l l positive real  k,  there  0  —  r  <  °°  -LiiLL )  2  For  r  |>r  numbers  e  and  exists a positive real  w  e  n  a  v  e  r,  yOR \B )iog(l + $(y)dy(y) + | y|  b)  numbers  U  y +  y  e >_ U  a l l positive  number  r^  r on  B^  r  .  such  integers that  for  37  -  -  Proof: a)  Consider  in  E \B 2  r  any  r  in  (0,°°)  .  r  y  Then f o r any  x  in  R  and  any  y  ,  x-y j | < —  1  M  +  +  so  $ ( x - y ) = - 27  l o g j |x-yI x  i " 27 l o g 1 + Thus f o r a l l  x  in  TR ,  y  U  r (x)  $(x-y)dy(y)  =  r  E \B 2  yOR \B )log 1  > - ^  2  r  +  $(y)dy(y)  y  b)  For  any  x  in  TR  and  r  !>  r  in  (O, ), 0 0  y U (x) y  y'  = U  r  (x)  + U  > U  r  (x)  -  r  2  r  Now  choose  r^  in  2  y  (l,  ^yOR \B  c o n c l u s i o n o f b)  then  +  0 0  )  >r  so  that  )iog(i+k)  +  log||yI|dy(y)  ^ y||  The  1  log||y||dy(y),  2IT  a).  (x)  yOR \B )loj  1_  by  +  > r  r  follows.  •  - 38 -  4.6.  Lemma:  2 be a compact n o n - p o l a r subset of ]R .  Let K  2 t h e r e i s a non-zero measure A on R A 2 such t h a t U i s bounded above on TR  Then  which i s supported by  K,  2 Proof: the  Let D  be a Green r e g i o n i n R  Green f u n c t i o n o f  ing  K,  in  D .  K .  D .  containing  (We can t a k e  D  A  be t h e R i e s z measure o f  u .  We s h a l l denote t h e o b v i o u s e x t e n s i o n of  by t h e same l e t t e r such t h a t  A .  = u + h  and l e t G  Then Then A  u A  i s a potential i s s u p p o r t e d by 2  t o a measure on  Now t h e r e i s a harmonic f u n c t i o n  in D .  Since  u  be  t o be an open b a l l c o n t a i n -  f o r instance.) Let u = bal(l,K,D) . Let  K  h  i s bounded i n D  R  in D and  h,  by  c o n t i n u i t y , i s bounded i n a neighbourhood o f K, i s bounded i n a A 2 neighbourhood of K . A l s o , U i s c o n t i n u o u s on ]R \K . I t now A 2 f o l l o w s from 4.3(b) t h a t U i s bounded above on R 2 Now here i s the s t r o n g d o m i n a t i o n p r i n c i p l e f o r R 4.7.  Theorem:  Let u  is a potential.  be a non-zero measure on R 2 be a subset of ]R  Let E  2  such t h a t  and l e t v  U  y  be a s u p e r -  2 harmonic f u n c t i o n on R a)  such t h a t : 2 l i m i n f ( v ( x ) - pQR ) * ( x ) ) > -«  b)  v >_ U  c)  uQR \base(E)) = 0 .  Then  y  on  E\P  where  P  i s a polar set  2  v > U  Proof:  P  on a l l o f R  2  .  The p r o o f proceeds by r e d u c i n g t o t h e case o f a Green r e g i o n .  For  each  at  0  r > 0  2 i n ]R  let  denote t h e open b a l l o f r a d i u s  and l e t u  4.5(b), there e x i s t s  r^  r  r  centred  denote in  u_ . Choose e > 0 . By B r (0,°°) such t h a t f o r a l l r i n [ r ^ , ) 0 0  - 39 -  U  + -| > U 2 —  y  u  _ B . r  on  r  2 N e x t , as  u ^ 0  but  not a p o l a r s e t . E\fringe(E)  yQR \ b a s e ( E ) ) = 0,  base(E) ± 0 . Hence  (Combine 2.6 and 3.11).  and, by 3.11, f r i n g e ( E )  Now base(E)  i s polar.  Thus  E is  contains base(E)  is  not p o l a r . Hence t h e r e e x i s t s r ^ i n [ r ^ , ) such t h a t base(E) n B i s not p o l a r . Now base(E) n B i s analytic; i n fact l i i t i s a c o u n t a b l e i n t e r s e c t i o n o f open s e t s , by 3.11. Thus t h e r e i s 0 0  r  r  a compact n o n - p o l a r s e t K  contained  i n base(E) n B  , by 6.23, l 7.32, and 7.33 o f Helms [ 1 ] . By 4.6, t h e r e i s a non-zero measure X r  2 TR.  on  s u p p o r t e d by  (0,1] .  Then  U  K  = aU  a X  such t h a t <_ -|  A  U where  y  r,a  = y  r  + aA .  such t h a t f o r a l l r  Then f o r any  r  in  r  As  in  t^,  on TR  e <_ y  X  .  2  y + e >• U ' —  y  y  U  2 on TR .  Choose  Thus f o r a l l r on  a  a,A. ^ 0,  a in  in  [r ,«0,  in  [ r , ,°°) 1'  0  _ B , r  there e x i s t s  r„ 2  [r^, ), 0 0  0 R ) > yOR ) • 2  r  a  2  ),  0 0  2 lim i n f (v(x) - y QR ) * ( x ) ) = +°° . i i ii r,a  Il l x  Choose  r, in  [r ,°°) .  [r„,°°)  2 such t h a t on 1R \B  9  J  Let y  =  M  •  Then t h e r e e x i s t s  r, i n  we have  r  4  2 v > y(]R ) $ and yOR )* + e > U 2  where t h e second e s t i m a t e  Y  f o l l o w s from 4 . 3 ( b ) , s i n c e  y  has compact  - 40 -  support. B  r  5  ,  Now choose  r,. i n  (r^,°°)  w h i c h we s h a l l denote by  on  Y  = in  Y  (Note t h a t t h e s u p p o r t of  y  i s c o n t i n u o u s on  B,  h|B  U  v + e >_ h  B .  Now  c o n t i n u o u s on  B  and  B  i s a compact subset of  on  8B  and  and superharmonic i n  B  .  + e- h  y  Thus  on  v + E - h B .  v + e - h ^ U  (EnB)\P - h  T  and  on  U  (EnB  3  r  r  contains  base(E) n B  by 3.11. r  Therefore  B .)  v + e - h  Hence  + e - h >_ U  y  ) \P .  Now  - h  - h  Y  on  base(EnB  3  Hence  is  Now  ,B) 3  r  y(B\base(EnB  ,B)) = 0 . r  Y  h  i s lower semi-  3  v+e-h>_U  Then  i s t h e g r e a t e s t harmonic m i n o r a n t of  a n o n - n e g a t i v e superharmonic f u n c t i o n i n B . v + £- h^U  region  8B  PI(U ;B)  in  t h e Green  B . Let U  h  and c o n s i d e r  throughout  B  3  by 4.1,  since  Y  U  - h  Y  i s t h e Green p o t e n t i a l of in ]R . 2  B .  y  r e l a t i v e to  As t h i s i s t r u e f o r a l l r,. i n  Now l e t t i n g  Next, l e t t i n g  r^  go t o  <*>,  e > 0  was a r b i t r a r y ,  Thus  [r^,°°),  we o b t a i n  v + e >_U  v + e >_ U  v + e >_  a d e c r e a s e t o z e r o we f i n d v + e >_ U on ]R . y  As  B .  u ^ y+a  on  Y  on  H  2  .  that  2  i t follows that  v >_ U  y  on  H ,  and we  2  a r e done. • 4.8.  Theorem:  Let  are p o t e n t i a l s and  u,v  be measures on  u 4 v .  b) below a r e e q u i v a l e n t .  Let  c  H  2  such t h a t  be a r e a l number.  U , y  U  V  Then a) and  - 41 -  a)  U  b)  i) U  V  + c >_ U  on  y  + c >_ U  V  K almost everywhere w i t h r e s p e c t  y  to  y 2  ii)  y ( Z ) <_ v ( Z ) f o r a l l B o r e l p o l a r s u b s e t s of 2 2 V0R ) £ y(R ) .  iii)  H  Proof: a) = >  b)  i)  i s obvious.  The p r o o f of i i ) w i l l be d e f e r r e d . 2  See 10.8.  i i i ) F o r each  x  in H  let y  be t h e u n i f o r m u n i t x  2 i i (y e E : j |y-x|| = 1} .  d i s t r i b u t i o n on  U  Then  + c dy  V  Y U  X  x dv + c  = U (x) + c V  and  V  1  dy  = U (x) + c y  x  Thus U  Hence  + c > U  2 2 vQR ) £ yOR ) •  b) = > a) P  V  Let  y  T h i s f o l l o w s from 4.3(a) ..  E = {U  V  + c >_ U }  and l e t P = f r i n g e ( E ) .  y  i s a B o r e l p o l a r s e t by 3 . 1 1 ( c ) .  sets contained  in P .  Let  Thus  8=yp,  y ( Z ) <_ v(Z)  a = y - B ,  Then  for a l l Borel  y = v - & .  Then  2 ct 6 y a,B,Y a r e ( 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 \ b a s e ( E ) ) = 0, and U + c >_ U on E e x c e p t p o s s i b l y on t h e polar set  {U  B  = °°} .  Also  2 2 yOR) <_aCJR )  l i m i n f ((U (x) + c) - a(R )$(x)) Y  I lx|  |-^°°  2  so >_ c,  - 42 -  by 4.4. R  2  Thus, i f a ^ 0,  by a p p l y i n g 4.7.  we may c o n c l u d e t h a t  Then by a d d i n g  i n e q u a l i t y we o b t a i n  U  v  + c >_ U  y  On the o t h e r hand, suppose 2 2 v0R) yQR)  so  <  4.9.  Corollary.  - a.e.  Proof:  Then  Take  y  Let  y ^ 0 .  p o t e n t i a l and y  y = v .  U  v = 0  on a  R  c  on  2  i s zero.  Then  But we a r e assuming  Let  + c >_U  6 t o b o t h s i d e s of t h i s  be a measure on  < c  y  U  U  R  y = B <_ v  y f v .  such t h a t  2  but  be a r e a l number.  • U  Suppose  is a  y  U  <_ c  y  everywhere.  i n 4.8. •  F i n a l l y we g i v e a proof  (1.5) t h a t i f y  F i r s t of a l l , r e c a l l U  y  Lemma.  potential. let  £  Let  5  (-°°,C]  i s finite. R  y  on  .  Then  Consider f >_ 0,  [£,°°),  and  £,  f  y .  U  as  y dy(y) .  (  T O ) X ]  just  [x [ —^- «> . Suppose  x-y|dy(y)  - |x-y|dy(y)  (x-y)l _  isa  i s i n c r e a s i n g on  f (x) = - j yOR)(?-x) + \ x - y  y  (If y = 0  Then  1 2  then  the f u n c t i o n  f (x) — • 0  y(IR)5  R  Now we prove a lemma.  such t h a t  be t h e c e n t r e of mass of  By t h e c h o i c e of .  i s a measure on  be a measure on  be any r e a l number.)  i s decreasing  Proof:  y  Let  f = y(R)$(£>*) - U f  x dy(x)  i s a potentiali f f  4.10.  of the d o m i n a t i o n p r i n c i p l e f o r R  (y)dy(y)  x  i sin  - 43 -  Hence as  f  i s n o n - n e g a t i v e and i n c r e a s i n g on  x —•  .  Also  f (x) —>- 0  by the Lebesgue dominated convergence theorem.  interval  i s treated  The  similarly. •  4.11.  Corollary.  Let  y,v  are p o t e n t i a l s , and l e t  c U  be measures on  TR  be a r e a l number.  V  + c > U  y  such t h a t  U,  U  y  V  Suppose  .  Then: a)  vCJR) <_ uQR)  b)  if  vQR) = uQR)  then  c >_ 0  and  y  and  v  have the same  c e n t r e of mass. Proof:  By  4.10, l i m ( U ( x ) - y(IR)$(C,x)) = 0 | x | -**> y  and l i m ( U ( x ) - vOR)$(n,x)) = 0 |x|-**> V  where  E,  ( r e s p . n)  i s the c e n t r e of mass of  y ( r e s p . v) .  The  c o r o l l a r y f o l l o w s i m m e d i a t e l y from t h i s . • Now  here i s the d o m i n a t i o n p r i n c i p l e f o r ]R .  4.12.  Theorem.  Let  y  i s a p o t e n t i a l and l e t v that  be a non-zero measure on  E.  such t h a t  be a superharmonic f u n c t i o n on  R  U  such  y  -  a)  44  -  l i m i n f (v(x) - y(E)$(x))  >  -»  | x | -*» b)  v  v  > U  Then  Proof:  A  > U  almost everywhere  y  everywhere  y  function  concave.  Note  on  that  TR  i s superharmonic  such a f u n c t i o n  E =  y(W)  =0  so  one,  this  j u s t amounts t o s a y i n g  graph  this p  U  of  U  let  and  let  (a,b)  Also  0  Case  1.  a  v(b)  >^U (b)  be  v(p)  >. U ( p )  Case  2.  u E,  b  both  i s open.  .  Since  we  are i n dimension  on  .  Also  [a,b]  v  .  each  By  of  component  W  i s concave,  {<*>},  Hence  v  >_ U  on  y  of  v(a)  and  of  U  and a  W,  of  of  containing  e E u  Then  component  Also  the c o n t i n u i t y  so a t l e a s t one  finite.  continuous.  W  of each  b  and  Then  that  t h e component  y 4 0,  and  on  W  the c l o s u r e  a e {-°°}  since  y  function  C  where  E 4  in  i f fi t is finite  .  i s a straight line.  y  y  i s automatically  W = H\E  i s harmonic  y  a c t u a l l y h o l d s on  e W  C =  and  y  to  TR .  Let  the  {v ^ U }  on  with respect  W p  >_U (a) y  y  .  .  Let Then  a < p and  U ,  < b  b  is  . finite  and  is a straight-line  y  [a,b]  .  In p a r t i c u l a r ,  .  P  a =  b  finite.  lim  By  a)  and  inf(v(x) - U (x) y  4.10,  > -°° .  x->-°°  Hence there  v  - U  y  i s bounded  i s a u n i q u e number  p =  below c(x) e  on  (- ,b]  [0,1]  ro  .  such  (1 - c ( x ) ) x + c ( x ) b  For each that  x  e  (-°°,p] ,  -  As  x —>• -<*>,  c ( x ) —>- 1  v(p)  since  U  Letting  .  45 -  Now  1  (1 - c ( x ) ) v ( x )  1  (1 - c ( x ) ) ( v ( x )  x  3.  c(x)v(b)  - U (x))  go t o  we  a  finite,  b =  0 0  +  U  i s a straight-line function  y  on  U (p), y  (-°°,b]  a n d v ( b ) _> U ( b ) . y  obtain  v(p)  Case  +  .  >D (p)  .  y  This  i s s i m i l a r to case  2 .  •  4.13. are  Corollary.  Let  u,v  p o t e n t i a l s , and l e t  P ^ 0  .  Then  a)  U  b)  vQR)  V  +  a) and  c >_ U  be a r e a l below a r e  on  TR  number.  such  that  Assume  U , y  U*  also  that  everywhere  with  V  equivalent.  on a l l o f TR  P  to  c  b)  <_ u Q R ) ,  respect  be m e a s u r e s  and  U  V  + c >_ U  almost  y  u .  Proof:  a)  ==>  b)  Apply  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 t h e n o t a t i o n and t e r m i n o l o g y  t h a t we s h a l l need f o r Brownian m o t i o n .  Due t o t h e 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 r e s e r v e t h e r i g h t  t o use these  l e t t e r s f o r o t h e r purposes l a t e r on. However, whenever they a r e used w i t h o u t e x p l a n a t i o n t h e i r meanings w i l l be as d e f i n e d i n t h i s  5.1.  F o r 0 < t < °°,  p  t  w i l l denote t h e f u n c t i o n on ]R  n  section.  defined  by , s  1 *n/2  f  -||x|| /4t 2  Then: a)  p  s a t i s f i e s t h e heat  P  87  b)  p  equation:  t  =A  p  t ,n  i s a probability  d e n s i t y on ]R  variance 2nt: p  > 0  t  P (x)dx  = 1  t  xp (x)dx = 0 t  x|  c)  p (x)dx t  2nt  ( p ) i s a c o n v o l u t i o n semigroup: t  p (x-y)p (y)dy g  t  w i t h mean  0 and  - 47 -  5.2.  will  The  point  denote  isolated 5.3.  at i n f i n i t y  the space  ]R  n  for  H  will  n  u {3}  where  will  denote  be  8  denoted  by  9,  i s considered  as  0 < t < °°,  P  the f u n c t i o n  on  H  AfHR P (x,A) =  i f  n  1 (3)  i f  A  (P ) .  i s a temporally P  o  (x,A)  should  x =  Brownian  starts  motion,  Note Now  let  homogeneous Markov  be  thought  of as  that  at  e ]R  n  at a certain  i t will  be  found  a particle  suppose x  x  i  transition  function  the p r o b a b i l i t y  that i f  .  We  wish  i s t o be at  = l,...,k  x  ...  at  3  < t^  to define  thought  Well,  A  t  does and  units  not  go  P  (x,A  of as s,  the p r o b a b i l i t y  i t will  c l e a r l y we  be  in  should  that  A.  have  ^t/^V• s p  of  to time  anywhere.  A^,. . . ,A^  S y t^ , ••., t^  a t time .  and moves a c c o r d i n g  i n the set  starting  0 <_ s < t ^ <  time  O  starts  3  t  a particle  which  x 4 3  \  t  Also  0  by  t  later.  Borel  0  P (y-x)dy  on  n  o  an  x  n  t  Then  ]R  point.  For  defined  and  e  Borel  ,...,A, ) , 1  i f a  at time  particle t.  1  K  for  x  ( x  V  'V  and P  .  s ,t^,.  . .,  j_  P l  J-  ,. . . ,A. )  ( x , d a )P '1  A  (x,A  ( a ,A 1'  2'  k  ,...,A.)  - 48 -  so by r e c u r s i o n we a r e l e a d t o d e f i n e  V  V (a da ) (x,da ) P A 1' 2 '1  P  A  5.4.  We d e f i n e  C  C  P  1  t  -t  ( a  k-l'  2  ft  t o be t h e s e t o f a l l f u n c t i o n s  u>: [0,°°] —*• E„  such t h a t : i s c o n t i n u o u s on =  a)(«)  (Then i f 3 .)  co(t) = 3  We w r i t e  3 t e [0,°°), io  f o r some  CJ  [0,°°)  f o r t h e element o f  i s i d e n t i c a l l y equal to  P. w h i c h i s i d e n t i c a l l y e q u a l  3  to  3 .  t e [0,°°]  F o r each B :  and we l e t  fi-*lR  we d e f i n e  by  3  B (u)) = u ( t ) , fc  be t h e a - f i e l d on  We a l s o w r i t e  8^  Q  g e n e r a t e d by  { B : 0 <_ s <_ t } g  f o r 8^ . CO  If  y  i s any  a-f i n i t e measure on  B o r e l R^  then t h e r e i s a  o  unique in  B^-outer measure  Borel E^ 3  and  P (B  P  0 < t  y  Q  n  < ... < t. l k  e A , B Q  w ( d x ) P  on ft such t h a t i f Ap,A^,...,A^ a r e  y  e  then  A^-.-.B  0;t ,...,t. 1  •v  (In p a r t i c u l a r ,  taking  1  ( s t  ,  V  » r-"» k • A  A  )  k  A^ = ... = A^ = R^  we have  P (B M  Q  e A ) = y Q  d  - 49 -  P  P  E  y  i s called will  one also  the Wiener measure  denote the i n t e g r a l  takes  y  y's  H o w e v e r , we  0 < s < t < ° °  measure  we  Q  e A)  P (B  t  e  The  second e q u a l i t y  y =  S x  P  on  Q  y  .  by  is  y  have o c c a s i o n  to  another v e r s i o n  =  for  P  s ; t  for  as t h e  A e B o r e l ]R  and  n  (B ,A) s  P  .  X  on  ^  B  : s  The 6„ t  [0,°°] t £.  s  9{.'  s  property. H e B°  If  If  then  and  <j>  we  a  r  called  t o be  define  the  0 :  translation  we  E  have  a-field —>•  t  by  operators.  Now  here i s  F o r any n o n - n e g a t i v e  (and any o - f i n i t e measure  B. t  e  property:  0 <_ t < °°,  E (4> • e | B ° ) . =  ^8  (^B,B^)-measurable. t  on  y  define and we  is  o f t h e Markov  measurable f u n c t i o n and  such that  d e n o t e d by  t e  , , s+t  = B  t  Q,  be d e f i n e d  X  Since  B o r e l ]Rg)  P  y(dx)P (H)  .  t  • B  shall  can a l s o  y  function  =  ( 0 c j ) ( s ) = u)(s+t)  s  Customarily,  i n which case  h e r e i s known a s t h e M a r k o v  F o r any  generated  B  .  y(A)  |B°)  i s a Borel  y  any  A  =  i s also  y  P (H)  for  P  on  P (B  y  X  y  y .  have  y  x \—y P ( H )  P  measure  w h i c h a r e n o t p r o b a b i l i t y m e a s u r e s , and w h i c h i n  8^-outer  then  initial  to  t o be a p r o b a b i l i t y m e a s u r e ,  some c a s e s a r e e v e n i n f i n i t e . unique  with  with respect  a p r o b a b i l i t y measure.  consider  Q  on  y  on  B  -  -  Now we to  introduce the  be t h e  is to  c-field  o - f i n i t e on  a-fields  of a l l  B^  50 -  B, y  B,  B,  y  P -measurable y  ( i n d e e d , on  Bg)  and  .  subsets  We  o f ft .  take  B  Since  and o u t e r r e g u l a r w i t h  y  P  y  respect  B°, B  = {Heft;  y  L c H c M  and  P (M\L) = 0  f o r some L,M e B°},  U  N e x t , we l e t  B  Finally  = {He  y  B: y  P (HAF) = 0  f o r some  y  F  in  B°},  we l e t  1 . . 8 "  y where t h e i n t e r s e c t i o n s Borel  .  sections  r u n over  closed For  (We w o u l d  under  any  a l l  a - f i n i t e measures  g e t t h e same t h i n g  probability  the Souslin  A c  r u n over  i f we j u s t  measures.)  o p e r a t i o n , and  These B  y  = B,  on  l e tthe inter-  a-fields y  +  y  B  area l l = B^  .  we l e t  o  is of and the  A  called .  T  If • are  completed  D  A  = inf{t  >_ 0: B  t  e A}  T  A  = inf{t  > 0: B  t  e A} .  t h e debut A  of  A  i s analytic  and  i s called  ( o r more p a r t i c u l a r l y ,  (B ) - s t o p p i n g t i m e s . a-fields.  T^  the h i t t i n g Borel)  This i s the reason  time  then  f o rintroducing  I f we d i d n o t do t h i s we w o u l d  have  D  , T A  A  - 51 -  being  (8^)-stopping times f o r  for  A  For  open,  each  t,  measurable,  y  and  8  =  but f o r and  A  each f i n i t e  Q:  {H £  .  L c H c M  It follows  that  u  measurable  y - a.a.  for  E % )  In  B  follows:  and  and  would  be  at a  loss.  ( B,8 )y  is  V  y,  then  measure  Q,  on  (<j>)  y  on  P (M\L) =  then  (j)  If  is  P  i s measurable  x  y  The  system  motion  5.5.  (in  8 |B ) t  t  (ti,B,B^,B^,B) n  is  with  (tj>)  i s u n i v e r s a l l y measurable.  B-measurable B o r e l E.^,  and  c a n be  stated  function any  t e  as  on  fi,  [0,°°)  = E (<f>) t  w i l l be  called  standard  Brownian  dimensions).  Consider a system  $  -  a E (<|) •  0}  P  .  t h e Markov p r o p e r t y  t >  with  and  x i—*• E  F o r any n o n - n e g a t i v e  a-finite  e °8  (8,8)-measurable.  x t — E  X  B  is  fc  y(dx)E ((j.)  =  B-measurable,  terms of  any  x  6  L,M  function  to the completion of  is  times  +  8  t h e map  f o r some  8 -measurable  $  y,  (B^ )-stopping  and  m e r e l y B o r e l , we  a non-negative  If  closed,  where  v = yP^  respect  A  (A,M,M ,X ,P) where: t  t  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 -  for  each  t e [0,°°],  for  each  co e A,  X^  i sidentically  X  i s an  t  M ~ m e a s u r a b l e map o f t  A  into  P  t 1—>• X ( t o )  A e Borel  M  which  and  [0,°°)  9  equal to  i s a measure on  if  i s c o n t i n u o u s on  t  i s a-finite  0 < s < t <  on  MQ  then  0 0  0  P(X  Such  e A|M ) = P  t  8  a system w i l l  be c a l l e d  (GBMP f o r s h o r t ) . that  t h e system  (A,M,M ,X ,P) t  (A,M,M  T + t +  ,X  T + t  ,P)  i s an  a GBMP.  (M  This  Q,  T  i sa  n B o r e l JR™  measure on  9  process  simply say  (BMP f o r s h o r t ) .  F C +  )-stopping  time  If then  i s known a s t h e s t r o n g  motion,  Markov p r o p e r t y i s : on  motion  we s h a l l  process  F o r standard Brownian  of the strong  a-finite  T  Brownian  P ( A ) = 1,  motion  and  i salso  B-measurable f u n c t i o n a  s  a generalized  i s a Brownian  Markov p r o p e r t y . version  (X ,A) .  I f i n addition  i s a GBMP  t  s ; t  a very  i f < f >  useful  i s a non-negative  (8 )-stopping t  time and  u  i s  then  B„  E(<j> . e |B ) = E(<f>) . y  T  T  (Note to  that  6  i s  T  the universal  T  (B,B)-measurable a n d  completion of  Borel  B^  i smeasurable  from  B  .) o  Suppose of  (A,M,M ,X ,P) t  t  i s a GBMP  .  Let  u  be t h e " d i s t r i b u t i o n "  X : Q  u(A)  = P(X  n  e A)  f o r A e Borel  .  - 53 -  Suppose  y  i s o-finite.  i s an element of ft . 4> .  l e t us c a l l i t P(* [H])  t  *• X (uj)  Thus we have a n a t u r a l map of  A  i n t o ft;  Also  y  (A,M,M ,X ,P)  t h e map  H e 8°,  F o r each  = P (H) .  _ 1  w e A,  F o r each  B  • IJJ = X  -1  .  and  Thus we may r e g a r d  (ft,B°,8°,B ,P ) .  as an "enlargement" o f  t  i^ [H] e M  t  y  t  In t h i s  manner we can prove t h i n g s about GBMP's by f i r s t p r o v i n g them about s t a n d a r d Brownian m o t i o n , where we can use t r a n s l a t i o n o p e r a t o r s and v a r i o u s o t h e r a s p e c t s of t h e s t r u c t u r e  of s t a n d a r d  Brownian m o t i o n which g i v e us more " h a n d l e s " f o r computations.  5.6.  Let u e C (R )  Theorem.  2  and suppose  n  two p a r t i a l s a r e bounded on E . . N  Rn  on  (M ) t  For  0 <_ t < «>,  i s a m a r t i n g a l e over  S k e t c h of P r o o f : that  E (M ) = t  and a l l i t s f i r s t  Let y  be a p r o b a b i l i t y measure t Au(B )ds . Then = u(B ) s t  (ft,8,B ,P ) y  t  .  U s i n g 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 udy  y  let M  u  fora l l t .  Now a p p l y t h e Markov p r o p e r t y . •  5.7.  C o r o l l a rryy::  ,,2^n L eett u ee (.C OR )  and l e t  y  be a p r o b a b i l i t y rt  n  .n  For  0 <_ t < °°, l e t M  = u(B ) t  measure on TR Suppose t h a t f o r a l l t e (0,°°),  we have  u(x) p ( x ) d y ( a ) d x <  00  t  E  TR  and rt 0 ^ n J E  v ( x ) p ( x ) d y ( a ) d x < °°, E  Au(B ) d s s  - 54 -  where is  v =  |u|, | |grad  a martingale  u||,  and  |Au | ,  respectively.  Then  (M ) fc  (fi,B,8 ,P ).  over  y  t  2 Sketch apply  of Proof: 5.6  Cut  u  by  C  functions with  t o t h e c u t f u n c t i o n s , and take  a  compact  supports,  limit.  •  5.8. M  t  Corollary:  = u(B )  -  t  and  y e  ft  Let  Au(B )ds  2  .  g  [1,2) such  C 0R )  u e  n  .  is  |u|, ||grad  a martingale  5.9.  satisfying  Let  E (||B  b)  E (T) i sfinite  -a|| ) 2  T  is  =  Part  |Au | ,  ,P ) ,  and l e t  T  (M ) t  a e E. . n  be a  (8 )-stopping  time  t  Then:  i f f tf = { | | B _  be over  f o r x e ]R  (B )-stopping t  Then  a  A l l martingales w i l l 2  n  respectively.  f o r any  a  t  e ]R ,  x  - a|| : 2  In this  0 < t < »}  c a s e , we h a v e  i n a).  a m a r t i n g a l e , b y 5.8.  bounded  and  for a l l  P -uniformly integrable.  ||x - a | | ,  [0,°°)  a,B e  < 2nE (T) .  a  equality  Proof:  n  3  is  u(x)  a eR  P ( T < °°) = 1 .  3  let  Y  I  X  3  u||,  a)  I I  <_ a e ' ' '  (f2,B,8  over  Theorem:  °°,  that  v(x) v =  0 <_ t <  Suppose t h e r e a r e numbers  II  where  For  Hence  time  a ) now f o l l o w s f r o m  n  S,  Fatou's  .  (Q,B,E^,P )  Then  E (||Bg a  a  Au = 2 n ,  . Let so  (u(B ) t  - a|| ) = 2nE (S) 2  a  - 2nt)  f o r any  by t h e o p t i o n a l sampling  theorem.  lemma.  i s uniformly  Moreover,  i f H  -  integrable we  find  E (T)  then  that  < «  a  E  .  -  S = T A t  taking a i i ( | | B^  Then  55  and  letting  i 12 a - a | | ) = 2nE  E (||B  - all )  a  t  go  ( T ) < °° .  < E (T)  2  r p  t  to  infinity,  Conversely,  for a l l  3  t,  suppose so  2 {| | B ^  - a||:  A  integrable. 5.8.  Now  Thus  sampling  0 <_ t < °°}  B  is  e a c h component - a = E (B  A  t  theorem.  But  T  then  of  (B  a|5  -  a  T  L -bounded  TAt  and h e n c e  - a)  t  )  i s a martingale,  for a l l  2  2  t  T  by J e n s e n ' s i n e q u a l i t y .  t,  by  the  by  optional  for a l l t,  l|B , -a|| <EM|B -a|| |B T  uniformly  Thus  H  T A t  i s uniformly  ),  integrable.  •  5.10. let  Corollary: T  be a  b)  E (T)  ( | | B  is  -  T  let  y  Then  be  5.10  <  satisfying  2nE (T) y  i f f  H  P (T y  on  H  and  n  < °°) = 1  .  Then:  .  = {| |B  integrable.  holds  even i f  Let  a Borel  y  subset  be  - B  || :  In t h i s  0 <_ t < » }  c a s e , we  y  Dynkin's formula:  y  does not have  finite  a p r o b a b i l i t y measure  of  E (T) < ^-(diameter(A)) — zn  5.12.  a p r o b a b i l i t y measure  have  in a).  Corollary: A  2  Q  P -uniformly  equality  5.11.  B || )  is finite  V  that  be  t  E  (Note  y  (B )-stopping time  a)  y  Let  ]R 2  Let  n  .  Let  T = inf{t  variance.)  on  ]R  ^ 0 :  and  n  B  t  i A}  .  D  be a n o p e n  subset  of  E.  N  and l e t  2 u  be bounded  bounded  in  D  and .  continuous Let  on  R = inf{t  .  D,  and  >_ 0:  B  fc  C I D},  in  D,  let  with a e D,  Au and  -  let  T  be  (B )-stopping  a  t  56  -  time such  that  T <_ R  and  E  (T) <  from  T  Then  (Note This We  that  i f  D  i s bounded  i s essentially  point  seriously overcome.  out  that  Theorem  then 2,  t h e p r o o f Rao  flawed f o r  a e 9D  .  E (T) d  paragraph gives  < » 1,  follows  c h a p t e r 4 o f Rao  i s correct  However,  this  for  a e D,  difficulty  can  <_R [1]. but be  .)  -  6.  