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Hyper-finite methods for multi-dimensional stochastic processes Reimers, Mark Allan 1986

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HYPER-FINITE METHODS FOR MULTI-DIMENSIONAL STOCHASTIC PROCESSES By MARK ALLAN REIMERS  •Sc.,  B . S c , The U n i v e r s i t y o f Toronto, 1978 The U n i v e r s i t y o f B r i t i s h Columbia, 1983  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Department o f Mathematics The U n i v e r s i t y o f B r i t i s h Columbia  We a c c e p t t h i s  t h e s i s as conforming  to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA  October 1986 ©Mark  A l l a n Reimers, 1986  In  presenting  degree  this thesis  in partial fulfilment of the  for  an  advanced  at the University of British Columbia, I agree that the Library shall make it  freely available for reference and study. copying  requirements  of  department  this thesis for scholarly or  by  his  or  her  I further agree that permission for  purposes  extensive  may be granted by the head of  representatives.  It  is  understood  that  copying  my or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6(3/81)  ABSTRACT  In t h i s t h e s i s we i n t r o d u c e use  Non-Standard Methods, i n p a r t i c u l a r the  o f h y p e r f i n i t e d i f f e r e n c e e q u a t i o n s , t o the study o f space-time random  processes.  We o b t a i n a new e x i s t e n c e  theorem i n the s p i r i t o f K e i s l e r  (1984) f o r the one d i m e n s i o n a l h e a t e q u a t i o n f o r c e d n o n - l i n e a r l y by white noise.  We o b t a i n s e v e r a l new r e s u l t s on the sample p a t h p r o p e r t i e s o f  the C r i t i c a l B r a n c h i n g Measure D i f f u s i o n , and show t h a t i n one dimension i t has  a d e n s i t y which s a t i s f i e s a n o n - l i n e a r l y f o r c e d h e a t e q u a t i o n .  We  also  o b t a i n r e s u l t s on the dimension o f t h e s u p p o r t o f the F l e m i n g - V i o t P r o c e s s .  Edwin  Perkins  ACKNOWLEDGEMENT  I would  like  t o t h a n k E d P e r k i n s f o r many h e l p f u l c o n v e r s a t i o n s a n d  o f my f r i e n d s f o r p u t t i n g up w i t h my  complaining.  iv  TABLE OF CONTENTS page ABSTRACT  i i  ACKNOWLEDGEMENT  i i i  TABLE OF CONTENTS  iv  TABLE 1  1  CHAPTER ONE  - Introduction  3  1.1  Why t h i s T h e s i s  3  1.2  SPDEs  4  1.3  The Dawson C r i t i c a l Measure V a l u e d D i f f u s i o n  7  1.4  The F l e m i n g - V i o t  9  Process  CHAPTER TWO - Non-Standard A n a l y s i s and P r o b a b i l i t y 2.1  Some D e f i n i t i o n s and N o t a t i o n s  2.2  Non Standard White N o i s e  14  2.3  Adapted S t o c h a s t i c I n t e g r a l s  16  CHAPTER THREE - The Heat E q u a t i o n  from Non-Standard A n a l y s i s  10  w i t h Non-Linear S t o c h a s t i c F o r c i n g  10  20  3.1  Scope  20  3.2  White N o i s e on the space  3.3  Hyper-Finite  3.4  Some U s e f u l I n e q u a l i t i e s  26  3.5  Bounds on Moments o f  27  U  Difference