57  -  PRELIMINARIES ON BROWNIAN MOTION AND POTENTIAL THEORY  I n t h i s s e c t i o n we s t a t e t h e r e s u l t s t h a t we s h a l l need on t h e c o n n e c t i o n s between Brownian m o t i o n and p o t e n t i a l t h e o r y .  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 g i v e c o n v e n i e n t r e f e r e n c e s f o r them.  F o r t h e g e n e r a l t h e o r y o f Markov  p r o c e s s e s and p o t e n t i a l t h e o r y , t h e r e a d e r may c o n s u l t t h e fundamental papers o f Hunt [ 1 ] , [ 2 ] , [ 3 ] , o r t h e books o f Meyer [2] o r B l u m e n t h a l and Getoor [ 1 ] .  I n these works, t h e p o t e n t i a l t h e o r y i s d e f i n e d i n  terms o f t h e p r o c e s s , and t h e c o n n e c t i o n between 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 t h e o r y i s mentioned, b u t n o t p r o v e d . F o r the  t h e o r y o f 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 t h e o r y , t h e  l e c t u r e n o t e s o f Rao [1] a r e e x c e l l e n t , b u t some r e s u l t s w h i c h we need f o r B o r e l s e t s a r e proved o n l y f o r compact s e t s . Throughout  this section,  R = i n f { t >_ 0:  B  6.1.  Let u  D .  Theorem: L e t S,T  D  i s an open s u b s e t o f ]R  n  and  I D} .  be  be a n o n - n e g a t i v e superharmonic f u n c t i o n i n  (B )-stopping times, with t  S <_ T .  Then f o r any  x e D, u(x) l E ( u ( B ) l X  s  { s < R }  Thus i f u ( x ) i s f i n i t e then  )  > E (u(B )l X  T  (u(B ) 1 , t  over  (ft,B,(8 ),P ) .  6.2.  Theorem:  ,)  { T < R }  ) .  i s a supermartingale  tt<K.r  x  t  Let u  be superharmonic i n D . L e t  ft^ = { CJ e ft: t *—> u ( B ( w ) ) t  and f i n i t e on  i s c o n t i n u o u s on (0,R(co))} .  [0,R(<JJ))  - 58 -  Then: a) b)  F o r any  (If  x e H^\D  all  x e  6.3.  B^;  e Souslin x e D,  then  {B  P (fi\n ) = 0 .  ( T h i s i s due t o Doob [ 1 ] . )  x  = x} c {R = 0}  so b) i s a c t u a l l y t r u e f o r  .)  Theorem:  Let  A  be an a n a l y t i c subset of l R  a n o n - n e g a t i v e superharmonic  function  b a l ( u , A n D, D ) ( x ) =  where  e B .  hence  T = T  in D .  E (u(B )l  be  Then f o r a l l x e D,  X  T  and l e t u  n  { T < R }  ),  . A  6.4.  Theorem:  T = T  .  A  Thus  A  be an a n a l y t i c subset of H , x e ]R , n  A  i s thin at  base(A) = {x e ]R : P ( T > 0) = 0} n  = {x e A: P ( T > 0) = 1} .  T  6.5.  Also,  X  Theorem:  Let  A  x  i f f P ( T > 0) = 1. X  and  X  i b a s e ( A ) , T < °°) = 0  X  and l e t  n  Then f o r each  fringe(A) P (B  Let  we  have  f o r a l l x e ]R . n  be an a n a l y t i c subset of ]R  n  and l e t T = T.. A  Then t h e f o l l o w i n g a r e e q u i v a l e n t : a)  A  b)  P ( T < °°) = 0  f o r a t l e a s t one  c)  P ( T < ») = 0  f o r a l l x e ]R .  Moreover, x e H , n  i s polar. X  X  i f n = 1 or  or  n  n  2  P ( T < ») = 1 X  x e ]R .  then e i t h e r for a l l  P (T < <») = 0  x e IR . n  for a l l  - 59 -  6.6.  Corollary:  Suppose  n = 1  or  2  .  Then  the f o l l o w i n g are  equivalent:  6.7. Then  a)  D  i s a Green  b)  P (R  < °°) = 1  c)  P (R  < °°)  X  X  Theorem:  region. f o r some  for a l l  Let  E  E  i s thin  at  b)  There i s a  G  e D  be a n y  the f o l l o w i n g are  a)  x  x e D,  or  D  i s empty.  .  subset  of  TR ,  and  n  let  x e H  and  P (T  equivalent:  x . set  H c ]R  with  n  E £ H  There i s a Borel  d)  T h e r e i s an a n a l y t i c s e t  Proposition:  ft  deduced and  e B°,  goes t o z e r o  6.9.  e  Y £ ]R  at  n  with  n  A £ R  ft:  >_ 3  l i m ] |B  P (ft ) = 1  E £ Y  and  with  n  1  D = ]R ,  .  E £  P (T X  A  y  >  0)  and  (to) | | = °°} .  using  n  Let  f o r any  X  with  that  D  {_  and  x e ]R  .  n  the strong  ( T h i s may  Markov  the p o t e n t i a l of a measure w i t h  be  property  compact  support  infinity.)  Theorem:  f unct ion of  Suppose  =  f r o m 6.3  the f a c t  set  > 0) = 1 .  A  A  ft  Then  U  H  c)  X  6.8.  > 0) =  X  o  P (T  .  n  Suppose .  Then  D  i s a Green r e g i o n ,  f o r any  x e D  and  and  any  f  G(x,y)dy A  =  P (B X  0  e A,  t < R)dt  G  i s the Green  A & Borel  D,  - 60 -  t_c__ h___o__f___ P_r__o_o__f_: _S_k__e__  I t ' s enough t o prove t h e theorem w i t h  1  replaced  CO  by  <f>, where  <j> i s a n o n - n e g a t i v e  support i n D .  L e t u = G<f> .  except f o r a p o l a r s e t o f n >_ 3  then  u ( x ) —»- 0  as  C  Then  f u n c t i o n w i t h compact  u ( x ) —>• 0  as  x —>• z e 3D,  z ' s , by 8.31 o f Helms [ 1 ] . A l s o , i f | |x| | —>- » .  Thus by 6.5, 6.6, and  6.8, u ( B ) —> 0 as t + R, t  f o r any  x e D .  Now approximate  D  P  X  - a.s.  by an i n c r e a s i n g sequence o f  r e l a t i v e l y compact open s u b s e t s and a p p l y Dynkin's f o r m u l a 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  the Green f u n c t i o n of 7.2.  If  y  then l e t  e A,  y  y  T  T < R) .  y^ .  "randomized"  Given  (see 9.2)  ( 8 ^ ) - s t o p p i n g time  d e f i n e d by y  i s thus o b t a i n e d by  T  y,  v  v = y^ .  satisfying  n  If  T  where o n l y [0,T]  i s a l l o w e d to be  y=<5,  Gv <_Gy  (B )-stopping  to be a genuine  D = H ,  T,  i t i s n a t u r a l to ask what measures  t  v .  v = y 6 + -j y  .  can be reached from  y  However, i f we  time then t h e r e i s a t  For example, i f we where  y  u n i t d i s t r i b u t i o n on the u n i t sphere c e n t r e d a t v  I D} .  fc  D f o r the whole i n t e r v a l of time  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 n = 3,  is  then Rost [1] has shown t h a t one g e t s  p r e c i s e l y the measures T  T is a  The measure  can be w r i t t e n i n the form  require  and  B  d i f f u s e under Brownian m o t i o n up to time  c o n t r i b u t e to  G  n  R = i n f { t >_ 0: D  Brownian paths which s t a y i n  v  i s a Green r e g i o n i n ]R ,  be the measure on B o r e l D  y (A) = P ( B letting  and  i s a measure i n  y^  T  D,  D  u s i n g a randomized  take  i s the u n i f o r m 0  i n ]R ,  then  n  s t o p p i n g time ( t h e  o b v i o u s one!) but not w i t h a genuine s t o p p i n g time as b) of the f o l l o w i n g r e s u l t shows. 7.3.  Proposition.  Let  a p o t e n t i a l , and l e t  y  be a measure i n  v = y^  where  T  D  i s some  such t h a t  Gy  (B ) - s t o p p i n g t i m e .  Then: a)  Gv <_ Gy  (so  v  is  i s f i n i t e on compact s u b s e t s of  D)  - 62 -  b)  F o r e v e r y 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 ( T = 0) = 1} X  (Note t h a t  E  i s u n i v e r s a l l y measurable.)  Proof: a)  Let  x e D .  Then v(dy)G(y,x)  Gv(x) =  = E (G(B_,x)l y  { T < R }  )  y(dz)E^(G(B_,x)l _ {  y(dz)G(z,x)  < R }  ) (by 6.1)  = Gy(x) b)  v(Z) - P ( B  e Z,  T < R)  = P ( B _ £ Z,  T = 0,  y  T  y  y(dx)P (B X  Q  e Z,  T < R)  (by 6.2)  T = 0)  y ( d x ) l ( x ) P * ( T = 0) z  = y(Z n E ) ,  since f o r every  x,  P x,(T = 0) = 0  or  1 •  S i m i l a r l y , we can show:  - 63 -  7.4.  Proposition.  a potential.  L e t S,T  Then  Gy  7.5.  Lemma.  g  be a measure i n D  be  (B )-stopping  such t h a t  times w i t h  t  Gy i s  S <_ T .  >_ Gy^  potential. Let  Let u  v = y  Let y  .  such t h a t  be an a n a l y t i c subset o f 3R  Let A T  be a measure i n D  n  Then f o r a l l x Gv(x) =  Gy  i sa  and l e t T = T^  i n D,  E (Gy(B )l X  T  { T < R }  )  Proof: E (Gy(B )l X  T  { T < R }  )  y(dy)E (G(y,B )l X  T  { T < R }  y ( d y ) b a l ( G ( y , - ) , A n D,D)(x)  (by 6.3)  y(dy)bal(G(x,«),A n D,D)(y)  (by 2.11)  y(dy)E (G(x,B )l y  T  =  )  E (G(x,B )l y  T  { T < R }  { T < R }  )  )  v ( d z ) G ( x , z ) = Gv(x) . • To p u t i t another way, 7.6.  Corollary.  potential. and l e t T =  y  Let y  Let U  T  = b a l ( y , A n D,D) . be a measure i n D  such t h a t  Gy  i sa  be a B o r e l ( o r j u s t c o a n a l y t i c ) subset o f .  Suppose  D\U  i s t h i n a t each p o i n t o f  D,  U .  -  (That  i s , suppose  U  i s finely  Proof:  A s we  is  a t each point  thin  64  have j u s t  -  open.)  observed,  of  U,  y  Then  y^,(U) = 0 .  = bal(y,D\U,D)  T  base(D\U,D)  £  D\U  .  .  As  Now  D\U  apply  3.4.  •  7.7.  Corollary:  potential, suppose  Gy  relatively there  closure  in  D  >_ v  of  By  > 0) = 0  compact  in  D  X  be  a superharmonic f u n c t i o n  .  Let  subset h  a measure  U  of  in  D  such  be a B o r e l  D,  and  let  that in  (or just  T =  Gy D,  such  that  Gy  >_ h  >_ v  .  on  U  .  is a and  coanalytic) Suppose  w h i c h i s h a r m o n i c i n some o p e n  .  - a.s.  and by  That  i s ,  Then  P  Also,  Gy  = Gy  T  (T < R)  on  = 1  x  e D  set  Then  t  Gy (x)  containing  Gy^  >_ v  since  U  (see 6.4).  i s relatively and  every  .  Hence,  >_ E ( h ( B ^ , ) )  .  But by D y n k i n ' s f o r m u l a  t  the harmonicity  of  E (h(B )) X  T  by  that  >_h(B (u>)) X  T  such  base(D\U,D)  f o r e v e r y • co e { B Q = x}  Gy(B (a)))  Thus  f o r any  T  .  [0,T(co)),  5.12,  be  G y ( x ) = Gy(x)  .  e U  P  D  U,  7.5,  x  £  in  compact  Suppose  t  v  y  .  Proof: X  let  i s a function  the  P (T  and  Let  6.2,  Gy(B^)  >_ h ( B > T  h,  = h(x) .  Thus Gy (x) T  We  h a v e now  shown t h a t  fringe(D\U)  i s polar.  >_ v ( x ) .  Gy^, >_ v Hence  on Gy^  D\fringe(D\U) . >^ v  throughout  B u t by  3.11,  D .  •  -  7.8.  Lemma:  potential.  Let Let  y  be a measure  N = {v: v  Then f o r any compact  Let  relatively  K  y' = bal(y,W,D) v = bal(l,V,D) R i e s z measure  subsets  .  Then  .  Then  of  v  D  such in  that  D  Gy  and  i sa  Gv <_ Gy}  s e t K c_ D,  be a compact  compact  in  i s a measure  sup{v(K):  Proof:  65 -  .  v  0 0  of  D  such  lives  on  .  D .  =  X  W,  lives  Let  that  V  be open  .  . Let  y'(D) < »  i n D  on  V,W  K £ V c w so  i sa potential  Then  v(K)  subset  of  y'  v e N} <  .  Let  . Let X  be t h e  Now f o r a n y  v e N,  GXdv  ldv = K GXdv =  GvdX  GydX =  Gy'dX  GXdy' < y ' ( D )  7.9.  Lemma:  potential. suppose  Let Let  y  be a measure  Q .  p o i n t w i s e on  a)  T  (B^.)-stopping  b)  F o r any B o r e l f u n c t i o n j<j>|Gy <  D  (T_^) be a s e q u e n c e o f  T_^ —y T  i sa  in  0 0  ,  we h a v e  D,  we h a v e  (B^)-stopping  Gy  i sa times and  Then:  H  t(>: D N>Gy_  such  that  <f>Gy_ .  j <(>  F o r any continuous  that  time  •' c)  such  function  with  compact  support i n  - 66 -  d>dvT. 1  Proof:  a)  i s a standard  from  the r i g h t - c o n t i n u i t y  For  t e  t  It suffices CA)  and f o l l o w s  (B ) .  of b)  result,  to consider  < J > >_  0 .  [0,°°]  and  e ft, l e t fR(o»)  <j>(B_(_))ds  Z (o») = t  t A R ( _ )  It  follows  (*)  J*Gy  6.9  = E (Z )  and  c o n t i n u o u s on -  dominated  w e {Z  <j)Gp <  < »},  Q  0 0  From b) i t f o l l o w s in  D  S,  Also  Z  fc  (to) i s f i n i t e  <_ Z^ from  and the Lebesgue  theorem.  that  i f  <j>  i s  C  w i t h compact  support  then  'S'dPn  By  .  Thus b) f o l l o w s  convergence  ) - s t o p p i n g time  t —* Z  t h e map  [0,°°] .  (B  f o r any  .  s  for  0  that  y  s  Now  E"(Z )  c)  from  7.7,  s u p y_ (K) < i  assertion  4>dy  n  0 0  f o r any compact  set  K _£ D  .  The  i  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 then  above  proof  a more d i r e c t  seems a l i t t l e  proof  roundabout.  c a n be g i v e n ,  using  6.6  If and  y  i s finite  6.8.  - 67 -  7.10. the  Lemma: form  harmonic  ft  .  u  be a m e a s u r e  U = V n { v < c } , in  D,  sequence of on  Let  and  c  where  Suppose  also  V  i s a real  (B^)-stopping  in  u_  .  Consider  i s open  number  t i m e s and  that  D  .  in  D,  v  U  of  i s super-  Let  (T )  be  1\  —*• T  pointwise  suppose  (U) = 0  a set  for a l l i  .  a  Then  i P-(U)  = 0 .  Proof: is  Suppose  relatively  u_(U)  compact  u-(U')  4 0,  finite  continuous  f  outside  = 0  negative on x  where  e D  .  Then  V  and a r e a l  function  in  D  function  g = 0 0  on  U,  V  on  (Let  and  6.8  f o r a l l i , E (<t>(B_ ) l X  of  D  is  t  .  Thus  on  P  On  the other  hand  {T >_ R},  on  P  0  n  r  V  be a  such  and non-  that  g =  1  <f>(x) = f ( x ) g ( v ( x ) ) f o r  X  =  -  „i)  (B  P  -  X  ):L  outside  any  {T<R} •  - a.s.  *  ( B  .  x e A  - a.s.  * T  = 0}  Let  .  ultimately  {T < R},  ^T {T.<R}  00  non-negative  on  g  that  Then  (A  is  {T.<R>  u(D\A) = 0  - a.s.  •<V  , ]  such  ) - 0 .  m e a s u r a b l e and)  (** )  be a  which  l  T. universally  f  Let  00  V'  c' < c  f = 1  .  set  so f o r a l l i ,  H l  {x e D:  Let  that  of  [c,°°] .  outside  y  A =  number  such  continuous  cf> =  i s an open  U ' = V ' n { v < c ' } .  E (*(B_  Let  there  subset  and  Now  in  .  some c o m p a c t  finite  (-°°.c']  4 0  T  ) : L  {T<R}  .  By  compact  6.6 subset  -  since  t  *- ^ ( B ^ )  is  P  68  - a.s.  A  -  continuous  on  [0,R)  by  6.2.  Thus  •  P  - a.s.  on  it  follows  that  ( B  T.  ) : L  1  .  By  {T.<R} l  ^ V ^ R }  the Lebesgue dominated convergence theorem,  E (<KB )1  )  X  T  As  this  i s true  { T < R }  f o r a l l x e A,  E (*(B H  and as  y  T  Now it  <j> = 1 must  be  on true  U'  .  that  { T < R }  Hence u  u  - 0 .  u  lives  on  A,  ) = 0 .  (U') = 0  .  As  this  i s a contradiction,  (U) = 0 .  • Remark.  'The  harmonic D  .  in  Also,  collection by  only  and  U  finely  At also  D,  of the form c e H)  [1].)  set.  Thus, using  l a s t we  We  from  (V  open £  a base f o r the f i n e  D,  v  super-  topology of  open s e t s  has a c o u n t a b l e  the union  of the whole  sub-  collection  ( S e e r e m a r k f o l l o w i n g V.1.17 o f B l u m e n t h a l  i n the statement open  n {v < c}  of f i n e l y  whose u n i o n d i f f e r s set.  V  constitute  any c o l l e c t i o n  a polar  Getoor  that  sets  shall  can prove  the " f i n i t e n e s s p a r t "  o f 6.2,  o f t h e a b o v e lemma c a n b e r e p l a c e d not need  this,  though.  the main r e s u l t  of this  one o f t h e m a i n r e s u l t s o f t h i s  thesis.  we  see  by any  s e c t i o n , which i s  Borel  - 69 -  7.11.  Theorem:  potential.  Let  Let  y(Z) <_ v(Z)  v  u  be a measure i n  f o r every B o r e l s e t  a  (B )-stopping  T  i s not randomized.)  that T  time  t  Proof:  Let Gy  S  P  y  T  such t h a t  on  i f f T e S  v = y_ .  t  S,  isa  Gv <_ Gy  and  Then t h e r e i s  times  S  (B)-stopping  that  such time  T <_ S e S  and whenever  {T < R} .  Gy  (Note w e l l  (B )-stopping  then  ( F o r the " o n l y i f " p a r t , use the f a  t h a t f o r any non-negative B o r e l f u n c t i o n s t o p p i n g time  such t h a t  such t h a t  One can then show t h a t a  y_ = v  - a.s.  D  D  Z £ {Gv = °°} .  be the s e t of a l l  >_ Gv .  satisfies  T = S  be a measure i n  f  in  D  and any  (8 ) -  we have fR  (Gy )f = E ( P  s  f(B )dt)  SAR  T h i s f o l l o w s from 6.9 and the s t r o n g Markov p r o p e r t y . ) One can a l s o show that elements.  S  does c o n t a i n such "maximal"  What i s more, t h i s approach i s r e a s o n a b l y c o n s t r u c t i v e  s i n c e we don't need Zorn's lemma to produce a "maximal" p r i n c i p l e of dependent  T;  the  choice s u f f i c e s .  N e v e r t h e l e s s we s h a l l d e s c r i b e another way of p r o d u c i n g a suitable  T .  We p r e f e r 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 t h a t i t i s n o t q u i t e as s l i c k as the method outlined Let  above. 1/  be a c o u n t a b l e open base f o r D  r e l a t i v e l y compact s u b s e t s o f on  D  D .  Let G  consisting of  be the weakest t o p o l o g y  which i s s t r o n g e r than the u s u a l t o p o l o g y of  D  and which  -  makes D  Gv  continuous.  the do  Then  topology  U  G .  Let U  occurs  H  = inf{t  ^  be t h e c o l l e c t i o n V e V  where  i s countable. (This  not e x p l i c i t l y  perspective  U  V n {Gv < c}  o f t h e form  rational.  Let  70 -  Also,  assertion  and  U  -  ^ i > i  ^  infinitely > 0: B  t  let  e  sequence  a  in  many t i m e s .  I V}  .  ±  Let  U  that  i n which each i ,  = 0,  and f o r  T  Q  sets  .  less  we  We  Then  = inf{t  shall  than  b y 7.9.  show t h a t  u .  Hence  i >. 1 l e t  . in  stopping  time  c  i  > T. : B „ I U.} i - i t l  .)  1  sequence o f s t o p p i n g  T e S  element of  otherwise,  T. , + H. • 6„, i-i l T. i-i  .  . is  T. = I  i s an i n c r e a s i n g  T  indeed  let  i f this  T  1  X = v  positive  but i t gives a  F o r each  + IL • 6  T. , i - i  T = l i m T. i-x»  below;  G,  i-1  (T^)  is a  on t h e p r o o f . )  T._^  (Note  c  i s an open base f o r  i s n o t used  make u s e o f t h e t o p o l o g y  of subsets of  That  X = v .  times  i s ,  in  GX >_ Gv,  By 7 . 3 ( b ) ,  X ( Z ) <_ v ( Z )  S  X  f o r every  Then  . Let where  charges  polar  Borel set  Z c {Gv = <*>} . Let  A = {GX > Gv}  Then t h e r e i s a p o s i t i v e  .  We  rational  X({GX  > c} n {c > Gv}) > 0 .  hence  i s a countable union  V  e V  such  claim  X(A) = 0 . c  But  such  {GX > c}  of elements  of  Suppose  that i s open 1/ .  on  V  and  in  D,  Thus t h e r e  that  GX > c  X(A) > 0 .  X(U) > 0,  and exists  where  U = V  n {c  > Gv}  .  Let  I -  { i _> 1:  U  = U}  .  Then  I  is  infinite. Before proceeding (B  )-stopping  (u ) x  =  Y  U X  +  Y  .  times •  Q  f u r t h e r , l e t us  then  Now  the  if  strong  i e I  note  = U  so  by  = T. . + l - l  7.7,  But  H. l  t h e n by  G(y  • 6_ T. l-l  then  Gy_  )  T  > Gv  > c  by  7.10,  Hence GA  2.2, GA  = Gv  u  A(Z)  <_ Gv,  .  .  in  D  are  implies > Gv  that  on  H e n c e by  From  (U)  = 0  .  7.6,  That  must h a v e <_ v ( Z )  by  the  this  ;  hence  1 y_ (U) T. l  n  a c o n t r a d i c t i o n , we and,  X,Y  i-1  1-1 T. l  if  Markov p r o p e r t y  X U.  that  A (A)  is, =  0  A(U) .  f o r every Borel  that  =  0  for  .  this  <_ Gv  A -  GA  polar  subset  A = v  4.2.  i e I  As  Thus  domination p r i n c i p l e  i t follows  = 0  Z  .  is a.e.,  of  D  .  Therefore  .  • As  we  shall  possible  result  possible  result,  7.12. is  see of  a potential.  (Note  There i s a  b)  Gy  Now best  I am  Let  >_ Gv,  following  and  y(ZnA)  =>  b)  holds  there  are  time  by  consider  the  D,  the  best  i s the  present  and  best time.  suppose  Gy  equivalent:  T  such  set  A £  polar  sets  i s a Borel  for a l l Borel  following  i t at  measures i n  f c  =  possible.  be  the  to prove  (B )-stopping  v(Z)  l e t us  above theorem i s not  I believe  unable  y,v  Then the  a)  a)  i t s type.  but  Conjecture:  s h o r t l y , the  that D  y_  such Z £  D  = v  .  that .  7.3.) an  example  that  shows t h a t  7.11  is  not  - 72 -  7.13.  Example:  Suppose  D = ]R  where  n  n = 3 .  (We make these  assumptions to keep the computations from g e t t i n g too messy.) Let  u = 6 .  d e c r e a s i n g to  Let 0,  ( j)  ^  r  e  a  s e c  l e n c e of p o s i t i v e r e a l  such t h a t t h e r e e x i s t r. m < —^— — r  numbers  u  < M —  m,M  for a l l  e (l, )  with  0 0  j .  3 + 1  Let  IIxlI  S. = {x e D: 3  1  J  = r.} 3  1  < r.} 3  r.,_ < l l x l l  A. = { x e D : 3  (s.)  1  IIxlI  H. = {x e D:  Let  1  be a sequence i n  1 1  3 + 1  [O, ) 00  with  a s s i g n s mass  s^.  up too f a s t around  to  S^. 0,  f o r each  D j .  Suppose  i n the sense t h a t  (Bj.)-stopping time  Proof: to  Gy  First .  T  such t h a t  l e t us show t h a t  For each  Gv  .  and l e t v  which l i v e s on v  u  ^—  Then  llxll  for  < r,  3  Gv(x) = < — r r  x  (It follows that  Gv <_ Gy  .)  Then t h e r e  = v .  3  4iTr.  for  Thus f o r  and  J  grows v e r y s l o w l y a t  i , l e t v. = v„ . 3 S.  s.  u S_.  be  does not p i l e  £js. < °° . 3  is a  <r.} j  £s. £ 1, j 3  3  the s p h e r i c a l l y symmetric measure on  -  1 1  1 1  x  1  x e A , fc  -  > r. 3  0  compared  -  k Gv(x)  =  4lT  73  s.  "3=0  l*lT  j  r  j=k+l  ^  so Gv(x)  v  =  Gy(x)  j=0  I|x|  . J  j  r.  r,  0 0  < I Let  us c a l l  the l a t t e r  Gv we  1  a^ .  oo  oo  r  k=0  j=0  on  satisfies shall  and each  there  v  co  2  j=0  J  j=0  J  assume  that  CO  k=j  1  ^  \  r  j J  £=0  l e t u s n o t make s u c h  L e t us assume,  v  r  j  of the proof, .  Then  A  k=0 /-  the p r o p e r t i e s  we  rather,  ( i.) a  that  special  v  i s such  have j u s t v e r i f i e d .  i s a measure  are positive reals  in  such  D that  Explicitly,  such that £ a, k  that  <  0 0  Gv  <_ Gy  and f o r  k,  Gv  Now  r.  quantity  the remainder  assumptions  j=k+l  s.  CO  For  A  j  .Jr..  ' j  f o r any  relative  to  < a. G k y  E _c D, D  .  let  in  C(E)  (See Helms  denote  the outer capacity  [ 1 ] , c h a p t e r 7.)  of  E  For each j , l e t  - 74  be a c e r t a i n c o m p a c t specified E. 1  .  subset"of X^  s h o r t l y , and l e t  Also  Gy = - , — — 4irr.  -  on  VV  which w i l l  = bal(y,E^.,D)  E., i  3  S..  .  be more  Now  A  explicitly  lives  on  so  =  Thus f o r  x 6  4-r.  H., J > VJA j. v,A7 ' r\  ( ) v  v  A (E ) J J /(ATr)(2r / . _ \ <")_ -\ ) C(E  C(E ) 3 /-i/-c C(S  ___ \) 8TT_ o_  )  C(^T ^ r T ^ N x l  =  C(E  |Gy(x)  )||x||  2C(S  =  Thus f o r  C(S.) ' J  =  J  )r  G  y  (  x  (  x  )  '  )  x e A.,  3  C(E.) G  Choose  k  so t h a t  j  (  x  ^2C(S*)S  )  a. < ^ —  G  y  •  for a l l i > k  .  Now  J - 2M C(E^)/C(S_.) i s a n y number b e t w e e n  be c h o s e n s o Thus f o r  A  i > k  -  we  can choose  E.  so  3  the  E.'s  can  j 0  and  .  Then  1 .  that  C(E.) 2C(S.)M  Let  E =  (D\KL ) u ( u  E.)  j__k on GA  DXH^., > Gv —  .  and Now  for  .  = a. " " j  Let  A = bal(y,E,D)  j  k,  A = y , U  GA  where  T T  •> GA_.  L  (  S  j  ;  >_ a^.Gy  U = T_, L  C(E.) j_k  GA  =  Gy  2  j>k  2  >_ Gv  b y 7.5.  on Also  A  .  Hence  -  so  E  this  i s thin  at  0  .  Ujj({0})  it,  because  polar such  = 0  .  (Once  {0}  sets.  test  7.11,  (8  Thus  leaves  0  is a  P°(U  [1]  > 0) =  i t c a n n o t come b a c k  Thus n e i t h e r  V v  v = p^,  )-stopping  by  time  =  v  p^  nor  v  ~ 1,  to  charges  (8 )-stopping time  V  property,  i s the  t  '  the s t r o n g Markov  U + V  • 9  t  where  T  .  Q  u  The p r o o f act".  We  have  bal(Gp,E,D)  p,  f o r thinness.)  o f Helms  that  then  then  10.21  there  (  But  of theorem  (B^)  i s polar.)  Thus, by  -  (See p r o o f  i s the Wiener-Brelot  so  75  E  to take  >_ Gv  would  so t h a t  o f t h e above  .  On  n o t be  7.11  would  the  example E  j '  the other thin not  at  0,  apply.  s  i s a rather delicate  b i g enough  hand, and  i f we  so  "balancing  that  took  bal(p,E,D)  them  too b i g  would  just  be  - 76 -  8.  EMBEDDING MEASURES IN  BROWNIAN MOTION IN  IT  OR  ]R_,  USING  NON-RANDOMIZED STOPPING TIMES.  8.1.  If  y  i s a m e a s u r e on  then,  f o r the purposes  on  B o r e l ]R  the  condition  of  this  d e f i n e d by  n  "T  < °°"  H  and  n  T  is a  section,  y^,(A) = P ( B y  T  y^  will  e A,  i s s u p e r f l u o u s ; we  (B^)-stopping  T  time  denote the  < °°)  .  (Of  measure  course  include i t just  for  emphasis.) As  i n the p r e v i o u s  measures Of  v  can  course  considered  s e c t i o n , we  be w r i t t e n i n t h e  i f  n  >_ 3,  this  i n the p r e v i o u s  consider form  is a  v =  special  section.  the  question:  y^  What  ?"  case  However, i f  of  the  n =  1  question or  2  then  n -K  i s not  under  the If  Rost  a G r e e n r e g i o n , so  [2]  one T  of  the  previous  i s allowed has  t o be  e x p r e s s i b l e i n the  of  a-excessive functions. Skorohod x  variance stopping course  e 1R and  [1] and  form  showed i f  v  mean e q u a l  randomized  T  x,  Brownian motion  formula  Note  t h a t by  c e n t r e o f mass e q u a l  y,v  ;  Rost's  case  then  criterion  n =  to  < °°  finite)  1)  U  to  i f  y =  H  with  i s a randomized  &^  for finite  (8  )-  v = u . (Then o f 1 o f - j a r i s e s b e c a u s e we  the  V  v  i s i n terms  c l e a n e s t form  a p r o b a b i l i t y measure  i f f  then  and  factor  to get  4.10, x  there  are  condition for  i s a p r o b a b i l i t y m e a s u r e on  time T such t h a t E (T) x 1 E (T) = y v a r i a n c e ( v ) ; t h e  5.12.)  (and  sufficient  ( i n the  have n o r m a l i z e d  has  and  v = y  that  to  subsumed  section.  given a necessary  be  some  the above q u e s t i o n i s then not  i s a potential  f o r Dynkin's v and  on U  V  H <_  U  y  -  where  y =  .  father  of a l l these  Dubins produce  v  .  then  of  8.2.  Definition. H  noted  that  then  that w i l l  T  be  that  U  R  V  and  D i s c u s s i o n o f 8.2.  U  y,v  result  i s the  be  a r e any  time  " a l l o w e d " , as  f c  and  Suppose  R,S  S  and  We  t  If  n  are p o t e n t i a l s  and  >_ 3  T  t h e n any  than  .  T  P  T =  T  or equal  to  U  y,  by  Hence i n d i m e n s i o n  one  than  or equal  n = 1) is  y  inf{t  <_ 2  to that  of  is y  - almost  surely  finite.  > 0:  =  and  B  t  1},  y-standard. .  Hence  4.8 or  (if two,  .  the  R U  is  y  say  be that  times  and  S > U y  y-standard,  Then  by  T  U  is a  potential  y_  has  t o t a l mass g r e a t e r  n =  2)  or  i f  Suppose  v = y_  that  let  shall  (B )-stopping  are  probability  follows.  y  and  i s less  n  which  such  y  7.4.  grand-  to r e s t r i c t  time,  i s a potential.  y  T  l e a d s us  (B )-stopping  a U  i f  (B^)-stopping  i f f whenever y  R < S < T  Skorohod's  .  there i s a  such  n  of  results.  [3]) has  Let  y-standard  8.3.  result  [ l j gave p r o o f s of Skorohod's r e s u l t T's  stopping times  a m e a s u r e on is  -  T h i s somewhat d i s q u i e t i n g  class  T  Root  (see Meyer H  this  embedding  [ 1 ] and  m e a s u r e s on =  course,  non-randomized  Doob  y_  Of  77  T  is  (if  y-standard  n = 1,  Then  4.11  y =  v =  &  .  1  then  6^, Thus  T  is  2 not y and  y-standard.  Now  suppose  i s the uniform u n i t v = y_  .  Then  not  a polar set.  on  S^  .  Now  T  n=2,  distribution is  Clearly  P  y  then,  S on  - a.s. v  = S-,  finite  {x  e H  :  ||x||=r},  T = inf{t by  6.5,  > 0:  since  i s the uniform u n i t  B S^  t  e  S^},  is  distribution  -  2TT  irV) =  78  -  log  2  2TT l o g  f o r ]Ix  I  I|x  <  for  2  x  > 2  while  U (x) V  Thus  T  i s not  8.4.  Lemma.  potentials.  =  |  | x | | <_ 1  0  for  1 2TT  log I|x||  f o r | |x j | >  1  u-standard.  Let  y,v  be m e a s u r e s on  H  such  n  that  U , y  U  are  V  Then  $  (x,y)dy(x)dv(y)  < °°  Hence  U dv  and  y  both  make s e n s e ,  Proof:  If  y  v  and  n ^  IyI  then  by by  V  are equal.  have n o t h i n g If  n = 1  (yOR)$a,y) -  A l s o , they  to prove. then  by  are not equal  If  n = 1  or  1.6,  to  2  -°°  then  4.10,  $(x,y)dy(x))  =  0,  _Hxj  is finite;  also  $ = -$  .  i f  4.3,  Ixl by  we  $(£,y)dv(y)  1.5,  lim  and  3,  are f i n i t e .  lim  and  and t h e y  U dy  2 (yOR ) $ ( y ) -  (y)dv(y)  $  ( x , y ) d y ( x ) ) = 0,  i s finite.  Also,  f o r any  n,  n = 2  - 79 -  $  y  (x,y)dy(x)  i s continuous,  and so i s bounded  on c o m p a c t  sets.  •  8.5.  Theorem:  Let  y  be a measure  such  that  U  i s a potential.  such  that  E ( T ) < °° .  y  y  y  First  function and  on  let E  n  ,  Au = - f .  T  f c  any n o n - n e g a t i v e  Proof:  Let  f(B )dt) =  E (  for  Then  Borel  f  T  (M )  f  .  i s a m a r t i n g a l e over  t  But  u + c*  on  4.10  for  is  n = 1 .)  a martingale  continuous where  g  i s bounded  x,y  E ($  (B )) =  is  over  (infact,  all  y  Thus  compactly  Then  u  supported  is  f (B ) d s ,  C°°,  0  for  in  dy  P -uniformly y  where  fc  y  continuous) on  < «>,  E  ;  Now  )ds < s —  b = sup f  .  f(B  n =  Hence  (M )  follows  Thus  and  {M_ : At  d  f  has  2  and  (M )  right-  u ( x ) = g ( x ) - c$ ( x ) ,  that  the fact  that  { u ( B _ ) : 0 < t < °°} TAt —  ( H e r e we u s e t h e f a c t  )ds  a  By  that  E ( T ) < °° .) y  fT f(B  r  $ ( x ) <_ | |x - y | | + $ ( y ) f o r  together with  i t then  g  a e IT  for a l l t .  Of c o u r s e ,  paths.  also  5.10,  integrable.  TAt Also  y  u = U  <_ t < » .  4.3 f o r  (See  E (|M |) < »  t  From  Q  c =  C  grad  f o r any  t  (ft,8,8 ,P ) .  time  E  (ft,8,8 ,P )  i s b o u n d e d , where  2)  T  0 5.8,  or  (8^)-stopping  - U ) f  y  +  t  be a  n = 1  (U  f  = u(B >  (where  n  y - s t a n d a r d , and  be a n o n - n e g a t i v e  M  E  i s  function  and l e t u = U Let  on  E ( M  f(B  0 <_ t < °°}  )ds) < bE (T) < o ° , y  is  P -uniformly y  ,  - 80 -  integrable. theorem.  Hence  E^CHj.) = E ( M ) , Q  That i s , U du  -  f  In p a r t i c u l a r , y  by the o p t i o n a l sampling  y  T  U dy^,  f(B  E ( y  is finite.  r  U dy .  )ds) =  f  Hence (as  f  can be non-zero)  T  U  i s a potential.  Now  i s j u s t i f i e d by 8.4) rT  E ( y  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  we o b t a i n f ( B )ds) =  y T (U - U )f . y  y  0  As t h i s h o l d s f o r a l l compactly supported n o n - n e g a t i v e f  on  R ,  we f i n d t h a t  n  U  >_ U ^,  y  C  functions  and then a monotone c l a s s  argument shows t h a t the f o r m u l a h o l d s f o r a l l n o n - n e g a t i v e functions Now  f  on  H  s t o p p i n g time which i s P -  <_ T,  (B  ( 8 ) - s t o p p i n g times and  R <_ S <_ T  t  U R , y  )-  s i n c e such a s t o p p i n g time w i l l  e x p e c t a t i o n . I t t h e n f o l l o w s t h a t i f R,S  y  Borel  .  n  c l e a r l y we can a p p l y the above argument t o any  have f i n i t e  (which  U S y  also are  then  are p o t e n t i a l s ,  and  U That i s , T  8.6. y  is  R  > u  s  y-standard.  Terminology.  •  A measure  has compact s u p p o r t and  U  y Y  on  ]R  w i l l be c a l l e d good i f f  i s f i n i t e and c o n t i n u o u s .  Note t h a t  -  if  <j)  y(dx) with  i s a compactly = <j>(x)dx,  a non-zero  8.7. and  good  y  i s good.  y  be a measure  potential  fora l l  supported  continuous functions  i e I,  U  dy  1  function  on  H  n  and  b y 4.6 o f t h i s w o r k t o g e t h e r  non-polar  on  be a n e t o f measures  T  U^  Also,  [ 1 ] , any compact  Let  (y.). l l e i  a)  Borel  subset of  ]R  carries  n  measure.  Theorem: let  s u p p o r t e d bounded  then  6.21 o f H e l m s  81 -  Now  suppose a l s o  that  for  some n o n - z e r o  y  on  n  H  H  v  ( U on  that  '  is a  f o r a l l compactly  y  converges  such  n  U  2,  Then  dv) 3R  or y  such  <j>dy  on  n = 1  f o r a l l good measures  the net  measure  <f>  where  n  cfidy.  and  dy  ]R ,  that  U  on  ]R  n  to a f i n i t e V  is a  limit  potential.  Then  b)  U  is a  c)  The n e t  (U  U  where  y  potential. u  + C,  U  d) (Remark:  C  )  U  dy  1  i  y  converges C  u n i f o r m l y on c o m p a c t  i s some f i n i t e  - C dy  non-negative  f o r a l l good m e a s u r e s  n e e d n o t b e z e r o ; s e e 8.15 f o r a n a t u r a l  n Proof: as  n <_ 2,  from is sup ^ 0  If  this,  y  i s a good m e a s u r e  i s also upon  compactly  <  H  supported.  of the theorem. Hence  constant,  y  on  example  ]R  n  .  of t h i s . )  y then  U  Part  +  i s continuous and,  a) f o l l o w s  i n t e r c h a n g i n g orders of i n t e g r a t i o n .  as i n the statement U_ dy  on  sets to  Then  f o r some  lim(sup y.({x e H : n  Now  immediately suppose  1^,  ||x|| > ^ r } ) ) = 0  v  -  But  t h e n , s i n c e we  finite on  limit  ]R ,  also  f o r each  know t h a t  (j<|>dy^)  the net  continuous compactly  there exists  n  82 -  i ^ >_ i ^  sup  (*)  such  y . (R )  <  n  converges  to a  supported function  $  that  0 0  ^ 1  Thus  u0R )  < °°  n  functions  f  and  on  H  Next,  f o r a l l bounded  i t i s easy  t o show  continuous  that  u.  (**)  U  Thus by F a t o u ' s  < lim infU  y  U dv  Hence  infinite. finite R,  U  n  <  y  differs  x  i n ]R  and  from  that  .  (*),  X  dv  v 4 0,  U  i s not i d e n t i c a l l y  y  o r 1.6  ( i f n = 2),  i s a potential.  y  vOR )$ n  (x,')  Also,  by a bounded  (U_ ( x ) ) c o n v e r g e s  - U^(y)|  x,y  U_  i s  x  in  f o r any  continuous  to a f i n i t e  function.  limit for  i n ]R , n  < y.0R )||x - y| I . n  i  y  Thus by  U  But f o r any  |u\x)  U  ( i f n = 1)  the net n  <_ l i m i  Therefore, as  H e n c e , b y 1.5  follows  each  .  0 0  everywhere,  V  1  lemma,  U_dv  It  fdy  | f dy,^  {U_  : i >_ i ^ }  i s equicontinuous (indeed, uniformly  u. Lipschitz) of  R  H  .  n  n  . Let  so i n f a c t Let  (-°°, ]-valued 00  u  u = U on  (U  )  converges  be i t s l i m i t , y  - u_ H  n  .  Then Also  which u  U  1  u n i f o r m l y on compact i s finite  subsets  and c o n t i n u o u s on  i s l o w e r - s e i n i c o n t i n u o u s and dy  udy  f o r a l l good  measures  -  Y  1R  on  In p a r t i c u l a r ,  83 -  this  convergence holds  for  y  of the  CO  form on  y ( d x ) = <j>(x)dx,  ]R  .  n  Thus  integrable, there  y = -Au .  i s a harmonic  x,  - h  Y  .  u(x)  Now b y (**) , function Helms of  U  on  H  h  over  = v(x)  [ 1 ] ; true  H  i s locally  Thus  f o r any  n  checks  n  .)  u = v  Hence  almost  where  n  6,  x e E.  centred a t  .  n  But a non-negative  (This i s Picard's Thus  H  that a c t u a l l y  h >_ 0 .  .)  on  ]R ,  of radius  supported  Lebesgue  such that  n  a ball  fora l l  i s constant.  n  and c o m p a c t l y  u  on  one e a s i l y  <_ u _ .  Y  C  t o Lebesgue measure on  By a v e r a g i n g 6-1-0,  is  a Schwartz d i s t r i b u t i o n  function  respect  and l e t t i n g  <j>  (Clearly  and so d e f i n e s  everywhere w i t h v = U  where  C  i s just  harmonic  theorem - 1 . 1 1 o f the constant  value  h .  • 8.8. such C  Lemma. that  Let  U , U Y  e [0,°°) .  y,a  are potentials.  0  Suppose a l s o  Proof:  If  Let  be t h e u n i f o r m  y  centred  be m e a s u r e s on  n = 1,  at  0  this  Y  Suppose  *  denotes  m = (u*y)0R ) = 2  U  yQR ) = aOR ) n  n  from  unit distribution  Y  n = 1  - C >_ U , A  .  Then  4.10.  or  2,  where  C = 0 .  Suppose  n = 2 .  on t h e sphere o f r a d i u s  1  Then  - C =  Y  (U  Y  - C)*y  > U°*Y  where  where  n  follows easily  2 i n TR. .  U *  that  H ,  convolution.  (a*y)0R ) 2  .  = U * A  ,  2 m = y(R ) .  Let  Also  Y  U * Y  Y  = U *y Y  Then  = -$~*u = -U  Y  and  - 84 -  U°*  Y  = -u°  .  T h u s b y 4.3,  |U  P  Y  as  ||x|  ( x ) - m*(x) |  0,  and  |u°* (x) - m * ( x ) |  0  Y  From  this  i t follows  that  C  must  be  0 .  •  8.9. n  Lemma:  = 1  or  Suppose  2,  such  M  i s a family  on  E. ,  where  n  that  u e M,  a)  For each  b)  sup{pOR ): u e M} < °° .  c)  inf{U (x):  U  is a  y  potential.  n  y e M ,  y  set  K c n  x e K}  > -»  f o r some c o m p a c t  non-polar  .  n  l i m sup y ( { x eE. :  Then  of measures  n  | |x| | >_r})  = 0 .  r-x» yeM Proof: that Then  By 4.6, U  Y  there  i s bounded  i s a n o n - z e r o measure a b o v e on  ]R  .  n  Let  y  carried  by  L = inf{U (x): y  K  yeM,  such x e K}  y e M ,  f o r any  U dy Y  =  U dy  > Ly(K)  M  Y  Then  from b ) , the upper boundedness  U (x) —• - c o Y  as  | j x | | —> lim  °°,  sup y ( { x  r-x» yeM  of  U  i t follows  eE. : n  ,  and  the f a c t  that  that  ||x|| >_r})  =  0 . Q  -  8.10.  Lemma:  collection a  Let  of  u  85  -  be a f i n i t e  (B^)-stopping  ( B ) - s t o p p i n g time  measure  times  satisfying  t  such  on  that  S <_ T,  R  .  n  i f  Let  T e T  then  S e T  T and  .  be  a  S  is  Suppose  that  lim  sup u _ ( { x e ] R :  ||x|| >_r})  n  = 0 .  r-x° T e T Then  lim  sup P ( T y  > t) = 0 .  t-*» T e T  Proof:  F o r e a c h n a t u r a l number  Then each T A R.  R  is  ±  e T  and  P  i , let  - a.s. f i n i t e ,  y  P ( T _> R.) y  = y_  = inf{t  b y 5.11.  If  ({X e ]R :  B  t  T e T  i^'  then  | |x| | >_ i } ) .  n  A R  I l l I _.  _L 0:  Thus  i lim  Fix  some  e > 0  t e  .  Then  sup P ( T TeT  >_ R.)  y  f o r some  = 0 .  1  i , sup P ( T TeT y  >_R.)  __ f" •  1  Next, f o r  1  [ 0, °°) ,  P (R  > t) <  y  ±  |  Then sup P ( T TeT y  8.11. such  Theorem: that  U  such that  y  and  Suppose converging  T  u  a, U  be a measure on ] R  (T_^)  B  T  Let  a r e measures  dy >_  U  dy  ft  (where  n  n = 1 or 2)  be a s e t o f good measures on on  ]R  n  such  for a l l y e f  i s a sequence o f  p o i n t w i s e on  is a  •  i s a potential.  whenever  potentials  then  Let  >_ t ) <_ £  y-standard  to a f u n c t i o n  (B^)-stopping time because  T  that  , then  U , > U  (B )-stopping t  : ft -*- [ 0 , °>] .  (B^)  are  a  u"  H  £  times  (Note  that  i s right-continuous.)  n  -  Consider a)  the following  There  86 -  statements:  i s a measure  a  on  ]R y  and y  b)  aQR )  = P QR )  n  , with  n  i s a p o t e n t i a l and i U dy +  for  T  a l l measures  is  is a  potential  >_ U  1  fora l l i .  a  f  y  D  T  dy  y e V  y-standard.  T h e n a ) => b ) =>  c)  a ) => b ) .  Let  H = {H H  y  Then  U°  T  U  T  Proof:  that  T  U  c)  , such  n  U  lim  sup  r+~  Heh  lim  sup  t-*»  i  finite,  > U —  a  : H  (B ) - s t o p p i n g  time  and  H <_ T.  T. x  is  Z  n  f o r some  s i n c e each  : | |x| | >_ r } ) = 0  , b y 8.9.  y-standard.  Therefore  Hence  H  P (T^. >_ t ) = 0 , b y 8.10. l —  Therefore  y  so f o r each bounded  continuous  <f> dy_  4>  •+  T  function  i s  <j>  on  P  y  H  - a.s.  ,  dy_ ,  i since  ( Let  unit  B t  )  i  v  be t h e u n i f o r m  sphere  s  continuous  centred  o>  at r  on  P -U  V  U  a  distribution  Then  T. 1  U  [ 0 , °°) .  unit  0 .  dv =  dy  T. x  dv .  i} .  1  fora l l H e H  y „ ( { x e !R  1  i s a  -U  U  V  dy  T. x dv  on t h e s u r f a c e o f t h e  - 86a  -  T. Thus t h e sequence  (  a r g u m e n t , u s i n g 8.7 b ) =>  c).  U  dv)  1  and 8.8,  i s bounded.  we  obtain  By a s u b - s u b s e q u e n c e  the statement  C o n s i d e r any bounded  (8 )-stopping t  <f>dpT.AQ  b)  .  time  Q  H n  a l l bounded  for  a l l  of  ( a =>  i  . b)  continuous functions  find  that  T.AQ U P  Now  <j>  T h u s , by t h e same m e t h o d , we  f o r each  i  ,  U  T.AQ 1  on  we  ]R  used  f o r a l l measures  dy  1  TAQ v  U Y  >_ U  We  have  TAQ  l  for  .  T. 1  as  T AQ  y > U  ±  , and  U  i n concluding the proof y e f ,  ,  dy .  T_^  is  y-standard.  It follows  that  U ^ ^  for  a l l In  R < S  y e T  , whence  particular,  i f  f  dy >  U  dy  ^ >_ U  R,  S  are  ( B ) - s t o p p i n g times t  then  u  y  T T  TARAt  y  y  > „u T A S A t > uT T T  y for >_  a l l  we  t e  [ 0 , °°) .  t h e t o t a l mass o f  equal  satisfying  and t h u s  T  (Also y  must  U  y  >_ U  T , s o t h e t o t a l mass o f  , w h e n c e t h e s e two be f i n i t e  P  have /RAt  >  / s A t  >  y  t o t a l m a s s e s must  - a.s.).  y  T  is  i n fact  If i n addition  S <_T  be ,  -  for  a l l  t e [ 0 , °°) .  87  Letting  -  t -*•  a n d a p p l y i n g 8.7  0 0  a n d 8.8  we  obtain  U 8.12. 2,  Corollary:  such  that  U  >U  R  Let  .  S  y  •  be a m e a s u r e  i s a potential.  y  Let y  time.  Then  T  is  y-standard  y U  Proof:  in  i f f  U  H ,  T  T  where  n  be a  n = 1  or  (8 )-stopping t  i s a potential  and  y  T  A  (==>)  > U  t  for a l l  T  t e  f o l l o w s immediately  [0,°°) .  from  8.2,  the d e f i n i t i o n  of  "y-standard". (<==) (if  Taking n = 1)  T A t  is  and  t = 0 we  a n d a p p l y i n g 4.8  find  that  y-standard,  ( i f n = 2)  y ( R ) = yOR ) n  .  n  T  b y 8.5.  Now  apply  or  For each  4.11 t,  8.11, w i t h  a = y  T  T. = T A i .  •  8.13.  Theorem:  such  that  Let  m  a)  U  y  be a m e a s u r e on  i s a potential.  y  be L e b e s g u e m e a s u r e  F o r any measure and  b)  Let  on  v  i t s m - i n t e g r a l over t h o u g h i t may  T  y-standard  is  K  H  on  sense,  U  Let  u  - U  T  dm  be  T .  n  ]R , n  where  n  be a  n = 1  (B^)-stopping  or  2,  time.  Then:  U  y  - U  any compact  i s defined  V  subset  of  H  subset  K  m - a.e., n  makes  -H>° .  i f f f o r each  is finite  H ,  compact  and e q u a l  to  y E (  1 (B K  0  of )ds)  ]R , n  -  Proof:  a)  and  U  88  -  i s everywhere d e f i n e d ,  M  i s m-integrable  over  any  does not  compact  subset  v either on  ]R  the  same i s t r u e  except  n  undefined.  of  possibly  This  U  ,  or  set  else  of  U  (=>)  possibly  on  a  As  T  polar  is  set, U  y-standard, b y 8.7. - U  y  T  to  °°,  i , we  and  any  obtain  A  0  where  TAi  The  i t is  y T  U  + U  except  l (B )ds)  E (  K  s  subset  desired  K  of  e q u a l i t y by  R.  Letting  the  monotone  y theorem.  -°°  Also dm  ±  y  compact  the  Now  TAi =  any  .  n  a).  K  for  3R  value  is identically  m-measure  M  b)  of  the  v  for a  proves  assume  finiteness is clear,  since  y  U  and  i  go  convergence  T  U  are  potentials.  (<=)  For  any  t  e  U  [0,°°)  - U - i  y  r  M  T  'K  A  and  any  compact  set  K  £]R",  dm  t  (•TAt =  E (  <  E (  l (B )ds)  y  K  (by  s  8.5)  l (B )ds)  y  K  u  IT  s  K  - U  T  dm  ;  K l^TAt hence  U  P<j»  PIJ»  >_ U  m  - a.e.  Also  U  Therefore  U  is a y  finiteness apply  8.12.  assumption.  TAt  p o t e n t i a l by y  >_ U  the  T everywhere.  Now  -  89  -  T Remark:  Note t h a t  1  E ( y  (B  0 spends i n K  time  that  between above  fc  (*)  of  the  proof  i s the  P -expected  amount  y  of  s  K  of  up  to  7.9,  time  and  T  .  Observe  the formula  the  analogy  established in  the  theorem.  8.14. 2,  (B )  )ds)  Corollary: such  times,  that  and  U  Let  y  be  a measure on  i s a potential.  y  consider  the  (8  Let  )-stopping  E. ,  where  n  R,S  be  time  n =  R  is  y-standard  or  (8 )-stopping t  T = R +  S  • 8  t Suppose  1  . R  and  S  is  y_,-standard.  Then  T  is  K  y-standard.  Proof:  Apply  8.13  i n conjunction with  the  s t r o n g Markov  property.  •  8.15.  In order  is  sufficient  not  that a  (8 )-stopping  time  t  T  be  y-standard, i t  that fT 1„(B  E ( y  0  be  finite  f o r each  following  example.  distribution  on  and  inf{t  a.s. the  let  T =  finite uniform  compact Let  {x  e H > 0:  (because unit  n  K  set  n =  V  K _c ]R  2  . =  B^  .  e A}  < 3),  n  Let  : ||x||  distribution  U (x)  )ds) s  by  2}  .  T h i s i s shown b y  y  be  .  As  Let A  A =  i s not  6.5.  Then  on  A  Thus  0  for  .  the uniform {x  unit  e TR : ||x||  polar,  clearly  the  v  T  is  = y_  | | x | | <_ 1  = - ^ l o g | | x | |  for  ||x||  >  1  = P  is  1}, P  -  -  90  -  while - ^ l o g 2 U (x)  =  M  Thus  U  let  ^_U ,  M  A  =  ±  so  V  {x  Then each standard,  T  e H :  T^ by  has 8.5.  _ 1_ ^log||x||  cannot  ||x||  2  for  be  = 1}  x||  for  y-standard.  and  let  =  <  ||x||  Now inf{t  i =  > 0:  B  P -expectation,  by  Now  E y^,  of a uniform  v.  consists  >  for  finite  y  2  5.11,  and  t  so  2  3,4,5,... e k  u  ±  is  A}  y-  distribution  i of  mass  A,  e. X  where  on I e^  e  A.  plus a uniform  (0,1)  i s determined  log  This is  f o l l o w s from  harmonic  in  i +  Dynkin's  (1 -  v.  be  i t i s clear  used  that  T. l  to give another  E. = l  proof  mass 1 - E .  on  1  2  the Now  1  fact one  that  easily  log||•|| computes  2 '  (The  T  and  2  log  of  equation  1 = log  2TT  +  the  5.12,  v _ log  4-  1  by  s^log  formula  2> H^\{0} . T h u s  U  Also,  distribution  fact  that  0  e.  can  then  x  that  T  is  P  y  - a.s.  finite.)  It  2 follows  t h a t f o r any  compact  set  K £ H  y  E ( y  l (B )ds) K  =  s  (  U  -  ,  (  D  T _ l o g _ 2  )  )  f  K  which of We  course also  example  _ log =  2 here.  2TT  finite.  remark t h a t the  natural  C  is  i n which  the  sequence of measures C  of  theorem  8.7  is  (v^)  yields  a  non-zero;  •  that  -  8.16.  Lemma:  T = inf{t U  Let  > 0: _  U  Proof: then  If  T  T  .  on  so  i s true  T  X  =  P  .  (if n  <_ 2)  U  claim  T  Case  1.  We  (x) =  that  E  V  such  n  Let that  that  U  V  i s a potential  b y 8.5  and a l s o  i f  (if n £  v =  6^  2) .  f o r any  also  have  y  for a l l  x,y  e ]R , n  ($(y,B_)) = E ($(x,B_)) y  be a b o u n d e d  x e IR \V n  .  open  Then  =  X  subset  *(y,x)  y e E. \V  .  subcase  b.  y e V  Let  W £ V  .  .  Then  ]R  n  such  Then  W  $(x,*)  = *(x,y)  that  H c V  so  X  a.  n  of  .  P ( T = 0) = 1,  subcase  and  i n order of i n t e g r a t i o n i s j u s t i f i e d  E ($(x,B_))y(dy)  E ($(y,B_))  H £ W  .  X  by  Let  TR  on  n  .  n  v-standard  v = y,  y  E.  E (4>(y,B_))y(dy),  interchange  we  is  i f  where the r e q u i r e d  Now  such  n  of  Hence  E (U (B_))  8.4  E.  subset  and  for a l l x e R  y  b y 5.11,  particular, this n  y-standard,  i s any measure  V  co-analytic  Then f o r any measure  is  X  E (T) < »  x e H  .  (x) = E (U (B_))  v  -  be a bounded  I H}  t  i s a potential,  y  In  H  91  .  E ($(x,B_)) y  = $(x,y)  be an open s u b s e t i s bounded  of  .  V  such  and c o n t i n u o u s  on  that W,  -  and  harmonic  Dynkin's  in  W  formula  .  Also  T  92  <_ i n f { t  =  y  y  Case  x,y  3.  e ]R \V  e V  .  minorant  of  Then  a.s.  .  f o r each  .  n  harmonic V  >_ 0:  B  t  £ W}  .  Thus  by  5.12,  E ($(x,B_))  C a s e 2.  -  This  For of  i s similar  each  $(z,«)  G = $ - h z e V  .  $(x,y)  on  z  in  in V  .  to case  V,  1.  let  h(z,') be  be  V  .  Let  G  x V  .  Also  T < i n f { t > 0:  the  the Green B  t  greatest function  I V}  P  Z  Thus  E (G(y,B_)) x  = bal(G(y,-),V\H,V)(x)  (by  6.3)  = bal(G(x,-),V\H,V)(y)  (by  2.11)  =  (by 6.3)  E (G(x,B_)) y  .  Also E (h(y,B_).) X  = h(y,x) =  h(x,y)  =  E (h(x,B_)) y  (by  5.12)  (by  5.12)  Hence E ($(y,B_)) = X  in  this  case  t o o , and  E ($(x,B_))  the c l a i m  y  i s established.  •  -  -  8.17. and  Corollary:  let  y  Let  H  be a m e a s u r e  93  -  be a b o u n d e d on  H  co-analytic  such that  n  U  subset  is a  y  of  H  ,  potential.  Let T = inf{t  Suppose  v  function  > 0: B  i s s u p e r h a r m o n i c on h  I H}  R. ,  v  n  .  <_ U ,  w h i c h i s h a r m o n i c i n some o p e n  closure  of  H,  such  Proof:  If  n _^ 3,  or  2,  we  of  7.5.  need  that  this  only  v  <_ h <_ U  i s just  y  and  y  there  is a  set containing  the y  in  H  a special  emulate the proof  .  case  o f 7.7,  Then  v  <_ U  o f 7.7.  If  using  8.16  T  n = 1  i n place  • 8.18.  Lemma:  analytic H  .  Let  subset  y  of  H  be a m e a s u r e such  n  H  ]R \H  .  n  Let  i s thin  n  H  be a c o -  at each p o i n t  of  Let T = inf{t  Then  Let  of  whose u n i o n  E.  n  R. l  = inf{t  to  Borel  Letting 8.19.  > 0: B  I H}  .  y_(H) = 0 .  Proof:  T  that  on  ( ^)  be  v  a  increasing  n  is  ]R  > 0: B_ i V.} — t I  V^, .  T h e n by  i — • ~, Lemma:  we  Let  be a c o l l e c t i o n  .  n  and  7.6,  obtain (A,F,Q)  of  sequence of bounded  For each let y.  y. I  P ^"(B- e H,  be a  [0,°°]-valued  subsets  i , let be  y_(H) = 0  open  the r e s t r i c t i o n  T < R ) = 0  of  y  for a l l i .  .  a-finite  ^ measure  space.  F-measurable f u n c t i o n s  Let on  A .  -  Suppose and  the l i m i t  T 4 0  Proof: chain  •  By C  Then  As  has  t h e same  to  T  the  full  and  f  fdQ  <  T ,...,T  let  M  Q  <_ T  and  ±  in  T  to  T,  [0,°°)  by  T  .  However,  of dependent on  lemma. choice.  A  such  (1 - e  g(T) = .  If  C belongs  to prove t h i s  <_ T e T }  Q  of  T = sup CQ  element of  needed  a maximal  CQ  subset  F-measurable f u n c t i o n  T  )fdQ  i _> 1  and  have been d e f i n e d , l e t <_ T e T }  g ( T ) _> M ±  .  belongs  contains  Then  maximal  = sup{g(T): T  Q  = sup{g(T): T  C .  the p r i n c i p l e  g : T —>  ,M ,...,M  Q  sequence  positive  T  T  some c o u n t a b l e  i s not r e a l l y  Define  and  principle,  supremum a s  which uses only  .  0 0  e T  TQ  T  a-finite,  Q-essentially  be a s t r i c t l y  sequence i n  Q - e s s e n t i a l l y maximal e l e m e n t s .  maximality  axiom of c h o i c e  Let  M  is  is a  i s a proof  Let  contains  Q-essential  Here  that  T  Q  T  -  of each i n c r e a s i n g  the Hausdorff  .  94  ±  and  - 2  _ 1  choose .  L e t . T = l i m T.  Then .  e T  ^  (T^  Then  such i s an  T e T  .  that increasing S e T  If  and  i-yoo  T  <_ S  g(S)  then <_ g ( T )  maximal  in  f o r any  i , g ( S ) <_  .  S = T  Thus  <_ g ( T ) +  Q - a.e.  2  Hence  <_ g ( T ) +  _ 1  T  is  2" ;  hence  1  Q-essentially  T .  • 8.20. are  Then  Theorem:  Let  potentials.  U  b)  u ( Z ) <_ v ( Z )  c)  yOR ) = vOR ),  there  n  be m e a s u r e s  on  ]R  n  such  that  U ,  U  y  V  Suppose  a)  y  > U  u,v  V  . f o r every n  on  i s a y-standard  Borel  set  Z £  {U  V  = »} .  n >_ 3 . (8 )-stopping t  time  T  such  that  y  T  = v  - 95  Proof: For  We  could  this  the sake of v a r i e t y T  proof.  Let  T  such  that  of  any  n  prove  though, l e t us g i v e  U  >_U  .  o r 7.9  element  A = v  course,  we  {U  T  ( i f n >_ 3 ) .  maximal  .  Of  > U }  A  V  exists  .  a number  0 e T,  Then  .  have o n l y  Let  claim  A(A) = 0  c  and a b o u n d e d  .  We U  shall <_ U  A  Suppose n o t .  open  set  Also  W  such  V  times  the  limit  (if  contains  .  T  7.11.  different  b y 8.11  T  t o show t h a t  We  A(H)  A = y  .  T,  to  T h u s , by 8.19, T  proved  (8^)-stopping  T 4 0  so  belongs  a s we  a somewhat  be t h e s e t o f a l l y - s t a n d a r d  essentially  A =  i n much t h e same way  i n c r e a s i n g sequence i n  <_ 2)  -  a  P M  show  .  that  Let  Then  there  that  > 0,  where  n {U* > c > U }  H = W  .  V  Let  S = inf{t  > 0: B  I H}  .  A  Then  S  is  the  strong  the  case  T u  <_T' T  Markov n  .  = u , T  A-standard  <_ 2,  But  contradiction A U  property, T'  Hence .  by 8.16,  is  A  P  T  = y  U ,  y-standard  P ( T 4 T') u (H)  and  > 0  = 0  while,  shows t h a t we  must  v  S >_ U  where  by 8.17.  V  T' = T +  by 8.14. .  y^,, (H) = 0  A(A) = 0  .  v -<_U  A-a.e.  If  Z  T'  S  • 0 e T  . .  Therefore  by 8.18,  have  Thus  Also,  i s a Borel polar  set  then  Thus  .  This  by In Now  -  A(Z)  96 -  = A(Z n { U  X  <_ U } )  =  A(Z n { U  X  =  00}  =  A(Z n { U  V  =  00})  <_ y ( Z n { U  V  = »})  V  {n  n  <_  X  u }) V  2.2)  (by  ( b y 6.5)  1 y(z) .  T h u s by t h e d o m i n a t i o n 4.8  i f n = 2,  o r 4.13  we a r e a s s u m i n g  that  8.21.  Baxter  Remark:  theorem: y U  principle,  "Let  i f  <_U  n = 1 ;  uQR ) = vOR ) n  y,v  n  and C h a c o n  .  V  note  ( S e e 4.2  that  i f n > 3,  i n the case  n <_ 2  .)  [1] h a v e p r o v e d  be p r o b a b i l i t y m e a s u r e s on  the f o l l o w i n g ]R  such  n  that  v , U  are potentials. a)  b)  U  y  > U  lim  V  Suppose  .  (U (x) P  - U (x)) V  = 0 .  I I I I" x  c)  U  ><J0  i s finite  V  (B )-stopping  time  second  so does  t  moment  T  and c o n t i n u o u s on such y,  that and  I xI I d v ( x ) =  If  n >_ 3,  condition  Now by Lemma 5 o f B a x t e r m e a s u r e s on then  ]R  n  such  that  T  If  Then v  +  there  has a  isa  finite  so t h a t  2nE (T) M  dropped."  and Chacon  U , U y  .  c a n be c h o s e n  2  b ) c a n be  U  y_ = v .  |x|| dy(x)  2  TR  V  [1], i f  y,v  a r e p o t e n t i a l s and  are f i n i t e U  P  >_ U , V  -  IxI I ^ d v ( x )  Also,  b y 8.13, i f  T  U (x)  (If  n >_ 3,  cited  y  P T  (x)dx  u s e 6.9 i n p l a c e  theorem  require  y-standard  - U  y  | x| |^du(x) +  =  is  of Baxter  and  v  and  Chacon has been c o m p l e t e l y  that  possible  (unless  I believe  8.22. V  time are  U  of 8.13).  T h u s 8.20 i s a g e n e r a l i z a t i o n o f t h e  In p a r t i c u l a r :  I f n >_ 3, we no  on  U  Condition  n = 1).  y - a.e.  This  Let  are potentials.  y,v  Then  that  y^ = v  U  i s made.  V  Also,  f o r example, i t  Of c o u r s e 8.20 i s n o t b e s t  ]R  y-standard  i f and o n l y  n = 3.  possible".  be m e a s u r e s on is a  need  c) has been  i s shown b y 7.13, i n t h e c a s e i s "best  there  on  has been r e l a x e d ;  V  they  n  such  that  U , y  (8 )-stopping t  i f the f o l l o w i n g  3  conditions  satisfied:  a)  U  b)  There  y  for c)  > U  V  .  i s a Borel  every  n > 3,  set  Borel  or  polar  A £ ]R set  y ( R ) = vCR ) . n  n  n  longer  b) o f t h e t h e o r e m o f B a x t e r  eliminated.  the following r e s u l t  such  y  Condition  be f i n i t e  V  Conjecture:  T  V  then  No c o n t i n u i t y a s s u m p t i o n  assumption  suffices  U  measures.  eliminated.  finiteness  - U (x)dx  y  t o h a v e t h e same t o t a l m a s s ; m o r e o v e r ,  e v e n be f i n i t e  the  2n U ( x )  = E (T) .  and Chacon.  not  partly  97 -  such  Z £ R  n  that .  v ( Z ) = y ( Z n A)  -  9.  -  RANDOMIZED STOPPING TIMES, AND  Suppose the  unit  y  If  v = u  only  P°(B  3R  uniformly  P°(T>0) = 1 have  i s the  m e a s u r e on  other h a l f  e  T  have  v = y^  point  w h i c h has  n  some  {0}  {0})  unit  distributed  for  T  so  "randomized  9 . 1 .  the  A (A,F)) every  1  time"  . T  A  in  topological  open s e t s  of  x ,  X  A  with  e H  , and  n  any  n  v =  0  is  and  the .  then  i f i t w e r e we .  T h u s we  , there "y  SPACES  v  = 1 }  ||x||  T  for  "  .  i s an  would can obvious  Namely,  let  1 / 2  probability  with  :  n  time  set,  that  1 / 2  probability  definitions.  (X,A)  and  (rrv,  probability  shall  {x  v({0}) 4 0  for  such  (A,F)  x(')(A)  that  is  for  measurable  short)  F-measurable.  X  , the  of  If  rrv's  a-field  spaces.  (X,A)  in A  m e a s u r e s on  speak simply  A = Borel  be  , such  X  is  in  X  (over  that  for  a ; it  g e n e r a t e d by  will the  .  9 . 2 .  Definition:  is  increasing  an  :  s p a c e , we  understood  But  random v a r i a b l e  i s a map A  a polar  in  i t s mass a t  stopping  n  =  0  of  S =  i f  Let  randomized  H  , contradicting  following  Definition:  mass a t  = 0  stopping  motivates  PROBABILITY  half  -  c a n n o t be  T  This  ENLARGEMENTS OF  on  (8^)  0  be  9 8  If  (A,F)  family  of  i s a measurable  sub  - a -  fields  of  s p a c e and  F  then a  ^t^Q<t«=°  randomized  -  (F )-stopping  time  t  (A,F)  over  such  measurable. we  s t o p by  for  time  t  9.3.  Examples:  t  depends o n l y on  (A,F)  Let  (X,A) x  Define  be  be  A -*• p r o b a b i l i t y  :  x(u)(A)  is,  For  any  x  Now  f  let  be  arising  stopping  a  from i f f  0  ]  -  probability  the  information  that  we  that  have  a measurable  by  x A  6  f(_)  space A  on  space. and  let  f  : A •+ X  Let  more o r  .  by  ;  .  (f('))  1  F-measurable  S  i f f this  f  i s a rv  c a s e we  i f f  X  (X,A);  in x  call  f  the  [A]  e F  .  that i s ,  rrv  arising  rv  T {T  be  an  [ O ,  in  .  increasing  For  <_ t }  0  0  ]  ,  any  e F  and  t e  •  family  let  x  of  be  sub  the  - a  rrv  x  is a rst i f f  fields  in  [0,°=) , x ( - ) ( [ 0 , t ] )  Thus  -  is T  is a  time.  (A,F)  and  Then r e l a t i v e  ® A  F  is  0  .  T  t  F  ,  ( F ) t 0_t<°°  let  F -measurable  be  X  In  the r v  9.4.  [ O ,  conditional  measures  (f(_))  i s measurable.  from  [0,°°]  1  (X,A)  f  ,  A £  =  is a rrv in  iff  F  in  the  another measurable  =  of  x  .)  Let  Thus  rrv  0_<t<°°,x(')([0,t])  (That i s , i n t u i t i v e l y ,  time  that  -  for short) i s a  ((F^J-rst,  that  99  less  (X,A)  be  to a measure  identified  , whose p r o j e c t i o n  on  measurable Q  on  w i t h measures A  i s equal  F  spaces.  , rrv's on to  in  (X,A)  the product Q  .  This  may  o-field i s made  -  p r e c i s e by the f o l l o w i n g  100 -  result,  Theorem:  a)  x  If  is  a  n  r  r  F-measurable = b)  x  and  i f we  of  define  y  y  Q  to  rise  to  Suppose  Q  F ® A  on  F  on  F ,  y(F><X) = Q ( F )  for  i s a measure  and  a  measure n  d  x'  such  y  Q(E) = 0 o-finite  that  in  i s countably (X,A)  F ® A  which  (according  the s e t  .  generated,  i  n  (X,A)  F , y  measure on  y(F*X) = Q(F)  t h e measure  x  on  A  and  are rrv's  A •-»• y ( A * A )  i s a measure  for a l l F  and f o r e a c h  t o a semicompact c l a s s a rrv  F  on  (X r" X M  =  and  is a  finite,  respect exists  F ® A  i n p a r t b ) ) then  i s countably  Q(F)  Q by  t o t h e same m e a s u r e  belongs  A  x  and i f  the f o r m u l a  on  and  Q(d_)x(o))(H(_))  i s a semifinite  E  d)  H(„)  .)  F ® A  on  i s a measure  generated, give  _  (Where  is  F .  in  If  F ® A .  in  (X,A)  x(^) (H(w))  to  then  over  i s an r r v i n  then  c)  H  H  y(H) =  F  (X,A)  n  f o r any  the s e c t i o n  If  i  v  F  in  that  F  F ,  with  i s inner regular with  contained i n  such  in  y(H) =  A .  Then  there  Q(d_)x(u)(H(_))  -  for  a l l  (A  e F ® A  H  semicompact  countable  £  .  x  Note t h a t  class  C  101  is a  , n  -  i  Q-essentially  s  collection  = 0  implies  C  of  n  sets,  = 0  u n i q u e by  such  c).  that  f o r some  finite  = c .) 0  We Pachl  shall  not  [1] h a s  prove  recently  this  shown t h a t  countably  generated  x( )  defined  only  e F  , H(co)  each for  w  each  H  co >->• x(w) Q  .  In  (H(co)) the  regularity  can  ® A  be  i n d),  dropped,  on  Let  the  a sub  - a  is in  A  -  respect  Pachl to  a  showed  semicompact  are A^  *  Q  that  remark  A  willing  to  A  of  i s the  the  that  that  Q -almost  co  Q -measurable, where  same p a p e r ,  we  field for  us  condition  provided  *  is  with  theorem h e r e .  have  , where  a l l  co  and  completion  requirement  class  be  is, in a  of  of  inner  sense,  necessary.  9.5.  Let  us  measurable a)  Every a  (X,A)  finite  (X,A)  (X,A)  of  Polish  space.)  for a  separated  following  m e a s u r e on class  A  are  on  regular with the  respect  measure).  isomorphic  to a u n i v e r s a l l y measurable  isomorphic  to a u n i v e r s a l l y measurable  a Polish  space.  i s u n i v e r s a l l y measurable i n every measurable  space i s a  generated  equivalent:  i s inner  (depending  countably  TR. .  i s Borel  separated (A  , the  is Borel  subspace of d)  that  semicompact  subspace c)  remark (X,A)  space  to b)  also  separable  space i n which completely  countably  i t is Borel  metrizable  embedded.  topological  -  Let (X,A)  us  say  9.6.  of  be  i s Borel  measurable  space i f f  isomorphic to a  s u b s e t o f 3R .  (A,F,Q)  Let  a right-continuous  be  a measure s p a c e and l e t  increasing  family  o f sub  - a -  fields  U ^ t "^  00  F  •  Assume x  Let are  -  i s a universally  space which  measurable  Proposition:  ). t  (X,A)  i s a measurable  universally  (F  that  102  Q  be  is  a-finite  a rrv in  [0,°°]  on  F^  over  . (A,F)  .  Then t h e  following  equivalent: a) b)  T h e r e i s an (F )-rst T such that x = X Q " a.e. Whenever t e [0,°°), f e L ( Q ) . s u c h t h a t f dQ = F  0  for  on  t  1  a l l  [t,°°]  c)  Proof:  a)  , and  t e  [0,°°)  F^-measurable  =>  b)  If  =>  c)  Let  g e C[0,°°]  such  that  g =  0  Q(du>) X(w)(ds)f(_)g(&) = 0  , then  For each an  F e F  ,  x( )([0,t])  is  #  Q - a.e.  equal to  function.  A e Borel  [0,t] , then  x(.)(A)  is  F  -  measurable. b)  \\) h dQ  =  ^  t e  [0,~)  and  let  ^ = x(»)([0,t))  .  Then  E(h|F )dQ t  E(ip|F )E(h|F )dQ t  E(i|>|F )h dQ  t  for  all  h e L (Q) 1  .  Hence  i|> = E(i^ | F ) fc  -  (Note F  that  we Q  of  c ) => A  on  r  0  A  is  r  on  ^  •)  t  = l i m x(*)([0,t + 2~ )) . k-*>° k  we  For  r  obtain  the desired  rational,  r  0 < r <  r  to e A  x  here  , there t e  i s a unique [0,°°)  we  can choose  functions  rst .  map  F e  Let  (to) . r  (The r i g h t c o n t i n u i t y  of  (F^) i s •  An e n l a r g e m e n t  o f a measure  f  \p : M + A s u c h t h a t  space  ( M , G, R ) ip(R) = Q  and a  (A,F,Q)  is  (G,F)-  ( i . e .R(^  _ 1  [F])  F) .  (M,G,R,IJ0  (A,F,Q)  let  = infA r>t  x(to)  too.)  measurable a l l  p r o b a b i l i t y measure  , x(to)([0,t])  p a i r c o n s i s t i n g o f a measure s p a c e  and  0 0  Q - a.s.  i s the sought  Definition:  space  conclusion.  -measurable  such that  Then  9.8.  U s i n g the r i g h t -  that:  = x(O([0,r])  [0,°°]  for  to  < r ' => A < A , — r — r  on  a  a)  F  f o r each  9.7.  (F^_)  such  Now  used  a-finite  with respect  < A < 1 — r —  r  A  is  x(')([0,t])  continuity  A  can t a l k of c o n d i t i o n a l expectations  because  Now  103 -  .  be an e n l a r g e m e n t  Let  be a r v i n  (X,A)  of a  a-finite  measure  be a u n i v e r s a l l y m e a s u r a b l e  (X,A) o v e r  (M,G) .  space,  = Q(F)  -  Define  : M ->• A x X  g  (G,  F ® A)-measurable.  of  u  on  unique  rrv  x  V(H)  =  In  i  in  Any  9.8,  this  (Mp)  .  Thus,  by  (A,F)  over  . f (p)) on  F ®  space.  of  One  R  checks  with respect  A  Then .  such  that  H e F ®  A  gives  rise  that  to  i f  , F  g  is  Then the  there i s a  for a l l  easily  •  9.4,  a r v o v e r an e n l a r g e m e n t  Q-a.a. _ e A  9.9.  -  p = g(R)  Q(d_)x(u)(H(_))  disintegration for  = Q  (X,A)  n  t h i s way,  the o r i g i n a l  g(p) =  Let  ijJ(R)  is  A  by  104  (R ) _  Q-essentially  to a r r v over .  is a  _eA  x(^)  , then  projection  ^R^)  =  .  r r v o v e r a measure space  arises,  f r o m a r v i n an e n l a r g e m e n t .  i n the f a s h i o n  L e t us  describe  described  explicitly  why  i s so. (A,F)  Let (X,A)  over x  how  (A,F)  arises  Well,  , (X,A) .  be  Let  measurable Q  be  spaces  a measure on  f r o m a r v o v e r an e n l a r g e m e n t  let  M  = A x  and  let  F > and of  x  be  l e t us  a rrv in show  (A.F,Q) .  x  G = F ® A  R(G) =  f Then  (M,6,R,^)  (X,A) in  9.8.  over  (M,6)  Q(d_))xCw) ( G ( _ ) )  = projection  of  M  on  = projection  of  M  on  i s an enlargement , and  x  arises  of  for  G e G  X .  (A,F,Q) , f  from  f  i s an r v i n  i n t h e manner  described  -  9.10.  Given  two r r v ' s ,  enlargement, the  joint  rrv's  distribution  Definition.  (A,F,F ) T  » where  increasing  (AJFJF^) which  ensure any  A filtered  F  measurable  ( A , F ) i s a measurable  This  i n 9.9.  measure  a-finite  space  However,  FQ .  conditional  enlargements.  and  ^t^0<t<°°  ^  S  F .  space and  (The l a s t  cannot  i s a system  (A,F,F  i s a system  measurable  on  space  of  i s why we  consider  space  o f sub - a - f i e l d s  t h a t we c a n c o n s i d e r  Q  ,Q) , w h e r e i s a measure on  assumption  i s made t o  expectations with respect to  .)  9.12.  Definition.  (A,F,F ,Q)  An enlargement  i s a filtered  T  a map  ip s u c h  space  (A » F , Q )  that  measure space  (M,G,R,ijj)  and such  of a f i l t e r e d  that  measure  (M,G,G^,R)  i s an enlargement i n addition,  space  together with  o f the measure  \JJ i s  (G^, F ^ ) - m e a s u r a b l e  0 £ t < °° . An  optional  (M,G,G ,R,iJ;) t  we h a v e  enlargement  such  E(g|*,F)  We r e m a r k  that  that  = E(g|i|;,F ) optional  enlargements"  introduced  notion  this  changes.  of  f o r each  "distributional  time  described  o f t h e two r v ' s i s n o t d e t e r m i n e d b y t h e two  i s a filtered is  a s r v ' s i n a common  i n g e n e r a l , b u t must a l s o  family  A filtered  for  to that  a l o n e ; i t depends on the e n l a r g e m e n t .  9.11.  F  t h e y may b e r e a l i z e d  by a p r o c e s s s i m i l a r  j u s t work w i t h r r v ' s  an  105 -  (A,F,F ,Q) T  t e [0,«)  i s an  enlargement  and each  g e l^CM.G ,R)  Q-a.s.  enlargements  are e s s e n t i a l l y the  o f B a x t e r and Chacon  to f a c i l i t a t e  the d i s c u s s i o n  [3].  These a u t h o r s  o f randomized  -  N e x t we  prove  notion  of o p t i o n a l  9.13.  Theorem: (F  that be  an  t  )  106  a r e s u l t which  -  clarifies  the meaning o f  enlargement.  (A,F,,Q)  Let  be  i s r i g h t - c o n t i n u o u s and  a filtered  measure space  F = a(uF  .  t  enlargement  the  of  (A,F,F._,Q) .  Then  )  t  the  such  ( M , G , G ,R,i|0 t  Let  following  are  equivalent: a)  (M,G,G ,R,i|j)  b)  Every  i s an o p t i o n a l  t  ( G ^ ) - s t o p p i n g time  c)  E  d) '  =  Moreover, w i t h function,  Proof:  described  =>  to  Q)  t  (E,E) '  (X^oijj)  regard  then  a)  i n 9.8,  T .  t  i s a measurable  b) .  i n 9.8.  in  (E,E)  has  =>  d), i f  x  (A,F,F  over  (^ )  n  a  x  Let  t e  be  the r r v i n  [O, ) 00  Q(d_)x(u)([0,t])-=  .  R(^  s  then  function.  [0,°°]  [F]n{T  Q)  transition  a  arising F e  T h e n f o r any  _ 1  t >  < o  (K,C,G^,R) .  t  t h e same t r a n s i t i o n  Let  (X_)„ t 0<t °  and  i s a Markov p r o c e s s over t o a)  (Y^)  space r  a Markov p r o c e s s  (Y^)  time  t  as  (X  Whenever  is  gives r i s e ,  i s a martingale over (A,F F t 0<t<°° t (X °i|i) i s a m a r t i n g a l e over (M,G,G ,R) .  Whenever (Y^)  T  (F )-stopping  a randomized  enlargement.  < t})  from  F  T  as  ,  .  ' F Thus  x(*)([0,t])  F^-measurable equal  =>  a) .  (G )-stopping t  {  _  Let time  t  <  f u n c t i o n by (F^)-rst  to an b)  = E ( l  ^U,F)  , which  optionality.  i s Q - a.e.  Hence, by  9.6,  equal x  to  an  is Q -  a.e.  T . t e  [0,°°)  T = t 1  £  +  and 0 0  L^.  let  G e  » one  G finds  .  By  considering  the  that  E(l_|i|>,F)  is  - 107  F ^ - m e a s u r a b l e mod  -  Q .  Before proceeding make t h e f o l l o w i n g  to the proof  0 <_ s < t <  (*)  F o r any  .  0 0  g  Then  in  E(g|*,F  (*.) a n d  F o r any  the p r o o f  of  f  s  below a r e e q u i v a l e n t .  Q - a.e.  S  i - ^ A . F ,Q)  in  ) = E(f|F W s  ,  •  by a n " o r t h o g o n a l i t y a r g u m e n t " s u c h  (b =>  c) o f  The e q u i v a l e n c e with  (**)  ) = E(g|i|»,F )  E(f°iHG  can be p r o v e d  we  ^(M.G^R) ,  L  This  implications,  observation:  Let  (**)  of the remaining  a s was  used  in  9.6.  ( a <=>  c) f o l l o w s i m m e d i a t e l y  from  this  together  F = a(uF ) . t t  a)  =>  Let  d).  Let  f  be a n o n - n e g a t i v e  0 <_ s < t < »  .  E ( f ( Y ) |6 )  Then  E ( f ( X W | G , )  =  g  f c  t  = E(f(X )|F )o^ t  (by (*) => (**))  s  = E(f ( X J | X t s  -  1  (E)W  = E(f(Y.)|Y (E)) t s _ 1  Thus  )  transition (Y ) t  .  i s Markov. function for  E - m e a s u r a b l e f u n c t i o n on  ((X,) t .  A similar (X ) t  i s Markov)  calculation  i s also  shows t h a t  a transition  any  function for  E  -  d) => process  a).  in  Let  (E,E)  (E,E)  over  F  Then  (X ) t  be a measurable  (A,F,F ,Q)  = X (E)  A  .  t  and l e t  Since  E-measurable  (X^)  that  -  (Y ) = (X °4>) t  f  = X ^(E)  function  h  i s also  be a n o n - n e g a t i v e  on  , f = h(X ) t  E  E(f°^|G  .  s  Markov.  Then  ) = E ( h ( Y )|G ) t s 1  ( ( Y , ) i s Markov)  _ 1  =  s  E(h(X )|X t  1 8  t  see  that  there  =>  (*)  the enlargement  i s a measurable  , and the f a c t  i s optional.  space  (E,E)  that  t  (E))o^  = E(h(X )|F )o^  (**)  function  f o r some n o n - n e g a t i v e  t  using  Let  F^-measurable  = E(h(Y )|Y (E))  Thus,  a  for 0 < t < » .  _ 1  t  s p a c e and  i s Markov.  <_ s < t < °°  on  such  t  Hence, by h y p o t h e s i s , 0  107a -  g  .  F = o(uF t  ) , we t  Of c o u r s e , we must  and a p r o c e s s  (X )  show in  that  (E,E)  such  = X~ (E)  F  that  X  -  108  for a l l  t .  -  E = Q x [0,~)  Let  E = the and l e t  X (_) t  =  a-field (_,t)  (F )-progressively  of  for  measurable  t  _ e A , t e  [0,°°) .  This  does  sets  the  trick.  •  Remark:  The  equivalence assumption  right-continuity o f a ) , c ) , and  i s not needed  d) i n t h e a b o v e  F = o(uF ) t t  that  ('F )  of  theorem;  i s n o t needed  f o r the  also,  the  f o r the e q u i v a l e n c e  of  a) a n d b ) .  9.14. of  Observe  (A, F , F  (A.F.F.+.Q)  that  Q)  where  (^ )  1  S  t  .  Let  G  (A,F,F  Let  M =  A x  =  F®  ,Q)  enlargement  enlargement of  be  R(G)  =  \p = p r o j e c t i o n  a)  T h e r e i s an  b)  (M,G,G  c)  (M,G,G  ,R,IJJ) T +  ,R,iJ;)  a filtered  let X  A  a rrv in  A = Borel  , where for  = {A e A  0 < t < : A  n  space,  [O, ] 00  M  (F )-rst  on x  A .  i s an o p t i o n a l  , where  0 0  Then  (t,»]}  G e G  for  such that  i s an o p t i o n a l  [0,°=]  (t,»] = 0 o r  Q(dw)x(u)(G(u))  of  be  measure  [0,»]  F ® A t t  t  A  let  i s an o p t i o n a l  r i g h t - c o n t i n u o u s , and  G  and  ,R,iJ.)  i s an o p t i o n a l  •  Proposition.  (A,F)  (M,G,G^,R,u))  (M,G,G  then  9.15.  over  i f  the f o l l o w i n g X  =  T  are  equivalent.  Q-a.e.  enlargement of enlargement of  (A , ^ , ^ » Q ) t  (A,F,F ,Q) t  • .  -  Proof: (F  As  )  from  i s an  t  enlargement  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  =>  a) . x  <_ t } = A  time.  Let  R(^  _ 1  a) then  [F]  x  then =>  b) .  for  (a  n T  T  =  T = p r o j e c t i o n of  [0,t]  Hence, by  that  But  -  =>  _ 1  One  T(»)(G)  Note time  T  9.16.  b)  of  9.13,  =  easily F  is  i n the  m  be  checks  follows  that and  above p r o o f ,  way  of  be  i f  are  u n i q u e maps  and  b)  f  i s left-continuous  c)  g  i s right-continuous  d)  y = f(m)  <_ h _< g  h  and  = h(u+)  .  g  for  )-rst  F e F  t e  [0,°°)  E(lJiJ>.F)  =  is realized  T  and  such A  e A  and  G  e G  Q -  a.e.  as  (G^-stopping  the  i n an  m e a s u r e on  : (0,1)  I n 9.18  below,  o p t i o n a l enlargement.  B o r e l sets of  f, g  .  t  x  the  [0,°°] , (0,1)  ->• [0,°°]  and  . such  that:  are i n c r e a s i n g  =  .  T(»)(G)  (M,G,G ,R,i(j)  a probability  f  i f  T  realizing  L e b e s g u e m e a s u r e on there  (G^-stopping (F  i s an  Then  •  G  a)  Moreover,  g(u)  c)  [0,°°] . is a  Q (dto) x (to) (A)  -measurable  y  Let  T  there  o p t i o n a l enlargement  another  Lemma:  Then  f  and  a.e.  t h a t i n the  describe  let  •  t  o f b)  on  0 _< t < °° , s o  [A])  Q -  M  t  we  (A,F>F »Q)  of  9.14. b)  {T  (M,G,G ,R,iJj)  Clearly  109  g(m)  : (0,1) for a l l  •+ [ O , ] 00  u  in  i s i n c r e a s i n g and (0,1)  we  have  f (u)  h(m)  = y  = h(u-)  then and  -no Sketch  of Proof:  For  f(u)  and  0 < u < 1 ,  l e t  = sup{t e [0,»] : y ( [ 0 , t ] )  l e t g(u) = i n f { t  (where  sup 0 = 0 )  e [0,°°]  : y([0,t))  < u}  > u} .  • 9.17.  Definition:  where  (F )  with  Let  (A,F,F ,Q)  i s right-continuous.  m(X) = 1 .  Let  be a f i l t e r e d  T  Let  (X,A,m)  measure  space,  be a measure  space  M = A * X  G = F® A = ( F ® A)  G  (=  FC  F  n  e>0 for  t  9 +  A)  e  0 _< t < °°  R = Q <_. m i> = p r o j e c t i o n Then c l e a r l y (A,F,F^.Q)  ;  w  e  (M,G,G ,R,^)  call  of  M  on A .  i s an o p t i o n a l  t  i t the product enlargement  enlargement o f of  (A,F,F ,Q) T  by  (X,A,m) .  9.18. with  Proposition: (F )  Let  (A.F,F ,Q) T  right-continuous.  m(X) = 1 , a n d l e t (A,F,F ,Q)  by  T  Let  (H,G,G^,R,M))  be a f i l t e r e d (X,A,m)  measure  be a measure  space,  space w i t h  be the p r o d u c t enlargement o f  (X,A,m) .  Then: a)  If  T  i sa  (F )-stopping t  is  ( G ) - s t o p p i n g time t  time f o r each  a randomized  then  T(',x)  x e X , and  (F ) - s t o p p i n g  time.  i s an  T = T(',m)  -  b)  Suppose  X =  I l l -  , A = Borel X  (0,1)  , and  m =  Lebesgue  i  A  measure on time, the  and  x Proof:  T(*)([0,t))  such  = m({x  e X  T(to,u) = sup{t  x(to)([0,t]) is rational  and  is  a rational  r e  < t} =  r  £  r  Let  normed If  f(m)  T(to,»)  = x(to)  f  be  of  (0,1)  ( s e e 9.16)  then  time.  {to  (to,x) e {T  e A:  f o l l o w from  e  [0, t )  $  t )  such  ^,u)  .  right-continuity  e M  t  .  and  The  , T(to,u) = s u p { r  Thus  T(to,u) < t  : u < T (to) ( [ 0 , r ] ) }  (F'  of  (Here  x ( t o ) ( [ 0 , r ] ) _> u  that  ,  t  : x ( t o ) ( [ 0 , t ] ) < u}  [0,~]  e F  < t}}  F -measurable.  is  the  i ( t o ) ( [ 0 , r ] ) < u}  F^-measurable,  is  (G ) - s t o p p i n g  9.19.  let  .  sup e  [0,°°) :  .  {T  < t}  e F^®  A  .  Thus  be  a measurable  i s a weak*-dense s u b s e t  dual  E  T  time.  (A,F,Q)  making a l l the  0 =  That i s ,  space  and  let  E  be  a  separable  of  the  unit b a l l  of  the  * topological  .  i f f there  space. Z  )  rational  x(»)([0,r]) a  we  i s right-continuous i n  r  is  A  : T(»,x) < t})  b)  As  in  l e f t - c o n t i n u o u s map  that  { T ( « , x ) < t} =  c o n c l u s i o n s now  {T  to  (G^-stopping  desired  As  (F^-stopping  i s a randomized  { T < t } € ' F ® A .  a)  Hence  T  increasing  [0,°°]  is a  If  i f f o r each  unique  into  .  of  elements  E of  then Z  the  smallest  measurable  o-field  coincides with  on the  E  0.)  - 112 a-field  generated  Thus function  there from  by  the norm-open s u b s e t s  i s no A  into  of a l l functions  f(w)||Q(du.) <  ambiguity  f  E  : A  We  We  wish  to describe  Now  E  write  E  such that  f o r the space  X  the  f  F-measurable  mean by a n  L (A,F,Q;E)  .  f •* | |f | | =  E .  a b o u t what we  , equipped with  0 0  of  F-measurable  is  and  seminorm  | | | f (oi) | |Q(d_) .  the dual  n e e d n o t be  of this  separable,  space. and  E  may  n o t be  the  dual  * of  E  , s o we  have  t o be  careful  a b o u t what we  mean b y a m e a s u r a b l e  * E -valued We iff using  function.  shall  say that  _ H- <x,g((~)>  a function F-measurable  is  the s e p a r a b i l i t y  function  g  of  u> •->• | |g(aj)| |  E  , we  : A -»• E f o r each  is  weak*-F-measurable  x e E  c a n show t h a t  the  .  In this  case,  numerical  F-measurable.  is  * Also,  i f  f  F-measurable,  : A -> E  then  _ •->• <f ( _ ) ,  p r o v e d by a p p r o x i m a t i n g functions. {g 4 g'} defines formula  e F  Also,  i f  , since  a continuous  <f,r> =  Similarly  g'  F-measurable  is  f  g, g'  by  g(t_)>  <f(_),  defines  g  : A ->• E  : A  E  F-measurable.  is This  a r e weak*-F-measurable  to  {||g T  functional  - g ' | | = 0} on  .  Now  L (A,F,Q;E) X  then , g by  g(_)>Q(d_)  T'  .  If  Q  can be  F-measurable  countably-valued  i t i s equal linear  is  and  i s semifinite,  then  the  -  ||r|| = Q({g  the  t g'))  Q-essential = 0  .  On  continuous  113  supremum o f  i s any  is  a weak*-F-measurable  linear  functional  function  g  ||g(w)|| 1 I M l <f,T>  We  shall  Ionescu which  f  =  <f(„),  not  Tulcea E  i n which  E  9.20.  s o one  if  function  on  :A  l?~(1\,T,Q;E) E  such  separable. assumes  f  here.  f o r a proof  Q  See  Y  X  and  Y  respectively,  * Y  a  Let  K  (A,^,Q)  C(K),  , equipped  Thus  C(K)  with  1  Tulcea  and  In  this  t o be  case  the  complete.  are sets then  lifting I n our  ( L e t us  s e p a r a b l e , the  in  case,  also  F-measurable range.)  and  f, g  f ® g  will  are  real-valued  denote  the  d e f i n e d by  compact m e t r i z a b l e Then  there  i n t h e more g e n e r a l c a s e  (f®g)(x,y) = f ( x ) g ( y )  9.21.  then  _ (A,F,Q;E)  Ionescu  are d e f i n e d to have s e p a r a b l e  If  and i f  that  in  i s unnecessary.  assumed  i f f  w in A  a 1 1  for a l l  result  r = I"  c-finite  E  X, X  on  is  i s not  Notation: on  Q  E  into  functions  this  i s separable, this  remark t h a t functions  prove  assumed  theorem i s used,  f o r  g(_)>Q(d_)  [1], VII.4,  i s not  ||g(»)|| , and  the o t h e r hand, i f  r  and  -  the the  be  a  o-finite  measure  .  space,  and  let  K  be  space. space  of continuous  real-valued  functions  supremum norm, i s a s e p a r a b l e normed  can p l a y the r o l e  of  E  in  9.19.  on  space.  -  If  X  X(co,»)  i s a real-valued  i s continuous  F-measurable dory  f o r each  f u n c t i o n ) then  to •-»• X(to,»)  on  X  i s an  for  then  in is  each  linear  p  in  K  functionals  Thus  f  , since on  l?~(A,F,Q;C(K)) X  f e L (A F Q) 1  }  J  A  and  that X(»,p)  i s called  C(K)-valued i s an  a  is  Caratheo-  and hence  function.  F-measurable  C(K)-valued  F-measurable r e a l - v a l u e d  f u n c t i o n s on  f u n c t i o n s on  A  x  K  .  A  function  c a n be  Under  becomes i d e n t i f i e d w i t h  on  sup  If  X  such  (Borel K)-measurable  C(K)-valued  identification,  '  in  K  C(K) .  Caratheodory  functions  to  x  the p o i n t - e v a l u a t i o n s a r e continuous  with  Caratheodory  A  (such an  F ®  i s an  F-measurable  identified  K  F-measurable  f(»)(p)  on  f o r each  On t h e o t h e r h a n d , i f function  function  K  p  114 -  A x K  this  the s e t of  satisfying  |x(»,p) |dQ < °° .  peK  and  g e C(K)  then  f o g  belongs  to this  set. Let in  K  RRV(A,F,Q;K)  over  be t h e s p a c e  (A,F) , equipped  with  o f randomized  random  t h e weak t o p o l o g y  variables  induced  by the  maps  Q(dto)  where  h  denote  this  If  ranges  x  e  over  space R  R  V  L (A,F,Q;C(K)) 1  by j u s t  > then  RRV,  X  (to)(dp)h(to)(p)  .  To s a v e w r i t i n g ,  f o r the remainder  h >->- <x>h>  l e t us  o f 9.21.  i s a continuous  positive  linear  -  functional f  L. (A,F,Q;C(K))  on  1  e L^"(A,F,Q)  properties  .  On  arises  Thus  {<X» 1  ball  Alaoglu  follows  linear  X •->• <X>h> is  elements  o f RRV  - o -  are  field  for a l l  these  b e l o n g i n g to  subset of the  RRV.  unit  the  Banach-  converges  RRV  Also,  i f  H  has  t h e maps o f t h e  t o p o l o g y o f RRV.  form  i s pseudo-metrizable  F  c o u n t a b l e s u b s e t o f RRV (A,F')  .  F'  where  I t follows  that  (and  two a.e.).  are a l l randomized  i s some c o u n t a b l y RRV  F  Hence i f  zero d i s t a n c e a p a r t i f f they are equal Q -  over of  Q,  space. then  the o r i g i n a l  mod  o f any  e v e n when i t i s n o t L e t us  i s a compact  induce  generated  the elements  9.22.  x  unique)  s o i s w e a k * - c o m p a c t by  L*~(A, F , Q ; C ( K ) )  in  (heH)  random v a r i a b l e s sub  functional with  i s a weak*-closed  , and  t h a t RRV  span  countably  Now  fdQ  theorem.  It dense  =  *  L (A,F,Q;C(K))  of  <x,f«l>  satisfying  (Q-essentially  X e RRV}  :  -  t h e o t h e r h a n d , any  from a  # >  115  generated  i s sequentially  compact,  pseudo-metrizable.  g i v e an e x a m p l e  to a randomized  o f a sequence  o f random v a r i a b l e s  random v a r i a b l e w h i c h  i s not a  which  random  variable. Let  A =  [0,1)  F = Borel  A F  Q = L e b e s g u e m e a s u r e on K For u j-1 j odd  =  [0,1]  .  i = 1,2,..., l e t  [ ^ r - , ^j) 2 2 1  1  .  X^  be  Then each  X. 1  the c h a r a c t e r i s t i c i s a rv i n  K  function  over  (A,F)  of ;  - 116 let  be t h e c o r r e s p o n d i n g r r v . Let  x  be t h e r r v i n  Then u s i n g the f a c t  9.23.  Now  here  The r e a d e r to  may  over  (A,F)  t h a t the bounded  continuous  i s a version of a result also  find  the review  d e f i n e d by  f u n c t i o n s are dense i n  due t o B a x t e r  of this  article,  and Chacon [ 2 ] .  by Meyer [ 4 ] ,  be e n l i g h t e n i n g .  Theorem.  Let  (A,F,F  right-continuous. randomized as  K  ,Q)  Let  be a f i l t e r e d  RST = R S T ( A , F , F Q )  RRV  times,  space,  with  be t h e s p a c e  t >  (F^)-stopping  a subspace of  measure  endowed w i t h  (F ) fc  of  the topology  i t inherits  = RRV(A,F,Q; [0,°°]) .  Then: a)  RST  b)  If  i s compact (x.)  (Z t'0<t< ) process  and s e q u e n t i a l l y  i s a net converging  CO  is  to  compact. x  i n RST, a n d i f  a real-valued cadlag quasi-left-continuous  satisfying  sup  |z | dQ <  00  0<t<°° then  <x. ,Z> -»• <x,Z>  ; thati s ,  Q(du>) x ( _ ) ( d t ) Z ( _ ) i  t  -  Partial a)  9.6  (F^)  RRV  then  T h u s RST is. We  remark  on  For  from  the the  a  t h a t the  general  rather  arbitrary  S i n c e we refer  (To  fact  see  reader  For  [0,°°)  satisfying  let  F  u  M be  map  then  T  the  of  ^'  s  T  such  just  r  t|  a)  c l o s u r e of x  that  =  of RST  t Q -  a.e.  RRV  s  for "continu a droit  topology  (Z^~) a  left  o f RRV,  r  stopping  i s merely randomized  e  random v a r i a b l e s  continuous  [2] o r M e y e r  M  denote  the  , endowed w i t h  stopping [O, ] 00  f u n c t i o n of  (t) = y ( [ 0 , t ] )  map  of  M  y  times, , is  case.) we  [4].  o f measures i t s usual  For  not  times.  processes,  Chacon  space  does  quasi-left  in  and  1  avec limits).  and  c o n s i d e r the non-randomized  y({0,°°)) £  i s a 1-1  with  randomized  e  only with  distribution  F  (b =>  compact b e c a u s e  i s a compact m e t r i z a b l e s p a c e .  y — *  the  continuous  the a  ^'  randomized  moment, l e t  the  apply  desired conclusion follows  i n which  concerned  F  The  (right  t h a t the  to Baxter  9.24.  topology.  the  case,  this,  s h a l l be  the  gauche"  definition  the  crucial.  to  sequentially  that "cadlag" stands  continuous, than  i n RST  can  9.21).  the  fact  belongs  T  and  i s continuous  t  immediately  there i s a  limites  (Z )  x  that i f  i s compact  (See  des  depend  i s r i g h t - c o n t i n u o u s , we  to conclude  in  If  -  Proof: Since  b)  117  each  y  on  vague y  in  M  :  (0<t<»)  .  onto  the  set of  increasing  ,  -  r i g h t - c o n t i n u o u s maps o f sequence  for  in  each  M  t  most any  and  in  standard  this  result  each  y e M  we  then  [0,1]  y . ->• y i F^  U  the  in  If  M  (P^)  i f f  i  F  s  a  ( t ) -> F  u  One  can  sequences.  collection  as  i s shown i n show  Indeed,  of sets of  (t) y  ±  theory.  as w e l l as  be  .  i s continuous,  on p r o b a b i l i t y  f o r nets  let  -  into  00  at which  text  holds  [O, )  y e M  [0,°°)  118  that  i f for  the  form  y {v  where  : |F  e > 0  then  , k  (U ) y yeM  which  on  M  (Note  i  e IN  v  i  , and  which  t h u s , by  .  (t )-F (t )|  constitutes  w  topology  M  e M  < e  t-,...,t, O k  do  i s weaker  not  i = 0,...,k}  are  continuity  than  p o i n t s of  F  , y  a neighbourhood base  c o m p a c t n e s s , must be  t h a t we  for  the  given  identical  c l a i m t h a t the  for a  Hausdorff  topology to  the  of  M  given  elements of  U  and  topology are  open  of —  y only  that they Now  of  let  [0,°°)  [0,1] into  H  neighbourhoods  be  into  by  the  [0,°°] .  means o f a n  in  H  for  each  and t  in  continuous  (h^)  is a  pointwise [0,°°) [0,°°]  .  h  Since  almost  space  , then  [0,°°)  sequence  f  may  r i g h t - c o n t i n u o u s maps  be  identified  at in  i n a n a t u r a l way.  h  -»• h  in  h  countably  many  H  Lebesgue  <j>  H  i f f  to  t's  in  h  e H  with may  If  is a  any  [0,°°) .  h  bounded  h^  continuous  i n t e g r a b l e f u n c t i o n on  [O, ) 00  e  H  Thus i f h  r e s p e c t to Lebesgue measure i s any  made  h(t)  Now  then  be  (IK)  h^t)  i s continuous.  converging  everywhere w i t h  any  .)  [0,°°]  at which  It follows that i f  and  y  o r d e r - p r e s e r v i n g homeomorphism, H  e H  except  of  s e t of a l l i n c r e a s i n g  a compact m e t r i z a b l e  net  is  are  on  f u n c t i o n on then  -  f(t)4>(h(t))dt  Using for  t e  f u n c t i o n on continuous  [O, ) H  It H  let the  functions  from t h i s  be the  form  a-field  {h e H (H^)  Definition.  let  H  H  a map  and  H  T : A  map  c a n deduce from  h i—> h(t)  is a  Borel  o f a sequence  H  of  evaluation  i s the smallest  maps m e a s u r a b l e . H  o-field  F o r each  t e  [0,°°]  g e n e r a t e d by the s e t s o f  a e [0,°°)  family  that  H .  H = Borel  where  H  this  c o u n t a b l y many o f t h e s e of  on  and  E e Borel  [0,t] .  of countably generated  sub-o-  (A,F,F  such  )  be a f i l t e r e d  A right-continuous that  f o r each  a e  measurable (F )-time  space  change  [0,») , T(»)(a)  is  time.  that t e  function  = H .  H  (F^)-stopping  fora l l  we  of subsets of  Let  T : A ->• H [0,»)  and a l l  i s a right-continuous A e H  , T  _ 1  [A]  e F  fc  (F )-time .  (Here  H  change fc  a s i n 9.24.)  9.26.  Proposition.  where  (F )  X  that  b e a s i n 9.24.  Observe  is  Now  i s an i n c r e a s i n g  9.25.  iff  .  : h ( a ) e E}  of  an"  H  to s e p a r a t e the p o i n t s  fields  is  on  making a l l the e v a l u a t i o n H  h's  ; indeed i t i s the pointwise l i m i t  follows  Clearly  and  o f the  , the e v a l u a t i o n  00  maps s u f f i c e  on  i s a continuous real-valued  the r i g h t - c o n t i n u i t y  any  119 -  Let  (A,F,F  i s right-continuous.  be a r r v i n  H  over  M = A x H G = F ® H  (A,F) .  ,Q)  be a f i l t e r e d  Let  H, H,  Also, l e t  measure space,  b e a s i n 9.24.  '  Let  -  120 -  (G ) = ( ( F ® H ) ) t  t  t  +  Q(d_)x(u)(G(u)))  R(G) =  ij. = p r o j e c t i o n o f Then  the f o l l o w i n g a)  (M,G,G  b)  For a l l  are  c)  i s an o p t i o n a l  t e  Whenever  [0,»)  mod  , x(*)(A)  and a l l A e H  t e [0,°°)  i  = 0,...,k  (A,F,F._,Q)  enlargement of  F -  is  f c  Q .  (f> e C([0,°°)) i  A .  on  equivalent:  ,R,iJ.)  measurable  M  G e G  for  , k e ]N , a , . . . , a Q  $  such that  , g  on  i s the function  on  [0,°°)  [t,°°) H  , e £ (0,°°) ,  for  defined  by  a.+e l  k g(h) =  = 0  ±  e  k  n F  <j>. ( h ( s ) ) d s ]  i=0 *  ,  U. I  f £ L (A,F,Q)  and  such  fdQ = 0  that  for a l l  F £  F  ,  then  Q(dui) x ( t o ) ( d h ) f ( u ) g ( h ) = 0  Proof: for  First  a l l  note  t , and  a ) => b ) = X(«)(A)  that  is  (G, F ) - m e a s u r a b l e ,  (G^, F ^ ) - m e a s u r a b l e  ^(R) = Q .  If  c)  A eH  F o r each  H^-measurable g  ij.  then  A x A e G  fc  .  Also,  E (_  A><A  | i j ; , F)  Q - a.e.  b ) => is  that  is  H  (indeed  i , H  -measurable.  = 0  on  [t,°°)  -measurable) Thus  so  f o r each  X(»)(dh)g(h)  h  <—*•  s  .  i(;^(h(s)) It  i s equal  follows Q - a.e.  - 121 F^-measurable f u n c t i o n .  to an c) can  => b )  conclude  measurable t €  , k eU  00  on  for  0 0  II  of  i s  (F )  is  t e  i f  still  i f B  f o r any  Now h e r e  Theorem.  Let  i s only  be a s i n 9.26.  (M,G,G »R,i|0 t  given  argument, is  (^ )  in  = E(1  i s a version  (A»F,F ,Q)  , and  Q .  E  >0  that  From  this  then  the r i g h t - c o n t i n u i t y  G  one f i n d s  that i f  F ^ - m e a s u r a b l e mod Q .  one f i n d s  that  this  conclusion  .  Let R  B  U,F)  Q - a.e.  o f a r e s u l t due t o B a x t e r  be a f i l t e r e d  t  Let  such  G,  B e  right-continuous.  whenever  e C([0,°°))  Then u s i n g  x (w) (B (co))  X(0(B(.))  9.27.  -  conclusion.  By a monotone c l a s s  true  that  F  t h e map  [0,°°) , A e H  the d e s i r e d  co I—•  \Ji  find  F - m e a s u r a b l e mod  +  then  i n c ) we  *  1  the r i g h t - c o n t i n u i t y of  But  is  is  (co)(dh)g(h) = 0  described  we  and  , then  F ^ ^ - m e a s u r a b l e mod Q .  b ) => a )  Using  that  e -+ 0)  [0,°°)  x  Q - a.e. to an  i s equal  i = 0,...,k  1  we o b t a i n  B e F^ ® H  o f the sort  (letting  i(j.(h(a.))  i=0  x(«)(A)  Then  , a^...,^ e  [t, )  can conclude  g  x(*)(dh)g(h)  that  x(co)(dh)  we  f o r any  function.  tO, )  \\>. = 0  First,  Q(dto)f (to)  Hence  H, H,  H  measure  space, where  )  b e a s i n 9.24, and l e t M, G,  be t h e s e t o f m e a s u r e s  i s an o p t i o n a l enlargement o f  t h e weak t o p o l o g y  and C h a c o n [ 3 ]  R  (A,F,F  on  ,Q)  i n d u c e d b y t h e maps o f t h e f o r m  G  such  .  G^  that  Let R  be  -  122 -  f(u)(h)R(d_,dh)  where  f e L (A,F,Q;C(H)) X  .  Then  R  i s compact  and s e q u e n t i a l l y  compact.  Proof: space is  L e t RRV d e n o t e t h e c o m p a c t , RRV(A,F,Q;H)  compact Let  A  is  Q  and S  , as d i s c u s s e d  be t h e s e t o f m e a s u r e s  d> : RRV -»- S  then  <Kx)  Thus  d>  R £ S the by  S  be t o p o l o g i z e d  be d e f i n e d  on  Mx')  maps  i f f X  closed  and, s i n c e  proof,  =  x'  subsets  < | >  i t suffices  i s onto  Q - a.e.  G  S  .  to  If  Clearly  H  on  R . Let  X  d) [ R ] X  x>  x'  S  i s Hausdorff.  e  RRV  subsets o f  , R = <f>[d> [ R ] ] .  Thus  S  .  Now  t o complete  i s closed  i n RRV.  But  (a<=>b) o f 9.26,  :  for a l l t e x(*)(A)  this  set i s closed  i n RRV, b y  of the form considered i n 9.24.)  [0,°°)  i s equal  F -measurable t  out  whose p r o j e c t i o n  o f RRV o n t o c l o s e d  t o show t h a t  1  g  since  Q(d_)x(o>) (G(oj))  4'~ [R] = " X e RRV  and  c a n do t h i s ,  analogously  i s a c o n t i n u o u s map o f RRV o n t o =  We  topological  by  *(X)(G) =  d>  i n 9.21.  compact,  metrizable.  , and l e t S  Then  sequentially  and a l l  A" e H  .  Q - a.e. t o an  function,  (b<=>c) o f 9.26.  i n 9.26(c) a r e c o n t i n u o u s  (The f u n c t i o n s on  H  , as  pointed  - 123 Remark:  In view of (a<=>b)  the d e f i n i t i o n i n 9.24, R right-continuous  of 9.25, and the observation following  may be regarded as the set of "randomized  (F^)-time changes" (where we identify pairs of time  changes which are equal Q - a.e.). right-continuous converging continuous  Thus from any sequence of  (F^)-time changes, we may extract a subsequence  ( i n the sense explained above) to a randomized r i g h t (F^)-time change.  -  10.  EMBEDDING MEASURES  124 -  I N BROWNIAN MOTION  I N A GREEN REGION, USING  RANDOMIZED STOPPING T I M E S .  10.1.  Throughout  Green  function If  time,  y  P  = V-  T  y_  will  i f  denote  D  will  will  be  D  and  be a Green  inf{t T  arises  the measure on B o r e l D  from a genuine  in R  with  n  I D} .  i s a randomized  P  T  region  >_ 0 :  P ( d u . ) x ( „ ) ( { t e [0,R(o))) : B  (8 )-stopping t  d e f i n e d by  (ID) e A } )  (B^-stopping  time  T  then  where  y_(A) More and  R  i s a measure i n  then  that  section,  G , and  V_(A) =  Note  this  i f  generally,  T  product  arises  = P ( B _ e A, T < R) . M  i f  (X,A,m)  i s a measure space w i t h  as i n 9.18(a) f r o m a s t o p p i n g time of  (ft,B,B ,P )  V (A) =  m(dx)P (B  enlargement  by  y  t  (X,A,m)  T  m(X) = 1 , over the  , then f o r  A £ Borel D ,  m(dx)y_ . (  s £ A  ,  y  T  ) X )  , T(',x)  < R)  (A)  X 10.2. in  Notation.  [0, ] 0 0  over  (A,F)  If  ( A , F ) , we w r i t e  a(„) ( ( t , ] ) 0 0  t  e  i s a measurable a <_ x  <_ x ( _ ) ( ( t , » ] )  space  and  a, x  are rrv's  t o mean  fora l l  u> £ A  and a l l  [0,«) •  10.3.  Lemma.  Let  p  be a measure i n  D  such  that  Gy  is a  potential.  - 125 Let  a, T  Gy > Gy —  be randomized > Gy  0 —  .  t  (It follows that  y_  T  Let  and  O  compact subsets of Proof:  (B )-stopping times such that y  a <_ x .  Then  are f i n i t e on T  D .)  A = Borel (0,1)  S, T  and l e t  be the  ((8  ® A) )+  stopping times associated with Then  S < T . Also —  Moreover,  S(-,u)  0 < u < 1 .  10.4.  Now  Lemma.  potential.  and  y  Suppose y  t  .  Let  anc  u  T(-,u)  app ly  Let  RST(fi,B,B ,P ) a)  a, T by 9.18(b). '1 du Vg^. y * s i m i l a r l y for y = o 0 are  y^  (8 )-stopping times for t  7.4.  •  be a measure i n (T )  D  such that  i s a net converging to  v = y  , y  T  ±  = y_ x  x  Gy  is a  in  Then:  For any compactly supported continuous function  <f>  in  D ,  <j> i n  D ,  4>Gv  b)  For any compactly supported continuous function 4>dv  <(>dy ,  Proof: a)  Let  function i n  <J> be a non-negative compactly supported continuous D , and l e t  (Zj) t 0<t<°° n  process defined by  Z  <t>(B)ds  t =  s  t A R  be the non-negative decreasing  -  Then  E  that  E (Z )  for  <()Gy  (ZQ) =  i s finite.  Y  Q  any  co  Thus  f o r which <T  randomized  b y 6.9.  , Z>  <o,  Since  Now  Gy  t  ZQ(CO)  i s a potential,  i t follows  i s continuous  Z^(IXI)  on  [0,°°]  i s finite.  + <T, Z>.  (B^)-stopping  126 -  ( S e e 9.23.)  time  But i f c  i s any  then  P (dco) a(co)(dt)Z (co) y  Z>  t  du  P (dco) y  Z., v (co) S(co,u)  *(B )ds]  du E [ V  g  S(-,u)AR  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  property)  4>Gy  o  where a  S  i s the  ,  ((8^ ® Borel  (0,1)) )-stopping +  time  associated to  by 9.18(b) . b)  From a ) ,  i tfollows  that  the conclusion  o f b) h o l d s  i f <f)  2 is all  a compactly  supported  i , by 10.3.  C  function  T h u s , b y 7.8,  an  of  D  approximation  The Our  K  next  method  .  We  D  s u p y . (K) i  subset  i n  can therefore  . <  0 0  But  Gy^ <_ Gy  for  f o r any compact  1  complete  the proof  o f b) by  argument.  result  of proof,  i s a particular  case  w h i c h was s u g g e s t e d  of a result  due t o R o s t [ 1 ]  t o me b y R. V . C h a c o n , i s  -  quite  different  10.5.  Theorem.  a potential.  from  Rost's.  Let  y, v  be measures i n  D  , and suppose  Gy  is  Then t h e 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 y  127 -  exists  a randomized  ( 8 ) - s t o p p i n g time  T  t  such  that  = v . x  Proof: b)  => a ) .  See 1 0 . 3 .  a ) => b ) . — in  D  such  F o r each  that  natural  Gv. = i A G v l  f o r each  T  i .  be t h e measure  (8^)-stopping  times  T_^  such  By 9.23, t h e r e i s a r a n d o m i z e d  that  (8 )-stopping t  i  1  time  T  such  that  RST(fi,8,8 ,P ) V  t  10.6.  .  some s u b s e q u e n c e Now  Corollary:  potential  and  Then  Gv  Let  v ( Z ) <_ y ( Z )  (T^)  converges  to  H e n c e , b y 10.4, y ^ = v  be m e a s u r e s  f o r every Borel  Let  be t h e  by  .  of  in  D  such  T  in  .  that  ^  Gy  is a  Gv <_ Gy .  By 1 0 . 5 , v = y ^  T  + Gv  ±  y,v  Proof:  T  i , l e t v. l  .  T h e n , b y 7.11, t h e r e a r e v. = y  number  ((B  fc  polar  subset  f o r some r a n d o m i z e d  ® Borel  of  D .  ( 8 ) - s t o p p i n g time t  (0,1)) )-stopping +  Z  time a s s o c i a t e d  T .  to  9.18(b). (-1 Then  of  D  then  v =  y  J  du y Q  ,  , . T(-,u) x  , (Z) < _ y ( Z )  Now  i f  Z  f o r each  i s any B o r e l  u  in  polar  (0,1) , by  subset  7.3(b);  -  hence  v ( Z ) <_ y ( Z ) .  10.7.  The p r o o f  large how  o f 10.6  t h a t we  f r a c t i o n o f what h a s  this  have j u s t  gone b e f o r e .  given  depends on a  Therefore,  l e t us  using  only  of a l l there  subsets  of  A  2.2, we  have only  Z £  = °°} .  measures  a, B  i s a Borel  and  v  to prove  Let in  A £ D  on B o r e l  that  H = A D  u  set  subsets  v ( Z ) <_ y ( Z )  n {Gv = °°} .  such  such  u  < v  rl  Now  that  a + u  = u  v < u —  on B o r e l  subsets  c e r t a i n l y s u f f i c e s t o show  GB  <. Ga  and  Now  a(H) = 0  = 0  by  Now  Borel set  are unique  , 3 + u  = v  .  (Note  rl  i s polar.  {Gv  = <*>}\H , a n d  8(H) = 0  .  u  Observe  n  <_ u  , so  that  .  (H  i s a polar  s e t , so  p o t e n t i a l theory,  fringe(H,D)  bal(Ga,H,D) i s  r e g u l a r i z a t i o n of  inf{bal(Ga,U,D) H  .  .  a( fringe(H,D))  to the lower  of  that  Hence, by a theorem o f c l a s s i c a l  But  on  D\A  f o r every  Then t h e r e  v _> u  .)  it  equal  indicate  classical  that  of  n  that  rather  theory.  First  {Gv  -  r e s u l t c a n b e p r o v e d more d i r e c t l y ,  potential  Borel  128  Thus  this  : H £ U o p e n £ D} infimum  i s equal  . to zero  except  on a  polar set. Now  i f  H £ U open £ D  then  U £ base(U,D)  so  3  lives  on  n  base(U.D)  .  Hence  b a l ( G a , U , D ) >_ G 3  U  by  the domination p r i n c i p l e  H 4.1. is  Taking  a polar  the infimum over  s e t , w h e n c e i t must  a l l such be empty.  U  , we Thus  find 3(H)  that = 0 .  {^8^  >  ^  -  10.8. it  Now  we  completes  establish  Let  superharmonic  functions  Let  W  o f (b =>  set  a) o f 4.2  be an open s u b s e t in  E = b a s e ( { u J> v } )  polar  -  the f o l l o w i n g improvement  the proofs  Proposition:  129  W  n W  of  with Riesz  .  Then  o f 10.6.  and  ( a =>  H  and  n  b) o f  let  measures  v ( Z ) <_ y ( Z )  Note 4.8.  u, v  y, v  that  be  respectively,  f o r every  Borel  Z £ E .  Proof: C a s e 1. in  W  Now  .  Let  If  so  x >  F,W)  Thus .  But  of  Let  the  relatively  such  u, v that  there  by  F  so  Borel  polar  subset  V  a r e bounded and  v+c  of  E  subsets  itself  in  subsets  set  g  , and below  .  and V  in  E  by  W  V  are non-negative  polar set  .  Hence  Z  of the p r o p o s i t i o n . i s a countable  compact  so there in  E  2.2.  , so i t s u f f i c e s  such  on  .  that  is relatively V  principle  is finite  of  .  2.15.  >_ b a l ( v , F , W )  Then  i s relatively U  , g = bal(v,F,W)  f o r every Borel  Z c W  of  are p o t e n t i a l s  , by  be a s i n t h e s t a t e m e n t  Borel  Z  for  G__6 w x  bal(u,F,W)  polar  u, v  the domination  then  g(Z) = v(Z)  a r e open s e t s  compact  u+c  x e W\E  and  u, v  compact  case i n which Then  Then  W,  be a B o r e l  relatively  , and s i m i l a r l y  i f  on  and  a = bal(y,F,W)  does n o t charge p o l a r  f o r every  2.  let  x  hand,  u •> v  _> g ( Z )  Z  , and  bal(<!> ,F,W) = 6^  ot(Z) = y ( Z )  Case Let  then  the other  bal(6  a(Z)  i s a Green r e g i o n  F = {u >_ v}  x e E  On  Z £ E  W  y(dx)bal(<5_,F,W) x  a =  4.1.  Assume  in  compact  .  consider  .  Z £ u  i s a real V  W  to  union  in  , U W  number  is . c  -  Let  f = b a l ( u + c , U,  f  and  g  y  and  coincide with indeed  f = u+c  Now x e U {f at  n U  i s thin  Remark.  (ft,B)  of the  to  U  Let  and of  f T  T  n U  at  i f f .  T  be  whose R i e s z  U  by  relative  {u >_ v}  U  subsets of  U ;  .  Thus,  to  measures  to  i f f  i s thin  T  and  V  3.10,  x  .  by  n U  i s thin  Hence  case  in  i f  i f f  {u >_ v}  at  y ( Z ) _> v ( Z )  the s e t of r r v ' s  3.9  1.  [0,°°]  over  form  ranges over  Q  +  (1-f(w))«  B^-measurable  ranges over non-randomized are randomized  "randomized  in  x  Therefore  V  on B o r e l  .  x relative  T(_) = f ( _ ) 6  where  g = v+c  i s thin  Z c b a s e ( { f _> g}, V)  10.9.  and  at  in  respectively  n U = {u _> v}  { f >_ g}  relative  v U  then  >_ g} x  { f _> g}  .  are potentials  in  -  V)  g = b a l ( v + c , U, V)  Then  130  only  (B^)-stopping  t  0"  .  We  .  [0,1]-valued functions  (B )-stopping  a t time  T ( a ) )  times.  t i m e s w h i c h we  claim  that  might  on  The  element  say  are  the f o l l o w i n g  are  equivalent: a)  The  conjecture  b)  Whenever  u, v  potential  and  7.12  is valid  f o r the Green  are measures i n Gy  >_ Gv  , there  D  region  such that  exists  T e T  „  D  .  Gy  is a  such  that  -  131  -  Proof:  a)  =>  b) :  Let  1)1  be  a  v(Z)  =  H^\E  is  v  f o r every  let  f =  <}>(B  0  <_ y Borel  Borel  )  on  .  Borel  subsets  function  set  Z £  Then  f  on  E  E  .  is  of  E  such  Extend  = {Gv  =  °°}  that <|>  t o be  B -measurable; u  0  on  indeed,  f  B^-measurable. Now  6 +  10.6,  [0,l]-valued  <f> dy  , and  o  By  v  2.2,  there = v  E g  .  are We  does not  conjecture  7.12,  unique measures  have  Ga  _> Gg  charge p o l a r there  is a  8  a,  , and  in  g({Gg  sets.  But  D  such  =«=})=  that  0  .  Hence,  then according  (8 )-stopping  time  t  T  a +  to  such  the  that  T Let  If  T(U>)  A  =  f(u>)6 + 0  i s any  y (A) x  (1-f(u))6_ , , 1 (.to;  n  Borel  subset  (to)l (B (to))l  y  A  +  Thus  y  = T  v  +  a (A) T  [0,R(to))  { ( ) < R }  +  =  Then  : B (u) t  T  { T < R }  P (da )l (B (to))l J  e  A})  (to)]  a  v(A)  T e  (to)  A  A  £  0  .  then  (l-f(to))l (B (to))l  y(dx)<j>(x)l (x)  v (A)  D  e  y  P (dto) [f  =  of  P (dto)x(to)({t  =  (to e Q)  /  A  T  { T < R }  (to)  T  v by  = y  ,  -  b)  =>  a):  potential,  Let  Gy _> Gv  v(Z) = y(ZnA) Let  (B^-stopping  v" = y" charge such  , and  such time  E = {Gy=°°} .  Then  polar  that Now  there  that  T  Gv'  sets.  y_j,, = v ' f o r any  polar y_  ,y'  = y_^_  according  y",, = v "  .  T  By  such set  Z £ D  = v  <_ Gy"  A £ D  there  do  is a  such  that  randomized  here.)  , v' = y^  .  Gy  .  will  , y " = y_  7.11,  that  ( A c t u a l l y , any  Gv"  Also  , and l e t  y'  does  (B^-stopping  is a  not time  T'  .  Borel  polar  set  Z £ D  v"(Z)  = v(Z) = y(ZnA) = = y"(ZnA)  Thus,  . y_  and by  sets  = v  that  _< Gy'  D  exists a Borel  such  Thus,  -  be measures i n  for a l l Borel  T e T  Let  y, v  132  y(ZnAnE)  .  to the h y p o t h e s i s the d e f i n i t i o n  ,  of  of b ) , there T  , there  exists  is a  T" e T  such  that  8 ..-measurable U  [0,l]-valued  function  T(_) = f(u)6Q + assume  that  Let  and  A  on  ° B  Q  .  cu  in  time .  T"  such  that  C l e a r l y we  may  . Borel  set  H £ D  ,  = y"(HnA)  H}  = y(HnAnE) = v(H  E)  = v"(HnE) =  for a l l  T h e n f o r any  y  e  t  {T = 0}  g dP " {BQ  (B )-stopping  a  (1-f(_))6_„^^  f = 1  g = _  f  y"(HnE) T P "(d„)[f(_)l  (since  v'  does not  charge  polar  sets)  (03)]  V  { B o  £  H  n  E  j  Q  <  R }  (_)  +  <l-f(o>»l _ { B  e  HnE,  T <  R}  dP  u  g  dP "  f {B  e  Q  follows  that  g  and  are both  Let  Then  T  10.10. reduce the  is a  Here  1  T =  <  0  on  {BQ  e E  n  A}  T"  on  {BQ  £ E \  A}  Q  u  -  B  in  Since  r  a.e.  E}  t i m e , and  o f 10.5  F  y  T  which  i s a point  = v  .  should  make i t p o s s i b l e  mass i n a t t e m p t i n g t o  to  prove  7.12.  Let  y, v  be  measures  i s a family  (v ) „ x xeD  Gy _> Gv  there  x e D  b)  For a l l  A e Borel  and  10.5,  v(A)  there  in  D  such  that  Gy  is a  .  For a l l  By  I  {B  a)  Proof:  g = f P  on  t  and  D  for a l l  M  T'  (B )-stopping  conjecture  Then  dP  0  to the c a s e where  potential  f  -  B -measurable,  i s a corollary  Proposition:  133  H}  It  f  -  =  , Gv  < x — D  G6  o f measures  in  D  such  that:  x  , x H- ^ ( A ) x  i s a Borel  function  in  time  such  y(dx)v (A) x  i s a randomized  (B^)-stopping  x  - 134 that  v  = v  .  By 9.15,  we c a n m o d i f y  T  on a s e t o f  P -measure y  T  0  so that  i t becomes a r a n d o m i z e d  Now l e t  v  = (5 ) X  10.11.  Here  G2U  Let  u, v  on  Let x  R  be measures  ±  stopping  = inf{t  . x  of  10.5.  >_ 0 : B  Thus  i n ]R  i n  b y 10.5,  Then  ( i = 1, 2)  ±  G^V 2. ^2  f o ra l l A  '  V  .  Now  .' I n p a r t i c u l a r , there  Green  u ( D 2 \ D ^ ) = 0 = vCD^XD^) ,  G^u >^ G^v .  I D }  t  with  n  - ^2 *  such that  T)^ , a n d  such that  G^ <_ G G^u  i s a randomized  i n Borel  2  is  (8 )t  ,  P ( d c o ) x ( c o ) ( { t e [0, R-^to)) : B (u>) £ A } )  v(A) =  Let  i n  in  time  time.  D .  Suppose  in  ; h e n c e G^u £ G^P  a potential  i n  be Green r e g i o n s  respectively.  i s a potential  Proof:  x  )-stopping  g  D^,  G^,  Let  f o r each  t +  X T  i s another a p p l i c a t i o n  Proposition: functions  (B  U  t  a ( u ) ( E ) = x (co) (En [ 0 ^ (co) ) ) + x (co) ( ( R ^ c o ) , »])6 (E) oo  E e Borel  [0,°°] .  Then  f o r any  t e  [0,°°]  and any  f o r co e n ,  co £ ft ,  a(co)([0,t]) = x(co)(H(co))  where  H = (ft * [0,t])  randomized  n [0,R) £ 8  (8^)-stopping  time  .  ® Borel If  [0,t] .  A £ Borel  Thus , then  v(A) = v(AnD ) x  P (dco)a(co) ( { t £ [0,R (w)) y  1  : B (u>) e A n t  D^)  a  isa  -  135  -  P ( d _ ) a ( _ ) ( { t e [0,R (_)) : _ ( _ ) e A}) y  t  P ( d _ ) a ( _ ) ( { t e [ 0 , R ( u ) ) : B (_) e A}) y  2  Thus  G v 2  £  G u 2  , by  10.3.  -  11.  I n 1 0 . 5 , we saw t h a t  i n n  such that  n  exists  a randomized  where  y  Gy  i f y, v  i s a potential,  (B._)-stopping  i s as d e f i n e d  i n 10.1.  y, v then  there exists  y_ = v  i f fU  In  latter  this  y  _> U  y  on B o r e l  Also,  i n 8.20, we saw t h a t i f  y_ = v ,  v  U  and  U  (8 )-stopping t  E  are potentials, time  T  such  that  .  In this  the analogue  o f the  c a s e , t h e example  given  9 shows t h a t we must u s e r a n d o m i z e d  E  this  section,  i s a measure on  E  i f T n  T  course i f  T  where  , then  y_  will  P (dco)x(u.) ( { t e [0,°°) : B ^ w ) y  arises y_  More g e n e r a l l y ,  from a genuine  i s as d e f i n e d i f  (X,A,m)  i n 9.18(a) from a s t o p p i n g time  (n,B,8 ,P ) y  t  i s a randomized  (B )-stopping t  denote  the measure  d e f i n e d by  n  P (A) =  of  that  y(R.) = v(lR) .  f o r measures on  Throughout  y_. = y _  such  i f fthere  times.  time and  as  and  V  Gy >^ Gv  region  T  that  y-standard  the b e g i n n i n g o f s e c t i o n  11.2.  Of  a  such  RANDOMIZED  time  s e c t i o n , we a r e g o i n g t o p r o v e 2  result  stopping  then  y  a r e measures on R  , USING  a r e measures i n a Green  T  at  2  EMBEDDING MEASURES I N BROWNIAN MOTION I N R STOPPING TIMES  11.1. D  136 -  by  (X,A,m)  m x  (  d  x  (B )-stopping t  y  time  T  then  i n 8.1. i s a measurable T  over  then  )  e A})  T ( . , x )  '  space and  the product  T  arises  enlargement  -  11.3.  Definition:  Let  y  137 -  be a measure  a potential.  Let  T  be a randomized  We  say  T  is  shall  on  y-standard  i f f whenever  (B . . - s t o p p i n g t i m e s s u c h t h a t  the  of  U  p  <_  here)  then  U  such that  n  (8^)-stopping  randomized meaning  ]R  and  P  p, 0  are  ( s e e 10.2 f o r  are potentials  0  i s  y  time.  p <_ a <_ T U  U  and  > U ° .  11.4.  D i s c u s s i o n o f 11.3:  If  n >_ 3  t h e n any  x  is  y - s t a n d a r d , by  10.3. Suppose by  4.8  n <_ 2  ( i f n=2)  more e x p l i c i t l y  and  x  is  o r 4.11 ( i f  y-standard.  n=l) .  Thus  Then "x  y ^ 0 R ) 2. 0 R ) n  is  P  y  u  n  - a.s. f i n i t e " ;  P ( { _ : X(CJ) ({«=}) 4 0}) = 0 . y  11.5.  a)  Let f  m  be Lebesgue measure  f(m)  maps o f on  [0,°°] .  a((t,»]) £  to  Clearly  for probability  [0,°°]  t h i s map  measures  8((t,°°])  b)  If  A  upper bound  l o w e r bound  i s a s e t and  measures  on  [0,°°]  i s an o r d e r  a, 6  on  fora l l t e  of probability  least  greatest  By 9.16, t h e map  by  left-continuous  onto the s e t of p r o b a b i l i t y  measures  G i v e n a n y two p r o b a b i l i t y their  (0,1) .  i s a 1-1 map o f t h e s e t o f i n c r e a s i n g (0,1)  ordering  on B o r e l  i s o m o r p h i s m , where  [0,°°] , we [0,°°) .  on  measures  say  a <_ 8 i f f  I t follows  [0,°°] a, 6  measures  that  i s a lattice on  i n t h i s o r d e r i n g by  [0,°°] , we  this  ordering. denote  a v B , and t h e i r  ot A 8 .  a, T  a r e maps f r o m  we d e f i n e  a A X  and  A  to p r o b a b i l i t y a v x  i n the obvious  -  pointwise  c)  d)  i s a measurable  (A,F)  over we  fashion.  (A,F)  If  then  F  take  = F  (A,F,F )  If  so a r e  a A T  i s a probability  a  A 6  +  a((t,°°])6  [0,°°]  If  t  f  and  Let  ) .  (x )  [O, ] 00  is  t  and  explicitly;  a A 6  t e [O, ] 00  t  Borel  then  = a , [0,t] r n  f u n c t i o n on  be a  <x(ds)f(sAt) .  a-finite  be a n e t c o n v e r g i n g t o be  F-measurable.  a . ->• a l Let  in  x  (Z )„  A  6  space.  RRV = R R V ( A , F , Q ; [ 0 , « ] )  a.(a))  = x.(co) A 5  ,  r  T(u)  b e a nJ y c o n t i n u o u s  t 0<_t<°°  sup  IZ  (to) I < »  0<t<°°  m  in  Let  satisfying  (Z ) t AT  measure  RRV  Q(dco)  Then  (F )  (F )-stopping  a r e randomized  a(co) = x(co)  (A,F)  s p a c e , where  i s any n o n - n e g a t i v e  (A,F,Q)  Let  T : A ->- [0,°°]  Proof:  (where  then  Lemma:  Then  [0,°°]  a v x , b y 9.18.  quite  t  Let  00  are rrv's i n  a v T , b y 9.18  T and  measurable  (aA6 ) (ds)f (s) =  11.6.  a, T  and  measure on  c a n be d e s c r i b e d  t  A  and i f a , x  then so a r e  a  If  O  f o r 0 <_ t <  right-continuous,  e)  space  i s a filtered  t  times  138 -  i s also  such  a process,  real-valued  process  over  Q(doj)  Hence  -  139 -  T.(_)(dt)Z  (_)  1  Q(du>)  That  tAT  x(_)(dt)Z  t A  _(_)  i s , by 11.5(e),  Q(dco)  a (_)(dt)Z (_) 1  t  o(_)(dt)Z (_)  Q(da»)  t  •  11.7.  Lemma: Let  Suppose  a)  b)  Let  (x )  (A,F,Q)  be a  be a n e t c o n v e r g i n g  ±  Q(d_)x  l i m sup t-*» i  to  x  measure i n  (_)((t,»]) = 0 .  space,  RRV = R R V ( A , F , Q ; [ 0 , » ] )  Then:  Q(d_)x.(_)({»}) = 0 .  If set  (Z )  i s any continuous  [0,°°)  (not  [0,°°])  Q(dui)  then  a-finite  Q(du)  sup 0<t<~  real-valued  such  process with  time  that  |Z ( u ) |  <  x. ( u O ( d t ) Z (ui) l t  Q(dco)  T(_)(dt)Z (_) t  Proof: a)  Let  f  be a s t r i c t l y  positive  F-measurable  function  on  A  - 140 -  f dQ  such that For each  t  in  [0,°°)  valued function on and  g  fc  = 1  Q(du)  on  [0,°°]  such that  g  = 0  [0,1]-  on [0,t]  00  T (u))(ds)f(o))g (s) i  t  t . But  lim sup t-*» i '  Q(dco)  Q(doj)  be a continuous  fc  [t+1, ] . Then  for each  lim  let g  +  Q(do))  T(u)(ds)f(a>)g (s) t  T_^((JJ) (ds)f (co)g (s) = 0 . Hence t  x(to)(ds)f(to)g (s) = 0 . That i s ,  Q(dto)f (to) T (to) ({<»}) = 0 . As  f  i s s t r i c t l y p o s i t i v e , a)  i s proved. For each  s e [0,°°)  function on h  s  = 0  Q(dto)  on  [s+1, <*>] . Then  x . ( t o ) ( d t ) h (t)Z. (to) X  for each  [0,°°]  let h be a continuous [0,l]-valued s such that h = 1 on [0,s] and  S  s . Letting  s  "tightness condition" on result.  T(t0)(dt)h (t)Z (t0) g  t  t  go to < * > , and using the (x ) , we obtain the desired  -  11.8.  Lemma:  such  that  and  xeT  i f  y  Proof:  11.9. such  that  U  sup  r-x»  T Tel  y  Proof:  A  | |x| |  :  that  r}) = 0  y  be a measure on  is a  potential.  X = (0,1) , A = B o r e l  time  Then  x  i s the  y  ( E ) = Q(B  n = 1  X , and  of m =  or  2)  (ft,8,B ,P ) y  t  Lebesgue  a  i f  e E  motion  a  R, S  x  t  by 9.18(b).  ° v) .  time a s s o c i a t e d f o r each B o r e l  a r e randomized a r e the  Then  process with  i s any randomized  , S < °°)  i f p,  to  time, and l e t T  be  Suppose  y-standard.  t , l e t B^ = B  (B^)-stopping  p <_ a , a n d i f  (B^-stopping  associated  is  Brownian  that  S  (where  be the p r o d u c t enlargement  be a r a n d o m i z e d  .  n IR  .  generalized  Also,  Suppose  T  Let  F o r each  Note  .  n  t h e p r o o f o f 8.10,  , where  x  0 0  n  ]R  y  (Bp-stopping  T dQ <  ({x e ]R  y  (ft' , B ' ,8J_,Q,if>)  measure on  a  lim  measure on  xeT  (X,A,m)  the  a <_ x  t  P ( d u ) ) x ( _ ) ( ( t , °°]) = 0  Lemma:  Let  (8^)-stopping times  ( B ) - s t o p p i n g times w i t h  a r e randomized  be a f i n i t e  Emulate  Let by  be a s e t o f r a n d o m i z e d  a e T .  l i m sup t-**>  T  a, x  then  Let  Then  Let  141 -  (ft' ,8' ,8|.,Bj.,Q)  initial  distribution  is y  (B ) - s t o p p i n g t i m e a n d i f to  a  by 9.18(b),  subset  (B^-stopping  (B|_)-stopping  E  of  ]R  n  then .  times such  that  times a s s o c i a t e d to  -  p,  a  by 9.18(b),  then  working  with  11.10.  Corollary:  a  R _< S (ft'  the process  Let  u  142 -  .  Now  emulate  ,8' ,BJ_,Bj_,Q)  t h e p r o o f o f 8.5,  instead  be a measure  on  ]R  n  (ft,8,B ,B ,P )  of  U  t  such  that  such  that  U  P  t  is  potential. Let  T  be a r a n d o m i z e d  ( B ^ J - s t o p p i n g time  P ( { u ) : x ( t o ) ( ( t , «>]) 4 0}) = 0  f o r some  y  is  t  in  [ 0 , °°) .  Then  x  y-standard.  Proof: Clear.  11.11. such  Theorem:  that  U  Let a,  B  •  be a measure  ]R  n  B  (x^)  x_^  a)  There  i s a measure  aCR )  = yQR )  is  y-standard.  t-x» f  , and  n  that  (where  ]R  U ,  U  a  U  Consider a  on U  1  ]R  a  >_ U  such  n  n = 1 o r 2)  that  whenever  a r e p o t e n t i a l s and  B  >_ U  x  6  in  RST(ft,B,B ,P t  the f o l l o w i n g  such  n  that  U  a  ) ,  statements:  i s a potential,  fora l l i .  01  P (dai)x.(co)((t,-])  l i m sup  c)  1R  i s a net converging to  each  n  such  f o r a l l y e T , then  U dy  Suppose  b)  on  be a s e t o f good measures on  a r e measures on  U dy >  and  y  i s a potential.  P  T  Let  = 0 .  M  i dy  x. I  +  f  dy^  f o r e v e r y bounded  continuous  function  y d)  U  T  < i s a p o t e n t i a l and  T  U  i  dy ->  U  dy  for a l l  on  -  x  e)  is  Then  Proof:  a) a  times  a)  =>  =>  b).  such  b)  =>  c ) , and  Let  S  be  a <_ x_^  that  a)  is a potential  y-standard.  The b)  Hence  statement  =>  c).  P (ft) = y(R ) y  the  d)  =>  set of  f o r some  =>  sup  r-x»  aeS  b)  e).  randomized  i  .  d).  Since  _. U  d)  implies  ({x  e lR  :  n  with  a)  =>  (Z ) =  c ) , we  i  | |x| | >_ r } )  =>  s  in  as  i n the proof  of  s  a  = 0  r  e  , by  11.8. (f(B..))  is a potential  y  '  T  and  n  have t h a t  .  (Note  <_ 2  y  (a =>  b)  of  8.11,  that  .)  -»• y l  statement  time  e S  x .  Then, r e a s o n i n g  each  t  a  Then  —  the  (B )-stopping  , s i n c e the  f o l l o w s by  11.7,  U  U y  then  as  0 0  and  lim  Apply  <  n  a)  =>  v  U  8.9.  -  y-standard.  y that  143  weakly. x  we  can  deduce  d).  e).  For  [ 0 , °°)  d e f i n e d by  each randomized , let  oAS  denote  the randomized  .  O(O))A6  (OAS)(_J) =  ( B ^ ) - s t o p p i n g time  Then f o r each  a  and  (B )-stopping t  s  in  [ 0 , °°) ,  s x ^ s  ->  XAS  in  RST  , by  u by  11.10.  x  r e p l a c e d by  y  11.6,  x . A  1  >u  S  Thus, a p p l y i n g  and  ( a =>  x A s , and  a  s  y  d)  with  equal  to  y. U  for  a l l measures  y  e  dy -»-  1  Y  .  Now  X . AS  u  1  y  y  X . As T T  U  f o r each X .  > u  1  ("^^  g  X As  , we  ,  dy  i ,  r e p l a c e d by find  that  (x^As)^  ,  - 144 since  T  is  1  y-standard.  T h u s we  u for  a l l  total  s  in  mass o f  Now <_ a  p  .  that  of  >  p  u °°  y  so  and  P ({u  a  f o r any  are s  y  u  , i t follows  that  two m e a s u r e s  and  >u  [0, )  the  have  U  = 0 .  (8 )-stopping  times  t  with  ,  00  V a A s  u  >u  T A s  8.7  and  applying  that  : x(w)({°°}) 4 0})  randomized  in  V p A s  conclude  s = 0  , whence t h e s e  e Q  y  y h e r e ) we  with  —  m a s s , and  Then  s  T  is  U  Letting  > u  this,  suppose  <_ T  s  From  y  same t o t a l  A  [ 0 , °°) .  T the  T  that  U  A  U  find  >u . T  8.8  (we  a = y^  can take  in  8.8  y and  D  U  are potentials  and  y  > u  0  .  0  (As  (b =>  s ->•  c).  0 0  This  , y  pAs  -*• u  p  weakly  and  implication applies  y  y  OAs  a  weakly,  p , a <_ x  because  by  and  » p ( = ») » = o .) y  T  11.12. such  Corollary:  that  time.  U  Then a)  x  b)  U  y  be  the f o l l o w i n g a r e is  T  a) =>  a measure  i s a PC potential.  P  (xAt  Proof:  Let  Let  x  on be  3R  (where  a randomized  n = 1 or  2)  (B)-stopping  equivalent:  y-standard.  i s a p o t e n t i a l and i s as d e f i n e d  b).  Clear.  U  T  A  t  > U  i n the p r o o f  T  of  for a l l  t  11.11(d).)  in  [0,™)  -  145 -  u. b)  => a ) .  We a l w a y s  and  n <_ 2 , we h a v e  (if  n = 1) .  by  n  TAt  T  such  Corollary:  that  U  in  Let and  let  let  y  Let  T  B' = B  RST(ft,B,8 ,P )  y  then  b)  T  H  Then  U  is  R  R  t .  ^ ^)^ x  t  ]R  (  =  i n  >^ U  P  T  [0,°°) ,  T  A  t  )  To c o n c l u d e •  t  (where  n  U  n = 2) o r 4.11  t -> °° .  n = 1 o r 2)  t i m e , and l e t from  the f o l l o w i n g  and  S y  and  U  by  S  T  (ft,B,B ,P ) y  t  y ( E ) = Q(B g  T  g  as d e s c r i b e d  time  S  e E , S < °°)  be (a v e r s i o n  of)the  i n 9.8 a n d  are equivalent:  (8J.)-stopping  are  of  (8J_)-stopping  F o r each  defined  n  arising  t  Whenever  and  since  be an o p t i o n a l e n l a r g e m e n t  (8 )-stopping  => b ) o f 9.13. a)  a = y^  (8j_)-stopping  be a  as  y  t  be a measure on  be the measure on  g  (if  y - s t a n d a r d f o r each  ° I|J f o r e a c h  t  , b y 4.8  Now  potential.  (ft* ,8' ,8j.,Q,<Jj)  randomized a)  y  •  n  n  i s  Let  is a  P  n  T  T  t h e p r o o f , a p p l y 11.11 w i t h  11.13.  y O R ) £ yOR )  y Q R ) >_ y Q R ) TAt  Now  11.10, a n d  have  time and R U  S >_ U  P  are potentials,  and  R <_ S <_ T  y  y-standard.  Proof:  a ) => b ) .  (co e Q)  .  F o r each  T A t arises  t  in  TAt  from  [0,°°)  ( T A t ) (co) = T(to)A<5  let  i n t h e same way  that  T  t  arises  y from  T .  Also,  y  = y  T P  potential,  and  and  m  y  T  T A t  TAt  P  U  = y  .  m  Thus  U  i s a  TAt  T  >^ U  fora l l  t  in  [0,°°)  .  Thus  T  is  y - s t a n d a r d , b y 11.12. b) stopping  => a ) .  Let  p,  times a r i s i n g  o from  be  (versions R, S  o f the) randomized  respectively,  (B ) -  as d e s c r i b e d  i n 9.8  -  and  a ) => b ) o f 9.13.  146 -  p <_ a <_ T  Then  Q - a.e.  Let y  p'  = pAa' .  Then  y y  potentials,  and  . = y  a  y  U  and  a  y  p  , = y  are  Theorem:  Let  potentials.  .  But  y, v  Then  y  = y  U  b)  There i s a  Proof:  P  > U  such  that  b ) =>  a).  a) => b ) .  U  a  y  and  U  p  are  y  H  2  = y  .  o  o  |-j  such  that  U , y  U  V  the f o l l o w i n g a r e e q u i v a l e n t :  and  V  and R  be measures on  0 a)  p  Thus  0  >_ U  p  11.14.  .  a' = O A X ,  0  p ( l ) = v ( l) .  y-standard y  x  randomized  (B^_)-stopping  time  x  = v .  Clear.  For  0 < r <  , l e t A  0 0  be the u n i f o r m  unit  2 distribution and  v U  r  let  v  r  on the open b a l l = v*A  v  v  = U *\  .  Thus v  of  Helms  , where  r  [1], U  y-standard  r  U  v  tU  r  of radius  *  denotes  i s finite  as  r+0  (B ) - s t o p p i n g  time  .  r  at  convolution.  such  0  y_  1R  ,  A l s o , b y 4.18  r , there e x i s t s  that  in  Then  (and c o n t i n u o u s ) .  F o r each  T_  centred  = v_  a  , b y 8.20.  r By  9.23, t h e r e  i s a decreasing  sequence  (r(i))  in  (0,°°)  and a  randomized ( 8 ^ - s t o p p i n g time x such t h a t r ( i ) -»- 0 and ( i ) in R S T ( n , B , B _ , P ) . Then y = v and x i s y - s t a n d a r d , b y 11.11 t x T  T  r  y  (we c a n t a k e  a = v  i n order r(0)  to apply ^  11.11 . ) g  -  12.  147  -  EMBEDDING PROCESSES I N BROWNIAN MOTION  In  this  Chacon  s e c t i o n we  [ 1 ] , we  call  the processes  define a class  of processes  potential processes.  Essentially,  which  i n an o p t i o n a l enlargement  to  be  of  B r o w n i a n m o t i o n b y means o f a n i n c r e a s i n g  stopping  Notation:  write  U  12.2.  Definition:  integer. is  c a n b e embedded  t i m e s - s e e 12.7, 12.16, and  12.1.  for  P  An  a system  If U  y  v  Let  n-dimensional (A,F,F^,X^,P)  (A,F,P)  i s a probability  b)  ^ j ^ i e i  "*"  c)  (X.). i s a family of J l l e i  n  c  of  n  process with  y  shall  to B o r e l  be a  positive  time  set  H  n  I  r  e  a  space. f a m i l y o f sub - a - f i e l d s  s i n g  ]R -valued n  3  random v a r i a b l e s  of  F .  over  .  For each  i e I ,  i)  X. l  . .. n )  Whenever finitely  is  F.-measurable, l  law(X-j) U --  TI  iii)  e)  i  T  (A,F,P)  d)  n  t h e n we  o  where:  a)  s a  ]R^  [-«=, °°] , a n d l e t potential  t u r n out  12.17.  i s the r e s t r i c t i o n  I £  these  family of standard  i s a measure on B o r e l  where  V  which, f o l l o w i n g  1  If  S  . xs a  . potential.  n <_ 2 , P C X ^ S ) = 0 .  and  T  are  (F^)-stopping  many v a l u e s , a n d  S <_ T  law(X ) U  b  , we  law(X > U  1  ) .  times  have  t a k i n g on o n l y  .  -  148  law(X-)  (We  remark t h a t  i t follows  f r o m d)  that  U  and  law(X_) U  are p o t e n t i a l s .  assumed 12.3. and  to take v a l u e s i n  Proposition:  Let  (A,F,F^.X^P)  let  Also,  I _ be  (A,F,F  b)  For each  ,X  ,P)  [-°°,  , let  n  be  a positive  satisfying  a)  t h r o u g h d)  integer, of  12.2.  n-dimensional p o t e n t i a l process.  , ($(X  n  ]  .)  equivalent:  i s an  y e ]R  00  a system  T h e n a) a n d b) b e l o w a r e a)  I  ( F ^ ) - s t o p p i n g time i s  an  i 9  y)l^  ^  x  9}^  i  s  a  s  u  P  e  r  m  a  r  t  ingale  i in  Moreover,  c)  Proof:  the extended  i f  n = 1  (X_^)  If  then a t h i r d  S  (F_^)-stopping  i s an  y e ]R  a) => wish  a) b) .  t o show  i s a consequence i ,j Z  that F  (X^jy)!^.^ ^  Let  S =  (A,F,F^,P)  time  3  j  .  taking  (y) = E ( * ( X - , y ) l  Let  condition i s :  on o n l y  finitely  )  U  (b =>  .  then  n  law(X  (A,F,F^,P)  equivalent  i s a martingale over  many v a l u e s a n d  Thus  sense over  Y  e I  f o r each  i  on  F  j  on  A\F  3 }  )  of the o p t i o n a l with Z.  dP >  1  ^  F  i <_ j dP  and  .  sampling let  theorem.  F e F_^  f o r each  y e ]R  and  y e lR  each  We  , where  3  k e I  .  n  .  -  and  let  times  T = j  taking  A .  on a l l o f  on o n l y  finitely  149 -  Then  law(X_)  zl  conclude  that  U  law(X_) " (y) <  Z  dP >  Z  Thus  dP  Y  Y  •  and  are S £ T  (F ) - s t o p p i n g so  law(X )  forall  Z . dP >  0 0  T  > U  dP  y  and  many v a l u e s  U  That i s ,  S  i f we l e t  y  R  e  n  f o r each  y, v  From  t h i s we c a n  y e TR  such  be t h e measures  that  o n TR  d e f i n e d by  y(A) = P ( { X  ±  v(A) =  e  A} n F )  e  A} n F ) law(X_)  then Hence  all  U  > U  y  U  > U  y  y e H Now  except p o s s i b l y  V  suppose P(X  c ) => b ) .  finitely  ±  = 9) = 0 Since  Let  Z  That i s ,  y  = Z .  dP >  Y  00  }  dP  for  U  i e I  n = 1, 3>(x,y) = - -|-|x-y| forconditional be  such that  law(X )  same c e n t r e  .  f o r each  S, T  many v a l u e s ,  Then  n  {U  n = 1 .  Jensen's i n e q u a l i t y a) => c ) .  (X^)  TR  set  , as d e s i r e d .  n  Then  apply  on a l l o f  V  on the p o l a r  .  Thus we n e e d  only  expectations.  (F ) - s t o p p i n g  times assuming  only  S <_ T .  law(X ) _. U  , so  o f mass, b y 4.11.  i s a martingale  over  Hence  law(X.) E  and  lawCX.,)  ( X g ) = E(X_) .  (A,F,F_^,P) .  have the  I t follows  that  -  12.4.  Definition:  a potential, (ft,B,8 ,P ) y  t  Let  Let  B' = B  (Bp-stopping  A  Q) i f f w h e n e v e r  R,  law(B )  S  for  will  Note  U  is  y  enlargement of  0 <_ t < » .  be c a l l e d  (Bp-stopping  standard (relative  times and  R <_ S <_ T R  and  U  to then  l a w (Bp  law(B ) are potentials  ^ U  o f 12.4:  that  the d e f i n i t i o n  definition  12.4 i s a g e n e r a l i z a t i o n  o f the  8.2.  By 1 1 . 1 3 , T  i s standard r e l a t i v e  (Bp-stopping c)  T  are  U  Discussion  b)  time  ° i|;  such that  n  be an o p t i o n a l  g  and  a)  fc  H  law(B )  R  U  12.5.  be a measure on  (ft',B',B',Q,ij;)  and l e t .  y  150  time a r i s i n g  One c a n show t h a t  T  every  K £ ]R  compact  set  from  to T  Q  is  i f f " t h e " randomized y-standard.  i s standard r e l a t i v e  to  Q  i f f for  ,  n  T(w) Q(du>)  is  finite  If  K  l a w (Bp I (U - U ) . This 'K o f 8.13, a n d we o m i t t h e p r o o f .  and e q u a l to  generalization d)  l (B'(o3))dt  (Bp-stopping  n >_ 3 , e v e r y  i s a simple  time i s s t a n d a r d r e l a t i v e  to  Q • e)  If Q  12.6. U  y  n <_ 2 then  and T  Proposition:  i s a potential,  y({8})  is finite  Let  y  and l e t  = 0  and  T  i s standard r e l a t i v e  to  Q - a.s.  be a p r o b a b i l i t y (ft' , B ' ,8pQ,iJ>)  measure on  ]R  be an o p t i o n a l  n  such  that  enlargement  -  of  (ft,B,B ,P )  Let  ° 1J1  = B  I c [ - o o o o ] and l e t  Let  -  (Bp-stopping  times which  jB^-Q)  (ft',8',8.j, i set  .  y  t  i s an  151  for  (T.). _ li e l  <  0 <_ t  00  be a n i n c r e a s i n g Q .  are standard r e l a t i v e to n-dimensional  family  of  Then  p o t e n t i a l process with  time  i  I .  Proof:  Clearly  definition  the i n d i c a t e d  12.2.  To f i n i s h  b) o f p r o p o s i t i o n Let  system  satisfies  t h e p r o o f we  a) t h r o u g h  shall  show  that  d) o f t h e i t satisfies  12.3.  i ,j e I  with  i <_ j  a n d l e t F e B_  . i  T. l Let  S =  on  and  \ T. J  (Bp-stopping law  F Note  that  S  is a  As  T  i s standard,  law(B^) law(B^) U > B  .  For  (Bp  y e {U  < °°} , we  * Y> (  r  y ) 1  1  Then, by t h e method conclude  .  S7'\F  on  time.  l e t T = T. J  that  can conclude  {B  used  T  1. X  ^}  d  Q  from  -  _  this  $ ( B  ?  T.'  1  that  y ) 1  {B_  * _}  •  '  d Q  i n t h e p r o o f o f ( a => b) o f 1 2 . 3 , we c a n  this inequality actually holds f o r a l l  y e ]R  n  .  •  12.7.  Theorem: Let  n  I = {0,...,k}  be a p o s i t i v e i n t e g e r ,  n-dimensional a)  Let  There  potential process.  where and l e t  Let  i s a product enlargement  k e U  , or l e t I = K  (A,F,F^,X^,P)  u = law(X^)  .  .  be an  Then:  (ft' ,8' ,B|.,Q,^)  of  (ft,B,B ,P ) y  t  -  and  an i n c r e a s i n g  which  have ° IJJ  B£ = B  If  for  the  a)  Let  ( f t , 8 ' ,B^,Q,IJJ)  A = Lebesgue  measure on  Suppose times =  (L,L,A)  B^ = B j e I  T  °  n  shall  and  same j o i n t  distribution.  Then probability  Y_,...,Y., 0' j  and  n  that  L =  have  e  °^  j  ^ x.^iel B  3  n  That i s ,  u-standard ^ i^iel  d  X  (0,1),  L = Borel  n  ave  of  L, and  Evidently  to that  we  can l e t T  standard  Q  = 0 .  (8*.)-stopping  (Y.,...,Y.) 0 j  t h e same j o i n t  distribution.  a  3  (Bj.)-stopping  B' T j  and  X.,...,X., X... 0' j ' j+1  Let  a  5  +  E ^ 2+  We  X  distribution  measure on  time  be the j o i n t of  E^ ^ ,t +  , and t h e p r o j e c t i o n  • • • > j ' j +1^ ' X  1  be t h e j o i n t  i s a probability  measure on  ( Q  a standard  l e t E = E." .  and l e t  X  ^ ± ^ ±  j+1 e I , and suppose  X ,...,X.  so t h a t  X_,...,X. 0 3  i s necessary i n a ) .  have been chosen  T.,, > T. J+1 ~ J  of  where o f c o u r s e  '  0 <_ t <_ °° .  one c a n c h o o s e  D = ]R  and  2  show t h a t  Let  (B*, ) . i  L .  that  < ... < T. 0 — — j  ( B ' ,...,B' ) 0 j  , where  for  such  that  be t h e p r o d u c t e n l a r g e m e n t  1  by  M  Q , such  times  distribution.  (ft,B,8 ,P ) t  (Bj.)-stopping  °^  distribution,  family  times such  same j o i n t  j _  0 <_ t <_ <=° .  i s an i n c r e a s i n g  f c  e  to  t h e same j o i n t  (8 )-stopping  Let  ^ ± ^ ±  n = 1 , no e n l a r g e m e n t  there  Proof:  family  are standard r e l a t i v e  (X.). _ 11 el  b)  152 -  ^ ^ ^TJ ' * * ' ' ^ j ^  map  have the  distribution  X_.,...,X. 0 3 is  a  , X.,., j+1  -  onto  carries  a  H e n c e b y 9.4, t h e r e i s a n family  (8 ) X  A e Borel(E  . . xeE j + 2  3  of probability  a(dx)B  measures on  $(x ,y)l (x )d8(x ,...,x j  D  (A(x))  such  that  for a l l  .  0  j + 1  )  >  l  (x.  + 1  ,y)l (x. D  + 1  )d8(x ,...,x. 0  + 1  )  FxE  a l l  F e Borel(E^  + ±  )  and a l l  y e D  .  ( a => b ) o f 1 2 . 3 , b y c h a n g e o f v a r i a b l e s . this  inequality  may a l s o  while  This Now  may be d e d u c e d  the f i r s t  from  integral i n  be w r i t t e n as  •KXj , y ) l _ ( x  ) d a ( x , . .. ,x..) Q  t h e s e c o n d may b e w r i t t e n a s  U  From t h i s element topology  J  of of  D  , that  for  .  by a v e r a g i n g w i t h r e s p e c t  a c o u n t a b l e s e t o f open b a l l s a-almost  a l l  which  to  forms  (XQ,...,X_.)  y  over  a base e  E ^  +  ±  each  f o r the we  have  • • • , x. J  c o u r s e we  (y)da(x ,...,Xj) Q  o n e may d e d u c e ,  XQ,  Of  E  measurable  i s a potential process,  FxE  for  unique B o r e l  ) ,  X.,...,X., X.,. 0' J j+1  j  a-essentially  l  8(A)  Since  153 -  1  should v e r i f y  •)l-(Xj)  that  U  on  D  i s a potential  for  a - a.a. x e  -  154  -  e As  the  $  's a r e f i n i t e  measures,  that  for  U  Then  .  x  a - a.a.  's  a r e OK.  We  need  only  x  , U  i s finite-valued.  F i x any  y e  D  (y)da(x)  ^"(xj  + 2  $_(  ,y)1 (x  + 1  D  Vl' {X. ) (y) < »  J  B  i n particular,  U_  x  j + 1  )dB(x ,...,x.,x. Q  + 1  )  * o)  y)1  law(X = U_  Thus,  U  B  X  check  the  +1  .  ( y ) < °°  for  a - a.a.  x e E  J  i+1  , for  this  v  particular finite  y  a t one  the  case  for  a l l  x  But  point  n = 1  Thus 6  .  , as for  i f  y  then i t i s f i n i t e  and  1.6  § >_ 0  f o r the i n this  a - a.a.  x =  - s t a n d a r d randomized  B (A)  -  x  a l l  in if  case  n =  2  ; i f  .  this n  E  The' m e t h o d c a s e we  < 2  then  this  used  e  [0, » ]  e E^  see  10.5  to prove  takes  care of  : B  B  is  see  1.5  for  8  n>_3,  x D  , we  can  T  such  that  x  =  (CJ) e  (6  f o r the  11.14  can get non-randomized this  U_  U_  =  0  choose  a  X  (more b r i e f l y ,  the p r o o f o f  and  case.  (x.,...,x.) 0 j  (u))({t  J  A e Borel  (For = 2  x. P (dco)x  E  at a l l p o i n t s ' —  t  J  n  m e a s u r e on  (B ) - s t o p p i n g t i m e  X.  for  is a finite  x. J  )  A})  T  )  .  x  case  a l s o works  n  >^ 3 a n d  for  n = 1  s t o p p i n g times; see  a - a.a.  x e E^"*"  11.4 ,  though  8.20.)  since  then  Now  -  155  -  B P(X  =  3)  for  a-a.a.  measure  = 0  B  x  Now that  the o t h e r hand  x e E ^ x { 3 } ;  obtain  a family  that  for  - s t a n d a r d and  8  3  the  x  x  (Borel  For  's  c a n be  E^  )  + X  x e E^  associated  to  ®  such  6„ 3  s o we  (x ) x  . ,.. -J+1 xeE  =  (6  x  the  can  take  chosen  let  + 1  x^  by  t  let  First  T... J+l  «•  so  have  T  = "0"  x  (8  of randomized  x =  U  <_ 0  X  probability  J  a - a.a.  x  t h e n we  (x-.,. . . ,x^)  .  )-  t  e E"^  ,  + ±  ) x. x 3 x  that  f o r each  o f lemmas w h i c h t e  (0,°°)  follow,  the  map  T (_)([0,t)_ x  B^-measurable.  e'co'  Finally  n >_ 3  i t w i l l become a p p a r e n t , f r o m ja s e r i e s  (X,U>)  is  i f  f o r any  equal to  times such  is 6 x .  x  On  must b e  x  T h u s we  stopping  .  be  9.18(b),  =  = T. + 3  o f a l l we  T  ( 9  S  claim  t  ((8  the and  for  ®A))-stopping _'  =  (_,u)  e  time  ft  let  1  _)  (for  0 < t < °°) —  .  that  S  (8^  is a  + t  )-stopping  time.  3 Well,  {S  f o r any  < t} =  t e  u  (O, ) , 00  {_'-(_,u) e ft' : u < x  re[0,t) r  rational  (See p r o o f o f Also,  Y  9.18(b).)  f o r any  r e  [0,t) , the  map  0  (  U  ) ' - - "  Y  j  f  n  (  w  M  ( 6 >  T  , j  (  ,^)([0,r])} w}  156  =  (B^  is  (_,_)•*  -  ((Y (_'),...,Y (a»')), 0  ( B o r e l E"^ ) ® B^)-measurable  and  +±  +  r  >  6-  j  the  map  j ( ( . , . . . , x . ) , w ) •> x u j X Q , .. .  (w)([0,r]) xj  X f  is  (Borel E It  B^-measurable.  ) ®  follows  5  that  {S  < t } e B_j,  +  for  t  0 < t <  .  Thus,  _)-stopping  time,  (Bp  , one  00  as  j is  right-continuous,  S  (B^,  is a  j  as  j  claimed. checks a  + t  But  that  then, using T_  + S  the r i g h t - c o n t i n u i t y o f (Bp-stopping  is a  time.  That i s ,  T  j  +  easily i  i  s  (8|_)-stopping t i m e .  Now  negative  let  Borel  Y  = B'  and  functions  on  E  suppose  .  f (x )...f.(x.)f. 0  0  f  + 1  Q  (x.  j + 1  f  o V"- j . j (  [ n i=0  f  ( x  ) [  + 1  1  are  non-  (x  )dP  j + 1  )dB(x ,...,x., 0  x.  + 1  )  - _ i ( -.i) 3 (x )]da(x , j+1 j+1 x ,...,x j+1 0 x  d  v  Q  Vi  (u,)  f.(Y.(u.'))]E 1  f  Then  f (x )...f.(x.)f 0  f  [  f j  +  l  (  B  T .  )]dQ( Y (u'),...,Y 0  (a)')  (*) [ n f . ( Y , (_'))]_,_,__ ( B l i=0  j+l  x  T Y. ( a ) ' ) , . . . , Y ( i _ ' )  „'))dQ(a>')  (6' j  -  157 -  f (Y )...f.(Y.)f. 0  Thus  0  Y „ , . . . , Y . , Y.,., 0' j j+1  and  + 1  (Y.  + 1  )dQ  X „ , . . . , X . , X.,. 0' j j+1  have  t h e same  joint  distribution. (The a "souped Meyer  step  (*) a b o v e , a n d a l s o  immediately  Now  (**) b e l o w ,  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  [ 2 ] . F o r the reader's  result  the step  convenience  we  f o l l o w i n g the present  l e t us show  that  T.,,  state  follow  from  described i n  and p r o v e  this  proof.)  i s standard.  We  shall  prove  this  J+1 by  applying 12.5(c).  Let  K  be any compact  subset  of  D  .  T. r  J  l (B')dt  dQ  K  law(B^ ) [U  - U  y  law(X.)  [U - U M  ] =  j  L  ]  K  K j+1 l (B')dt  while  dQ  K  T'. J  T  v ( Y (oo Q  IN  ),...,  T Y  , , * (6 ' io') (co ) 1\  J  V t B  ( 9  T  a  j  )  '  )  )  d  t  A  W  )  (**){ Y.(co')  Y (co'),...,Y.( ') 0  W  1 IN.  (B')dt]dQ(co*) L  ;P ® A) law(Bj, X. x_» • • • > x. U J - u J  6  0  2  ]da(x ,...,Xj)  K  X K  U  j  X (y) - U  j • • •j  °'  x j  (y)da(x ,...,xjdy 0  Q  Then  -  U  158 -  law(X.) law(X 2 (y) - U  ) (y)dy  J  K  1  Thus  IN.  (B')dt  dQ  L  law(B^  [u  y  - U  [U  P  - U  2  ) +  ]  1  K  law(X.^) 2  +  1  ]< - .  K  It  follows  (b) except  that  The p r o o f  that  T.,, J+1  i s standard.  of this  i n place  part  i s quite  of the family  (T )  t i m e s , we a p p e a l  . xeE-  X  stopping  similar  to that  o f randomized  (Bp-stopping  t o 8.20 t o g e t a f a m i l y  series for  o f lemmas w h i c h  each  t e (O, )  times.  Itwill  follow,  that  (T )  ( B o r e l E ^~) ® 2+  Now h e r e in  the proof  12.8.  of  3 + i  from the  T 's c a n b e c h o s e n s o t h a t x  t h e map  00  (x,a>) + 1  is  . xeE  become a p p a r e n t ,  the  (8 ) -  5  X  genuine  of part a ) ,  {  T  <t  j(u)  B^measurable,  i s the v e r s i o n  •  of the strong  Markov p r o p e r t y  mentioned  o f 12.7.  Proposition.  Let  (W,M,M  t >  Z ,9 t  t >  P  )  be a s t r o n g  Markov  process  -  with  translation Let  be  M  Z  E  More  operators, with  be an  T  (  w  Let  u  space  time,  (E,E)  and l e t  f : W  be any p r o b a b i l i t y  u  )  -  state  (M^-stopping  ® M-measurable.  T  Then  T  159  (f(w,«)) = E ( f ( w , » ) M  ,  ° 6  |M ) ( w )  for  x W  [O, ] 00  measure  P  u  on  E .  - a.a. w £ W  M  precisely: Z (w) T  a)  The map  w  b)  F o r any  M-measurable  we  E  (f(w,«))  is  function  M-measurable.  g  : W -> [0,  0 0  ]  ,  have  Z  (w) (f(w,.))dP (w)  g(w)E  P  g(w)f(w,6 w)dP^(w) T  Proof: b  It suffices  to c o n s i d e r  are non-negative functions  f  on  of the form W  which are  a ® b  , where  M-measurable T  Z (w)  which  is  M^,-measurable,  Then  and  g a E  x  E  T (b)  dP  integral  x  y  T  g(w)a(w)b(e w)dP (w) H  T  follows  i n b) i s  y  y  the second step  (b) ,  y  g a E ( b o 0 |M ) d P  where  and  T  (f(w,«)) = a(w)E  the f i r s t  and  z (w)  T  M-measurable r e s p e c t i v e l y .  a  from the f a c t  ,  that  g a  is  M-measurable.  - 160 -  Before concluding, l e t us emphasize that, i n a s l i g h t  departure  from the usual formulation of the t r a n s l a t i o n 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 i n a more general setting  than t h i s , since we want to apply this proposition to the process (ft* ,B' ,8^,Bj.,epp » A) considered i n the proof of 12.7(a). X  • Now we take up the proof of the measurability results used i n the proof of 12.9.  Lemma.  12.7. Let  and  and Y  Y  be sets, l e t  of  subsets of  5  such that each set belonging to  let  X  X  g  is  S  range(F) and  i s of  S-measurable and  F(x)dy  is  be  a-rings  be a measure on  o-finite  y-measure. f : Y  H  F : X ->• L L  X  1  = L (Y,S,y) . ±  , and  R-measurable for each  S e S  S b)  range(F)  i s a separable subset of  F [U] n {F 4 0} e R X  c)  There i s a that  Proof:  a) => b).  , and U £ L  f : X x Y  X  .  TR  such  g dy  is  for a l l x e X .  One f i r s t v e r i f i e s that  R-measurable for each bounded  X  for every norm-open set  T-measurable function  F(x) = [f(x,«)l  L  x ^  f(x)  S-measurable function  g  on  Also.  , let  y({f 4 g}) = 0} .  i s a separable subset of  x  y  S  5-measurable function  Then the following are equivalent, for a)  and  respectively, and l e t  T = R ® 5 , and for each  [f] = {g : Y -*-]R|  R  Y .  -  Next,  as  r a n g e (F)  y-measure such  f o r any  S-measurable  i s separable,  c {f  bounded  L  e  a l l  now  follows  linear  function  x e X  g  b) that  F  => 1  ; thus  from  space valued  i s a set  on  A  functions;  c).  y - a.e.  functional Y  ° F  A  such  F(x)  the g e n e r a l  S e S  of  a-finite  is  g  a sequence  theory  functions  from  X  such  is a  bounded  dy  The  conclusion for  b)  Banach  [1]. o f r a n g e ( F ) , and  open  ball  B  in  of countable-valued  into  .  L"^ , t h e r e  of m e a s u r a b i l i t y  f o r each  (F_^)  on  Y\S}  R-measurable.  see P e t t i s  [ B ] n { F ^ 0 } e R  on  that  U s i n g the s e p a r a b i l i t y  construct  in  there  : f = 0  1  A(F(x))  for  -  that  range(F)  Thus  161  that  f o r each  L  the 1  fact  , we  can  R-measurable i  , and  f o r each  x  X ,  ||F(x) - F.(x)  ||  < 2"  ±  1  .  L Now  f o r each  on  X x Y  i  , there  such  is a  T-measurable  a l l  x e X  .  Moreover,  j [  of the r a t e  of  [f.(x,  for a l l  f (x,»)  because  function  that  F.(x) =  for  real-valued  •)]  x e X ,  F(x)  L^-convergence  y -  of  a.e.,  (F^.(x))  to  F(x)  .  f.  - 162 -  Let and  E =  define  {(x,y) e X x Y  f  : X x Y -> E  : (f\(x,y))  f.(x,y)  E e T  c) Let  =>  and  a).  to  RQ £ R  the  R e R„  and  0  and  let  and  f o r each  .  Thus  S^ £ S  properties.  part  Z = u S  n  0 .  x e X  .  such that  Let  Then  follows  part.  S.,  , f(x,«)  of the form  , so  is  there  o-ring  are countable  Z  is  where ,  a-finite  u-measure,  i s of  , which  RxS  respect  generated by  S^-measurable  range(F) £ " L ^ Z , S ^ y ) "  theorem.  i s measurable w i t h  be the  Z e S  from F u b i n i ' s  Well,  f  g e n e r a t e d by the s e t s  S e S  (x,y) I E  (x,y) e E  has the d e s i r e d  the s e p a r a b i l i t y  and  o-ring  i f  The m e a s u r a b i l i t y  us e s t a b l i s h  sets  Z  f  i f  = \  0  Then  }  by  film  f(x,y)  does n o t c o n v e r g e i n E  and v a n i s h e s  outside  separable.  •  Remark: really is  The a s s u m p t i o n t h a t i s needed  f o r the proof  shown b y t h e f o l l o w i n g  R = S = Borel  X  f  is  o f ( c =>  example.  , y = counting  f(x,y)  Then  each s e t i n  i s of  a)  Let  measure  S  o f the above  X = Y =  on  1  i f  x = y  0  i f  x  a-finite  S  y-measure  lemma.  [0,1] ,  , and l e t  = {  R ® S-measurable, but  4y  {[f(x,»)]  : x e X}  i s not a  This  -  separable  12.10. P  subset of  Lemma:  generated such  (A,F,F  P)  be  a filtered  = RST(A,F,F  Let  RST  .  Then  P  t >  ,P)  t h e r e i s a map  .  measure space, F  Suppose  T •+ x  where  i s countably  o f RST  into  itself  that a)  For every  b)  The  map  for  each  Proof: be  mod  -  L"*" .  Let  is finite.  163  , x = x  P-a.e.  (x,to) H- x ( t o ) ( [ 0 , t ) ) i s ( B o r e l RST) t e  For each  d e f i n e d by  x  (0,°°) .  positive  f  F-measurable  ®  rational  number  r  , let  f  : RST  x A  [0,1]  (x,co) = x ( w ) ( [ 0 , r ) ) .  Then f o r each  x  , f  (x,«)  F^_-measurable; a l s o ,  is  F e F  i f  ,  then  f F Thus  for  (x,-)dP  x(a3)(dt)l  P(dco)  =  F e F  , t h e map  f  x F  Indeed on  i t i s the p o i n t w i s e l i m i t  RST.  A l s o note  that  generated  mod  countably  Thus, by such  (t)l (to) F  .  12.9,  is  Borel(RST)-measurable.  of a sequence of continuous  functions  ,P|F )  F  is  : RST  x A  L^(A,F P  (x,«)dP r  i s separable since  r  .  f o r each  r  we  can  select  a map  that i)  For  [ 0 ) r )  r  g  r  is  ( B o r e l RST)  ii)  For  every  each  t e  (0,»)  x  , g  , let  F  ®  r  -measurable  (x,«)  = f (x,«)  h  sup g 0<r<t  =  P -  r  r  .  a.e.  Then each  h  t  is a  [0,1]  -  ( B o r e l RST)  ® F-measurable  each  e RST  (x,co)  * A  left-continuous. P -  x(co)  =  x A  into  T e  (0,°°)  [ 0 , 1]  , and  i s increasing  t  f o r each  12.11.  = h  (x,co)  (E,E)  .  defined  be  a measurable  For each  x  , h  (x,»)  =  for  and  x(»)([0,t))  in  a finite (E,E)  be  [0,  , let  00  ]  such  that  . properties.  measure s p a c e , and l e t a measurable (A,F)  process over  RRV  on  the r e q u i r e d  be  Let  by  y(x)  space,  with state  be  and l e t  space  E  the measure on  by  y(x)(A) =  T h e n t h e map e E  .  u  0 < t < »  has  (A,F,P)  Let  defined  measure  for  x •-»- x  = RRV(A,F,P;[0, «])  (X V t 0<t<°°  be  probability  t h e map  Lemma:  T  x , let  the unique  Then c l e a r l y  Let  co e A  co  x(co)(H(co))  : X  . is  (u>) e  A})  i n the sense  i s a Borel(RRV)-measurable  function  that on  f o r each RRV  .  (w)  e  , and l e t  (Borel[0,  each  [0, » ]  i s measurable,  = {(co,t) e A x  for  alright.  y(x)  A e E  H H e F ®  P(dco)x(co) ( { t e  x  , u(»)(A)  Proof:  Then  RST  t H- h ( x , c o )  , t h e map  Also,  f o r each  y([0,t))  A  of  -  a.e. Now  RRV  map  164  Note  «=]) that  }  [0, o o ] : x  and  H(co) =  f o r each  F-measurable,  so  (u>) e A}  {t e  x e RRV  .  [0, o o ] , the  the d e f i n i t i o n  X  :  A}  map of  y(x)  is  -  Let  y  (Y )_ t 0<t«=°  be over  (A,F)  such  this  map of  RRV  monotone  class  bounded measurable 1  H  e V  .  That  Lemma:  c o n t i n u o u s sample the d e f i n i t i o n  so i n t h i s  0^)  case  that  V  (A,F)  paths  of  the  .  By  e V  consists  processes over  a  of a l l .  In  particular  map  P(dw)x(a.) (H(u.))  H-  subset  )  mod  P  of  RST  space  and  let  state  space  (A,F,F ,P)  Let  is finite,  generated  Let  .  Let  RST  (E,E)  a measurable  map  a filtered  .  a  be  be  Z  any  Z^  i s measurable  b)  If  a(Z\Z.) = 0 I  , and  let  T  process over  u(x)  and  x e T } .  where  i s countably be  E  .  a  measurable  as  with  i n 12.11.  z H- V ( Z )  is  Let  Then: a  to the c o m p l e t i o n of  there i s a family  ( F ^ - s t o p p i n g times such  Borel  (A,F)  be  suppose  m e a s u r e s on  with respect  , then  F  , let  measure space  f o r some  a)  ,P)  measure s p a c e ,  countably generated  x e RST  to f i n i t e  Z ^ = { z e Z : v ( z ) = u ( x )  and  a measurable  For each  a-finite  from  randomized  = RST(A,F,F  (E,E)  (X__)„ t 0<t<°°  be  be  i s right-continuous,  .  (Z,A,a)  Let  has  Borel(RRV)-measurable.  12.12. P  map  c o n t i n u o u s by  i t follows  X  is  (Y )  real-valued  i s , the  processes  t  (see 9.21),  argument,  the  real-valued  x(o))(dt)Y ((_)  If  i s actually  topology  that  P(dco)  Borel(RRV)-measurable.  then  -  the s e t o f bounded measurable  T  is  165  that:  (x ) „ z ze_  of  .  -  i)  x  ii)  For each  z  e T  and  u(x  t e  A ®  Proof: the  Let  M  smallest  form  be  ) = v(z)  (0,°°) , t h e  of subsets  of  M  measurable.  z H- v ( z )  are  the  and  RST  9.23(a).  Hence  s e n s e o f Meyer  Souslin M  .  Souslin A  ; part  Now there  Then  p  z  z  p  : M^  RST)  E {u(x)  such  (Souslin(M|M  be  separated  x >-> u ( x )  and  (A,M)-measurable  i s a Blackwell : x e T}  Now  space i n  belongs to  belongs to  assume  251 o f D e l l a c h e r i e  f o r a l l m e M^  is a  : v ( z ) e M^}  from t h i s .  -> T  M  let  a(Z\Z^) = 0 .  and M e y e r  [1],  that: ))  , Borel  RST)-measurable  and  .  ¥(v(z))  i f  z e Z  "0"  i f  z e  = z  e T  and  i s (Borel  particular,  ¥  field  y(H'(m)) = m  M^  Z^ = {z e Z  t h e theorem on p .  i s (Borel  Let  Hence  , and  p s e u d o m e t r i z a b l e s p a c e , by  (RST, B o r e l  a) f o l l o w s  i s a function ¥  z H- p  [1].  Therefore  by  The maps  i s a compact  E  (M,M)  M)-measurable and  ( B o r e l RST,  on  w h i c h makes t h e maps o f t h e Then  generated measurable space.  9.21  z e Z .  map  measures  countably  Now  a - a.a.  z  the s e t o f f i n i t e  (A e E)  respectively.  for  F^-measurable.  a-field  m H- m(A)  -  T (a))([0,t))  (z,w) His  z  166  z H- p  u(p  z  ) = v(z)  field is  Z\Z,  for  ( S o u s l i n A) (A,  Borel  a - a.a. , Borel  z e Z  , and  the  RST)-measurable.  In  X  is  R S T ) - m e a s u r a b l e , where  map  -  the  completion  countably we h a v e  of  A  generated,  that  Let  a  =  z  a  = p  z  for  for  i f  z e  i f  z e Z\Z  Now  with  on  Z  z € Z , and  z  Borel  RST i s  a(Z\Z^)  = 0 ,  .  Z U A  a - a.a.  Finally,  a l l z e Z  .  e A  A-measurable  z  a  {  z  measurable.  to  is  '0"  Then  respect  s o f o r some  z »-»• p  p  with  167  .  let  x ^  Then c l e a r l y  0  x  a  is  z  (A, B o r e l R S T ) -  b e a s i n 12.10 a n d l e t x^ =  the family  (x ) „ z zeZ  5  z  has the d e s i r e d  properties,  12.13.  •  Lemma:  Let  potential.  Let  y-standard}  .  Then  If  or  Then  2 .  be a measure  RST = R S T ^ B . B ^ P ^ )  T e Borel  Proof:  y  n >_ 3 y  on  H  n  such  that  U  is a  y  a n d l e t T = {x e RST  : x  is  RST .  then  T = RST  is finite.  , a n d we a r e d o n e .  F o r each  x e RST  Suppose  , l e t y_  n = 1  be as i n  11.2. u  Now for for  x  is  y-standard  i f f  e a c h n a t u r a l number i . the meaning o f x A i .) Now  U  U  i s a p o t e n t i a l and  This  i s e s s e n t i a l l y 11.12.  |$(x)|dy  i s a p o t e n t i a li f f x  y Also,  U  y  X  .  y >_ U  T  i f f  u  T  A  1  >  (x)  U  . XAi  y  A  T  >_ U  ( S e e 11.5  i s finite,  >1  U  whenever  V  i s an open  ball  -  whose r a d i u s Thus of  the c o n d i t i o n  countably  12.14. is  countably  =  x  that  Let  be  and l e t  RRV  X  { Y e RRV  mod  f  u-standard  be a  coordinates.  c a n be e x p r e s s e d  P  .  a-finite  Let  = RRV(A,F,P;H)  P - a.e.  =  has r a t i o n a l  i n terms  conditions.  (A,F,P)  generated  -  a n d whose c e n t r e  many m e a s u r a b l e  Lemma:  sp a c e ,  RV  i s rational  168  H .  measure  be a c o m p a c t  s p a c e , where  F  metrizable  Let  f o r some  F-measurable  function  : A -»• H}  Then: a)  b).  Proof:  For  x e RRV  and  x = \ X  RV  a)  (=>)  (<=) Let is  U  Define  a  + \ x"  x e RV  " - -  p  a  e  t  i f f whenever h  e  x'»  x"  e  RRV  X = x' = x" P - a . e .  n  RRV  o  i s clear. x e RRV\RV  .  be a c o u n t a b l e o p e n b a s e  not a point  some  1  have  G -set in  Suppose  0 < x(w)(U) for  is a  > we  mass}  < 1}  .  .  Then  Thus  A = {to e A  A e F  U e U  , P(B) > 0  : A  [ 0 , 1]  by  for  .  As  , where  H :  .  Let  f o r some  x I RV  :  .  But  then  : 0 < x(^) (U) < 1}  Define  x(w)  U e U ,  , P(A) > 0  B = {u> e A  a(w) = x(^)(U) •  A = {oo e A  a , x e RRV  .  by  a (GO) ( E ) = y(u>) (E\U) l-a(cj) }  (UJ e B  , E e Borel  x(oo)(E) = y ( o j ) ( E n U ) a(oo) a(u>) = x ( u ) =  x(w)  (_ e  A\B)  H)  - 169 -  Then  {a 4 x}  P-measure.  a (to)  _  w  x'»  x"  L\ >  E  R  x  and  x ( ) = [1  Also  Now d e f i n e if  a  = B , so  R  V  differ  on a s e t o f p o s i t i v e  a(co)]a(co) + a(co)x(co)  co e A .  for a l l  as f o l l o w s :  let  X ' (co) = o(co) X"(ai) = [1 - 2a(co)]o(co) + 2a(co)x(co) ; a(co) >_  if  ,  let  X'(co) = [1 - 2(l-a(co))]x(co) + 2 ( 1 - a(oi))a(ai) x"(co) = x(co) . (x*  Then  4 x " l = B , so  P-measure.  b)  Also,  x  ^ x'  =  By 9.21, R R V  x'  + "J x"  onto  K  Let  .  D  a-compact.  a,  For  on a s e t o f p o s i t i v e  •  and l e t  ij;  K x K  iji : K x K -»• K  by  .  Then  i f a = x  is  a-compact  12.15.  i n K  At last  .  Hence  we c a n p r o v e  <()(x,y) =" y  Let  (Z,A,a)  be a  R V = \p " * " [ ^ [ R V ] ]  .  measure  P - a.e.  isa  Thus  space.  K\\JJ[RV]  G^-set  the m e a s u r a b i l i t y a s s e r t i o n s  a-finite  K  x + -| y " . T h e n  the p r o o f o f 12.7.  Lemma:  let  (K><K)\D i s  i s c o n t i n u o u s , a n d b y b ) , <|>[(KxK)\D] = K \ ^ [ R V ]  ()>  space;  b e t h e c a n o n i c a l map o f  x e R R V , \p(o) = \J/(x)  be t h e d i a g o n a l i n Define  differ  i s a compact p s e u d o - m e t r i z a b l e  be i t s m e t r i c i d e n t i f i c a t i o n , RRV  x"  and  i n RRV  made i n  -  Let  into on  D E  D = 3R  and such  , E =  n  g  .  o  for  map  a - a.a.  U^ ^  is a  Z  and  U  IT  Suppose  i s measurable  that  170 -  6 T r ( z )  i s a measurable  from  Z  map  to p r o b a b i l i t y  of  Z  measures  z e Z ,  potential  > U  6  (  Z  .  )  Then: a)  There  i s a family  times  such  i)  for  a - a.a.  TT (. Z )  b)  If  ii)  First  t h e lemma.  (Bp-stopping u_  denotes  t e (0,°°)  t  T  , t h e map  x (w)([0, t ) ) z  -measurable, (T ) 2  (B  of genuine  Z £  )t  _J  that: z e Z , T  t e (0,°°)  is  6  , ^ - s t a n d a r d and  z  TT(Z)  , t h e map  (z,o.)  1^_  t  ^  A ® 8^-measurable.  l e t us e x p l a i n If  (z,u>) ^  z  f o r each  is  . . - s t a n d a r d and  TT(Z)  Z  a - a.a.  (.z;  6  = S(z) .  times such  for  TT  of  is  n = 1 , there i s a family  i)  (8 ) - s t o p p i n g t  X  A ® B  stopping  Proof:  z e Z , x z  f o r each is  o f randomized  that:  (« /_0_ ii)  (x ) _ z zeZ  u  some o f t h e n o t a t i o n  i s a m e a s u r e on B o r e l E  time, and  T  t h e m e a s u r e on  i s a genuine Borel E  , x  used is a  (Bp-stopping  d e f i n e d by  i n the statement randomized time,  then  -  V (A)  while  y^,  denotes  t h e m e a s u r e on  = P (B y  T  z e Z  f o r each  , so  -TT(Z)  z .  If  , let  n >_ 2 times.  randomized P  If  T  if  ; a p p l y 12.13  T  Now  for  = v(z)  .  n = 1  .)  a  A ® B  n = 1  .  t  by  translated  by  z  )  t £  6-standard randomized  be  x(co)  =  6  function  f  : ft •+ [ 0 , °°] .  that  RST(ft,8,8 ,P n = 1  x e T  , 11.14  i f  there i s a family  (a  ,  a  z  e T  (O, ) , t h e map 00  .  t  , .  for  (co;  (Apply  12.13  .)  , there exists  z £ Z  )  t  i f  n >_ 3  6-standard  the s e t of  such  (Bp-  6  that  n = 2  , o r 8.20  )  such  z  and  such  i f  that:  Z£Z  a  = v(z) .  z o  (z,co)  (co)([0, °)) c  z  is  -measurable,  z , let (x  family  x  12.14  i f  a - a.a.  T  set i n  z e Z  12.12,  f o r each  each  B(z)  be  B-measurable  and  ( A p p l y 10.5  for  ii)  For  , let  i s a Borel  - a.a.  Thus by  i)  defined  .  the s e t o f  (8 ) - s t o p p i n g t i m e s t  case, T  <5  be  n = 1  In e i t h e r n >_ 2  E  6  co e ft , f o r some  - a.a.  £ A)  v(z)  , let  < u  v ( z )  a - a.a.  stopping  T  Borel  that  u for  : B (a>) £ A})  y  y (A)  Now  -  P ( d t o ) x ( c o ) ( { t e [ 0 , »]  =  T  171  x  be  z  then has  For each  z  "the  translation  the p r o p e r t i e s  , let  T  z  be  the  of  asserted (B t  a  z  by  i n a).  +TT(Z)" Now  .  The  suppose  ) - s t o p p i n g time d e f i n e d  by  -  T Then  2  x ( u ) = 6_ z T  z  172 -  (GJ) = i n f { t  : T ( u ) ) ( [ 0 , t ) ) > 0} .  , for P (_)  - a.a.  2  Z  co e ft , b y t h e c h o i c e o f  Also  {T < t } = {x ( » ) ( [ 0 , t ) ) > 0} . z z  (T_)  has the p r o p e r t i e s  12.16.  Theorem:  be a n and  an  Let n  right-continuous  i s an o p t i o n a l  increasing  standard a)  relative F o r each [0,-)  b)  t  to  integer,  sample p a t h s .  (T )p  < t <  _  Q , such  weft'  the family ~ l  process, with  enlargement  family  then,  i n b) .  be a p o s i t i v e  n-dimensional p o t e n t i a l  with  there  asserted  Clearly  T .  (ft'  time s e t  , 8 ' ,8|.,Q,i>)  t >  X ,P)  .  Then  (ft,8,8 ,P )  of  t  [0,°°) ,  L e t u = law(X^)  (B^)-stopping  of  (A,F,F  and l e t  y  t  and  times which a r e  that:  , t H- T ( w ) t  i s r i g h t - c o n t i n u o u s on  . (xp  The p r o c e s s e s  dimensional j o i n t J  and  (B_j_ )  have  distributions,  t h e same  where  finite  B' = B ° ii f o r s s  0 <_ s <_ »• .  Proof:  Let  (ft,8,8 ,P )  (ft",8",8J],R,d)) by  P  t  (L,L,A)  A = Lebesgue measure Then by 1 2 . 7 ( a ) ,  on  be t h e product  , where L .  L = (0,1)  Also,  let  enlargement o f , L = B o r e l L , and  B'^ = B  f o r e a c h n a t u r a l number  ° <j)  k , there  f o r 0 <_ t <_  00  .  i s an i n c r e a s i n g  k family to  R  (S )  ^  k  such that  of  ( B ' p - s t o p p i n g times which a r e standard  the processes  (X ) ' tel. k  and  (B", ) K S tel  relative  have t h e  c  t  k  -k same j o i n t  distribution,  where  1^ = { j 2  : j = 0 , 1, 2,...} .  Now  173  define  X  , T  k  k  fora l l  X? = X j2  = T j2  Then  (X ) t 0<t<°°  joint  distributions,  Let [O, )  H  (H ) „ t  by  for  (j-l)2~  k  < t < j2  k  , j = 1, 2, 3,  -k  ( B ' \ )„ 0<t<°°  and  and  X  have  -»• X^ t  t  as  t h e same f i n i t e  dimensional  k -»• °° , f o r e a c h  t .  be t h e space o f 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  [0, ]  Also,  let H  and  ft'  defined  by  into  00  »)  -k }  T  [0,  t e  -  0 0  , topologized  a s i n 9.24.  b e a s i n 9.24.  0<t<?° Let  = ft x H  ft'  B' = 8 9 H (Bp ijj  = ((B 9 H ) ) t  Y  each  k  , l e t T  r  each  t .  (B") t  i s right-continuous.)  (co  (B",  o n ft  b e t h e map  Then  i s  ft'  (0<t<°°) .  r,  k  +  = p r o j e c t i o n of  B' = B o ib t t  For  f c  (co)  =  of  ((f) (co)  ft"  ,  into  T.(co))  B ' ) - m e a s u r a b l e , and a l s o  T^(co)  is  (B'p  (B", t  H p -measurable  Next,  f o r each  .  k  B')-measurable f o r t  f o r each , l e t Q, K  t , and be t h e  -  measure on  8'  non-negative  174 -  Q (A)  d e f i n e d by  = RO^CA))  FC  .  8 ' - m e a s u r a b l e f u n c t i o n o n ft' .  B'^-measurable,  and f o r any  f ,-1  Then  f  is a  f ° r  is  A e 8 ,  f°r,  dQ - 1 - 1 r "[A]]  [A]  ip  Suppose  dR k  k  f ° r , -1  dR .  k  <j> [ A ] X  Thus  8) = E ( f o r |<f>, 8)  E, ( f | ^ ,  , (ft' ,8' , B ^ , Q , i j j )  Thus f o r e a c h  k  (ft,B,8 ,P )  F o r each  .  y  t  = h(t) .  T  = T  t  o  .  fc  dimensional  Then  to  Each  T  k  8 -measurable  [0, ») , d e f i n e  each  F o r each  joint  (relative  t e  is  i s an o p t i o n a l  k  T (u,h) t  , which  T  , the p r o c e s s  distributions,  relative  (B^ )  to  Q  i s standard r e l a t i v e  to  Q  .  measure  Q  enlargement  i s a subsequence on  8'  such  of  (ft,B,B ,P ) K  f e L (ft,B,P ;C(H))  a l l  T  are standard r e l a t i v e  1  y  has  t h e same f i n i t e  as  (X ) . t 0<t<°°  (\(^))  that  t  for  fc  .  by  time  , as  Also,  t  there  ft'  satisfying  h a s t h e same  K  V  (X^)  finite  has  P ).  t H- T ( _ ) , h ) i s i n c r e a s i n g a n d r i g h t - c o n t i n u o u s 9.27,  on  fc  P  enlargement o f  (Bp-stopping  is a  fc  T  mod  to  f  (\)  (ft',8',8^,Q,IJJ) f (u))(h)Q  and . Q  We  claim  and t h a t  dimensional joint  has ( r e l a t i v e  o  to  P ).  on  that  a  n  f o r each [ 0 , °°) .  d  By  probability  a  i s an o p t i o n a l k ( £ )  f (u) (h)Q(doo,dh)  (doj,dh)  the stopping  the process  distributions This  (co,h) e ft' ,  times  (B_ )  relative  Q  <  t  <  a  o  to  i s somewhat e a s i e r t o  Q  -  n 21 3 , s o l e t u s c o n s i d e r t h i s  prove  when  every  s t o p p i n g time  i s standard,  joint  distributions  are right.  Let  175 -  D = ]R  n  , E =  a n d we n e e d  , l e t 0 <_ t 0  g  be a c o n t i n u o u s  subset  of  D  3  .  f u n c t i o n on  case  E  first.  In this  o n l y check  that the  < ... < t . < ° ° , a n d l e t J  1  which vanishes  3  case,  outside a  compact  Then g(B'  ,...,B'  )dQ  l j  fc  fc  re  1  lim  0  e  e+0  )ds  ,B;  g(Bl  t +s  dQ  t.+s 3  ±  (*) lim  e4-0  lim I-**  lim  lim  e+0  £->~  1 e  re g(Bl  0  1  ,...,B'^ Jds  t +s ±  g(B"  e  lim  0  ;  e+0  0  g(X  ,...,X 1  t o show t h a t  i  £  g(X  lim  suffices  ..B«  (  i  )  )d.  dR  t.+s 3  +  lim  e+0  This  j  k(£) ' t s 1  d Q  +  s  ,...,X V  ) d s dP 3  S  )dP 3  (B^, )  h Q  <  t  <  O  0  a  s  t  n  e  r  i  g  h  t  finite  k  W  -  dimensional further above of  joint  distributions.  explanation.  calculation  the  step  (*)  (We  hold  that n  n  , but  , and  l  i  +  s  the  a l l the the  step  method  requires  steps  in  the  of v e r i f i c a t i o n  when  n _> 3  .)  (_,h))ds  2  l  (*)  other  i s simplest  ( o ) , h ) , ...,B'  g(B'  0  remark  on  -  However,  f o r any  depends  Well,  176  + S  re g  (  h ( t  B  Q  Now  i f  h  all  but  countably  on  t  of  D  m  -»• h  in  , except and,  J  Thus  H  as  for  t =  ||B  w  )  ' - - "  9.24). g  (  Now  w e f t ,  the  ,  (B^  )  )  d  s  i =  l,...,j  depends  by  - a .s.  P  a  for  i s supported  | | ->- °° P  t  h ( t . + s )  B  ( t . + s ) •+ h ( t . + s ) m i l  ; but  0 0  (  (see  - a.a.  y  h  s  n__3, P  s )  +  then  many  at  1  a  as  continuously  compact t -»-  , for  subset  .  0 0  map  re 8  (  h ( t  B  Q  is  continuous Now Let  Define  on  H  .  suppose  n  =  D, f  on  E,  This 1 or  t^,...,tj  ft  x H  1  s )  +  (  w  )  ' - - "  justifies 2  B  the  h ( t  j  +  s )  step  (  w  )  (*)  )  d  s  i n the  case  , and  g  be  as  before.  Fix  e e  [ 0 , °°)  by  =  g  (  B  Q  and  f o r each  a e  h(t +s)  (  1  [ 0 , °°)  define  t  ° '"-' h(t. s) )  B  (  w  )  )  d  s  +  f  on  cL  ft  x H  by  re f  (0),h)  8 0  (  B  >_ 3  .  re f(oi,h)  n  h ( t  1  +  s ) A a  (  w  )  ' - - - '  B  h ( t  j  +  s ) A a  (  w  )  )  d  S  .  -  Then in  f , f (0<a<°°)  i t s second  177 -  are uniformly  v a r i a b l e , and  bounded,  f = f  each  o n ft x H a  H  a  = {h e H  : h < a —  generalization  But  i f h  then  h  Hence  lim  sup  a-*>°  k  i n  h < a -  on  Q(S7*\ftxH  a  f  But  +  Now small in  I  I •+  0 0  f dQ  dQ -  slight  h  i s bounded by  Thus e a c h  (fixH ) .  W  t  of H  a h  on i n  [ 0 , t . + e) [ 0 , t ^ + e) ,  i s closed  a  I t follows  i n  H .  that  a  f  f  k(£)  - f dQ a  a  a  dQ -  "  f  the f i r s t  by t a k i n g  f  d C  \(i)  d  a  W)  term on t h e r i g h t hand  a large,  by t a k i n g  a  , f o r each  a  justification  .  k  8.9 a n d a  ) = 0 .  f  +  oo  using  that  [ 0 , t . + e) . 3  i+  Now  continuity point  Q(ftxH ) _> l i m s u p Q  , where  (f2'\(fixH ) ) = 0 .  and each  f o reach  a  lim  H  Q  i s continuous  a  [0, t . + e)} . J  o f 8.10, o n e f i n d s  h ( t ) <_ a  whence  on  f  the t h i r d  term  side  Thus  o f the step  (*) a b o v e  c a n b e made  c a n b e made s m a l l  l a r g e , w h i l e the second .  here  f  term dQ  i n the case  goes  .  This  to  uniformly 0  as  supplies the  n <_ 2 , s o we  now  -  know t h a t  (B^ )  has the r i g h t  178 -  finite  dimensional j o i n t  distributions  t relative times t  e  T  to  00  i n a l l cases.  I t remains  are standard r e l a t i v e  to  .  .  t  [0, )  Q  measure on  Let E  o = law(X  and  t+1  a(D) = 1 .  t o show t h a t  Q  when  Then  a  the s t o p p i n g  n = 1 or 2 . L e t is a  probability  Now re  law(B' L  ;Q) = l i m  t  e+0  the second  law(B' T  ;Q  t+s  , %)  k  (The  limits  any  a e  step follows law(B'  W  T  ;Q)ds  0  e4-0  = lim  where  law(B'  -  lim £-*=°  t+s  —  o  £  l a  from the f a c t  ;Q)  f o r each  "  ( B  i  ; Q t + 6  M^)  ) d s  •  that I , a s was  shown  above,  t+s  are with respect  t o the vague  topology.)  Now  consider  [ 0 , °°) . re  Then  law(Bl t T  ;Q) = l i m e+0  a  lim  e  law(B' 0  lim  l  e+0 l-**> where i n t h i s f  case, the second  i s a bounded  is  continuous i n Now  for  0  <_  a  w  (  on  E  f(B, , , s (u))ds h(t+s)Aa  h s <_ —  and  B  T  step follows  continuous function  (o),h)  A a  ;Q)ds  t+s  k ( £ ) >_ 1 ,  t  +  s  Aa  ;  Q  k(£)  )  d  from the f a c t t h e n t h e map  S  '  that i f  -  179  -  law(B U  Applying  8.7  and  8.8,  one  law(Bj,  A a  finds  ;Q) >  11.12,  T  is  a  > u  that  law(B^,  U  Then, by  )  ;Q. t+s  ;Q)  U  standard r e l a t i v e  to  Q  •  In  the  case  n = 1  , the above  c o n t i n u o u s m a r t i n g a l e c a n be Brownian  motion  by  standard  s t o p p i n g times  theorem  embedded  just  i n an  says  optional  that  a result  due  to Monroe  [1].  stopping times, rather  ones.  a s t a n d a r d s t o p p i n g time  Monroe i n e f f e c t though is  an  he  t o show t h a t proves  the  does n o t e x p l i c i t l y  interesting  embedding, standard  that  result  i t i s perhaps  s t o p p i n g times  converse  define  i n i t s own easier  t o work w i t h  than w i t h  remark  i s minimal. n = 1 ,  standard stopping times. but  This  f o r the purpose  the d e f i n i t i o n  the d e f i n i t i o n  of  than standard  i s t r u e when  right,  of  family  (We  t h a t Monroe works w i t h m i n i m a l I t i s easy  right-  enlargement  means o f 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 —  any  of  of  of minimal  stopping  times.)  12.17.  be  an  there  Corollary:  Let  I =  {0,  n-dimensional p o t e n t i a l is a probability  measure  -1,  -2,...}  process with u  on  ]R  n  and  time such  let  (A,F,F ,X ±  set  I  that  U  . y  ,P)  Then is a  0  potential  (and  p({9})  =0  i f  n = 1  o r 2 ) , and  an  optional  enlargement  -  (ft',B',B',Q,ifO  (ft,B,B ,P )  of  180 -  , such that  U  t  there  BJ.-stopping  are  txmes  T  which  are standard  and  (B^ )  have  0 ^ -  relative  - l -  T  T  - 2 ^  Q , such that  to  t h e same j o i n t  the processes  distribution  (where  (X^)  B| =  • f  I  that  i for  0 <_ t <_ °° ) .  Proof:  There are i n d i c e s  i ^ > i ^> i ^ >  for  y  i s some m e a s u r e o n law(X.)  Now 8.8  (if  U  law(X i  1  E  f  U  o  r  check  that  measure this  y  on  y ( E ) <_ 1 .  every  i  .  Thus,  b y 8.7 a n d  ) dy ->  2  on  D = ]R  n  .  U  y  dy  ( I f n _> 3  i tis trivial  convergence holds.)  law(X.) But as i  U  law(X increases  1  i -> -°° .  as  i  ( H e n c e we a l s o h a v e  decreases.  that  law(Xj  Thus ->• y  -»• -°° .) If  n = 1 or 2  lim r-x»  E = 3Rg ,  i s a p o t e n t i a l and  y  U  e v e r y good  f  )  u  n = 1 o r 2) ,  satisfying  law(X  for  such  j  every compactly supported continuous f u n c t i o n  where  to  in  f(x)y(dx)  f(x)law(X. ) ( d x ) 1  .••  t h e n , b y 8.9,  ssup up law(X.) ({x e D i  : | |x| | >. r } ) = 0  )  U v a g u e l y as  + U  -  In  this  Thus i f  c a s e we  a l s o have  n = 1 or 2  then  y  181  -  law(X^)({9})  = 0  is a probability  for a l l  measure  i  .  and law(X.)  y({9}) = 0 we  .  can deduce  as  9  On  the o t h e r hand, i f  that  i s an i s o l a t e d  !->•-«>.  f o r each  (A,F,F.,P)  sup  each  i such  surely But  y e D  and  from  X  this F  is —oo  L  case  E  too,  .  a potential  i  is a probability  , ($(X^,y))  .  P  1  also, as  measure.  Thus by  V,  T21  o f Meyer  1  that  , as  U (y) y  < »  $(X^,y)  t  !->•--, t o an  i t follows  that  F  where  =  Clearly  law(X  nF.  p r o c e s s , where  =  F  0  .  -oo  X  , and  and  =  Y 0  X  X.  such  that  P —  (A,F,(F.) Now  .  -*• X  i  J = Iu{-°°} •  almost  function  : A ->• E  —GO  2.  ^  ) = y  converges  integrable  there exists  —oo  G  >_ U  y  i s a supermartingale over  E($(X.,y)) = U (y)  -measurable,  1 -> - «  U  •*• y({3})  , law(X )({9}) y  —oo  as is  in  X  of  then from  (see p r o o f o f 7.8);  i  y e D  , and  1  _> l a w ( X > ( D )  point  Thus i n t h i s  Now  for  u(D)  n j> 3  a.s.  00  (x  )  P)  let  - »  and l e t  G  t  G and  so  in  00  )  F0  -  1=_  Y  v  Y  t t  (A,F,G ,Y »P) t  t  family  X  n  for  0  = X_  x  for  1 < t < |<  t <  0 0  1 ,  is a potential  process, with  , 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  an o p t i o n a l enlargement  bution  =  on.  Then [0,  t  =  law(Yg) = y  o f Brownian  motion  so  time s e t  c a n be  embedded  with i n i t i a l  distri-  , b y means o f 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  o f s t a n d a r d s t o p p i n g t i m e s , by  12.16.  I t i s now  c l e a r how  to  [1]  -  complete  In  the p r o o f .  the above  182  -  <  g  corollary,  i t i s necessary  ment o f B r o w n i a n m o t i o n , e v e n i n t h e o n e shown by  12.18.  the f o l l o w i n g example,  Example:  Let  let  by  (L,L,A)  , where  t h e measure  on  L  B' = B ° ij; t t  T!  on  be  (ft',B',8j.,Q,40  and  Let  y  ft'  be  which  for  0 < t < » — —  and  T^  _> T  _>  1  - 1 , -2,  that  shall  standard,  i = 0,  Chacon.  in  ]R  ,  n  (ft,B,B ,P ) M  T  , and  and  1,  V.  is  to  {1}  2,...  A  is  o  .  define  : ||B.(a>)|| = t  —  > 0  : | |B. (u>) | | t  (B  , by  TQ  deduce  is a potential  1  2  1  }  1  i s standard  relative  and  process with  time s e t  12.6.  T ^ >_ ...  )  T  =  2 " }  i  (B^,)  i We  {0}  L  This  • • • .  ...}  Suppose such  to  ( B ^ ) - s t o p p i n g time which  is a  i {0,  For  0  enlargement of  > 0  —  (ft',B',B^,,B^,,Q)  Thus  .  b y R.  set of  enlarge-  by  T! ( W , 1 ) = i n f { t x  each  mass  an  case.  mass a t  , L = power  assigns  T:(OJ,0) = i n f { t x  Then  1}  invented  point  the product  L = {0,  dimensional  w h i c h was  the u n i t  to consider  (B^-stopping  are  y-standard  have  t h e same j o i n t  times  distribution,  i a contradiction.  E^(T.) x  (U  y  y. - U ) 1  W e l l by , where  8.13,  as  y.. = l a w ( B x  T  T  ±  is ) =  y-  law(B^,)  to  -  But  law(B^,,) = j ±-i  It as  follows  °±  w  distribution  on  the sphere o f r a d i u s  2~*  as  T. 4- 0 P i  that  i -> -°° .  8 = n8 y  y  .  E ( T . ) -> 0 ^ l M  Let  T = lim  B  Now  y  P (T=0) = 1  = {E  and  y  to  8  Now  f o r each  y  +  n  e  r  e  f  8  y  r  e  a  c  J  h  i -»•-<» .  T_.. .. i  e  o  : E  Hence  (8 (8) ) t  As As  y  y  {T £  n  > °j  is  t}  e  i  s  t  h  e  centred  at  -  P  a.s.  right-continuous,  B  y  for  0 £  t < »}  .  But  u  since  i  since  , let  every  i s a point  E  = {||B  s e t of^ mass,  P -measure  8  || = 2 } 1  zero  y  =  y  .  {0, ft} mod  Then  belongs  P  P (E ) y  ±  .  y  = j  i But  we  a l s o have  P (E. I y  n E . ,) x-1  = Q(||Bl,||  = 2  , ||B',  1  i  = Q(ftx{l})  T h u s , mod  Then  0  i  1  since  -  \  a  uniform unit  183  P  y  E e n B i  triviality  , E = E 0  = 8  y  y  i of  B  y  .  -1  2 " ) 1  1  i-1  = |  = E = -2  , but  || =  .  •••  .  Let  n  E = i  P (E) y  = ~  , which  =  0  u j  =  i  E. l  contradicts  .  the  .  -  13.  184 -  APPENDIX OF MISCELLANEOUS NOTATION AND  13.1.  We  positive  take p o s i t i v e m e a s u r e we  increasing greatest  If  and  13.2.  13.3.  really  j u s t mean n o n - d e c r e a s i n g ) .  f  f~  i s any  v  , read  join,  [- ,°°]-valued (-f)vO  denotes  ]N  denotes  the s e t  Q  denotes  the s e t o f r a t i o n a l numbers.  llxll  numbers.  denotes  the usual  x : =  use the symbol n £  g2 — j  1=1  3x.  A =  of natural  then  n  l 5  ||x||  We  denotes  +  numbers.  {0,1,2,...}  ( x . . . , x ) e ]R 1 n  norm o f  f  upper  .  the s e t of r e a l  euclidean  denotes  least  then  denotes  x =  (though by a  , r e a d meet,  function  00  denotes  A  ]R  If  13.4.  positive  j u s t mean a n o n - n e g a t i v e o n e ; a l s o , b y  lower bound, w h i l e  bound. fvO  we  t o mean s t r i c t l y  TERMINOLOGY  2  ( x j + ...+  A  2  1  /  2  x )  .  2  i n two w a y s ;  , and t o denote  t o denote  symmetric  the  difference  Laplacian:  of sets:  l AAB =  13.5.  Let  (A\B)u(B\A)  X  .  be a t o p o l o g i c a l  the  interior  of  E  , E  the  f r o n t i e r of  E  (3E = E \ i n t (E))  function  on  open  X  i n  remark  X  denotes  space.  t h e n we  f o r each  i s t o emphasize  U  shall open  For  the closure  say i n  f  .  If  of f  E  , int(E) , and  is a  0 0  say  ; the point "f  8E  denotes denotes  [-°°, °°]-valued  i s continuous i f f  [-°°, ]  t h a t when we  E c X  f ''"[U]  of this  i s c o n t i n u o u s " , we  is  -  are  not  suggesting that  not  use  the e x p r e s s i o n  is  another  continuous  i t assumes o n l y  functions  into  Y  u  from  the  is a  [-°°, u  semicontinuous  function  , denoted  always  , we  that  x  i s the  13.6.  An the of  or  speak  i f  power s e t o f of  x  i s an  i n f u(y)  the  v <_ u  of  lower  lower; a  largest  collection  of  For  isolated point  of  X  i f  x  denotes  sup Vet/_ x  of neighbourhoods  a set such  and  point  inf yeV\{x}  of  o f a measure, w i t h o u t  X  X  is a limit  of  X  x  "real-valued",  o u t e r m e a s u r e on  subsets  then  supremum o f any  x  collection  When we  "signed"  the  subsets  largest  satisfying  s e t of  i s again lower-semicontinuous.  l i m i n f u(y) -y y  f  X  , i s the  Y  C(X^R) .  closed X  If  = y  remark  u  on  shall  have  lim  We  on  functions  u(x) u(x)  by  the  denotes  the  function  exists since  lower-semicontinuous x e X  v  ; C(X)  We  sense".  denotes  g e n e r a t e d by  °°]-valued  of  each  X  o-field  regularization  function  values.  " c o n t i n u o u s i n the extended C(X,Y)  X  such  finite  then  denotes  If  -  t o p o l o g i c a l space,  Borel X .  185  u(y)  , where  x  a qualifier  such  we  shall  always  X  is a  [ 0 , <=°]-valued f u n c t i o n  t h a t whenever  R c u S  we  have  as  mean a p o s i t i v e m e a s u r e .  S  i s a countable  y (R) <^  \  y(S)  ;  y  on  collection (In  SeS particular, If  y  i s an  taking  R = 0  S  and  o u t e r m e a s u r e on  X  =  0  , we  , then  find  M  denotes y  of  y-measurable  subsets  of  X  :  that  y(0) the  = 0)  .  a-field  -  M  = {R c X  : f o r any  186 -  S c X  , y ( S ) = y (SnR)  + y(S\R} .  y H  If  i s a collection  measure on  for  every  X  i s an outer  R c X space  measure  y  on B o r e l X  compact  set  defined  13.7.  A  .  Meyer  13.8.  then  an  on  X  such  that  : R c H e H}  that  (X,A)  A e A  D  explanation,  We  on  refer  denotes  y. A  denotes  the reader  region  centre  X  and  is a  , we mean a f o r each  space  closed  and  y  is A  the measure on  then  i s an open b a l l  t h e symbol  .  Then  PI(f;V)  i n lR  x e V of  measure on  f o r which V  3V  , r  =  denotes  n  r  3V  a  , used  and  f  is a  = the "surface  without  [- , ] - v a l u e d 00  ~  Ii  x  "  p  |  l|x-zl|  n  1  0 0  2 f (z) d a ( z )  t h e i n t e g r a l makes s e n s e , w h e r e of  theory.  D .  2  i s the radius  , and  G^  of this  the f u n c t i o n d e f i n e d by  r  1  a  n  and o f  [ 1 ] , D e l l a c h e r i e and  [1] f o r e x p o s i t i o n s  V  3V  t o Meyer  f  the  H c M ~~ y  of Souslin sets  the Green f u n c t i o n of  PI(f;V)(x)  a l l  X  i s a measurable  tf-outer  If  i s finite  some u s e o f t h e t h e o r y  i s a Green  Suppose  for  .  (or i n )  y(K)  then  [ 1 ] , and B r e s s l e r and S i o n  function  X  y . ( B ) = y(BnA) .  sets.  If  13.9.  such If  and  We h a v e made  analytic  y  of  then by a measure on  K c x  by  measure  , y(R) = inf{y(H)  topological  a measure on  of subsets  V  , a  area"  p  is  i s the "surface  area"  o f the u n i t sphere i n  n ]R  n  .  with  Note respect  PI to  stands a  f o r "Poisson  over  3V , t h e n  integral". PI(f;V)  If  f  i s integrable  i s harmonic i n  V  and  -  and  lim PI(f;V)(x) = f(z)  187 -  f o r each  z € 8V  a t which  f  i s  x->z continuous —  13.10.  s e e 2.4 a n d 2.7 o f Helms [ 1 ] .  We f o l l o w  the convention that  a d d i t i o n , we a d o p t  0  the convention that  times 0  0 0  equals  0  .  In  times undefined equals  0 ;  s e e 6.1 f o r a n e x a m p l e o f t h i s .  13.11.  Suppose  F , and  measure on law(i(j)  (A,F)  : A -»• E  , and law(^;P)  by  y (A) = P (ip [ A ] )  and  least  valued  t h e measure  function  h  on  A , then  E  such  u  notation  i n a given situation.' on  spaces, P  (F,E)-measurable.  is  a l l denote  F-measurable f u n c t i o n  E-measurable  a r e measurable  ; we u s e w h i c h e v e r  X  ambiguous  (E,E)  and  If  on  Then E  i sa ^(P) ,  defined  i s most s u g g e s t i v e  g  i s a  E(g|i|;,E)  [-°°, °°]denotes any  that  h(x)lawGjj) (dx) =  g dP  A  for  a l l A e E ; i f l a w Op)  or i n t e g r a b l e , Finally, (P ) x xeE and  each  i s  such a f u n c t i o n  a disintegration of probability  of  h P  exists,  and  g  F  i s either  non-negative  lawGjO .  a n d i s u n i q u e mod  with respect  measures on  such  to that  , E  i s a family  f o r each  A e E  F e F ,  x B- P ( F ) is x  P(Fn^  1  E-measurable, and  [ A ] )=  lawdlO  is  a-finite,  (F)law(i|0(dx) .  P JA  If  a-finite  X  F is countably generated, and  P  is  -  inner  regular with  gration  of  P  with  respect respect  188  -  t o a semicompact to  \\i, E  class,  exists.  then a  (See a l s o  disinte9.4.)  -  189  -  REFERENCES  R.  B a x t e r a n d R. V. 1.  Potentials  Chacon  of Stopped D i s t r i b u t i o n s .  111. J . M a t h .  18  (1974)  649-656. 2.  Compactness verw.  3.  M.  o f S t o p p i n g T i m e s , Z.  Gebiete,  40  Enlargement of  (1977)  o-algebras  Can. J . Math.,  29_ ( 1 9 7 7 )  B l u m e n t h a l a n d R.  K. G e t o o r  1.  Markov  Processes  Wahrscheinlichkeitstheorie  169-181. and C o m p a c t n e s s  o f Time  Changes,  1055-1065.  and P o t e n t i a l  Theory, Academic P r e s s ,  1968.  Brelot 1.  Minorantes Math.  2.  Centre  Sorbonne,  Bressler 1.  I X , 24. ( 1 9 4 5 )  E l e m e n t s de l a T h e o r i e (1965),  W.  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Die Stoppverteilungen eines Markoff-Prozesses mit 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 M a t h . , 3 (1970) The S t o p p i n g M a t h . , 14  A. B.  Distributions  (1971)  o f a Markov  Process,  321-330.  Invent.  1-16.  Skorohod 1.  Studies  i n the Theory  o f Random P r o c e s s e s ,  A d d i s o n - W e s l e y , 1965.  -  INDEX OF S E L E C T E D  analytic B  192 -  NOTATION AND  sets  TERMINOLOGY  13.7 5.4  t  8°,  B°, B\ B j ,  8,  B , °B , J t  B  5.4  balayage, b a l ( , , )  1.11, 2.  base(  , )  3.1  base(  )  3.10  D  5.4 A  disintegration  13.11  E(  13.11  | , )  enlargement  9.7, 9.12  filtered  measurable  filtered  measure space  fine  space  9.11 9.11  topology  3.12  fringe(  , )  3.1  fringe(  )  3.10  G  13.8  D  Gy  1.10  generalized  Brownian  motion  process  5.5  good m e a s u r e  8.6  Green  function  1.9  Green  potential  1.10  Green  region  1.9  law( lower  ), law( , ) regularization  13.11 13.5  - 193 13.6  measure optional outer  enlargement  9.12  measure  13.6  5.1  p P  5.3  t  5.4  P ,  P  PI(  ; )  y  polar  X  13.9  set  1.9  potential  1.4,  potential product  process  12.2  enlargement  9.17  randomized  random v a r i a b l e ,  randomized  stopping  reduite,  rrv  time, r s t  9.1 9.2  red( , , )  1.11  R i e s z measure  1.2  RRV(  , , ; )  9.21  RST(  , , ,)  9.23  Souslin  1.10  sets  13.7  standard  randomized  standard  stopping  stopping  time  time  11.3 8.2,  12.4  5.4  T  A thinness  3.1,  U , u j , U_ y  1.4  y  12.1  U  3.10  - 194  universally 6,  measurable  6  9.5 1.1 5.4  t  u  space  -  13.6  A  $  1.3  Q  5.4  (ft,B,B ,B ,e ,P ) x  t  t  t  5.4  5.2  

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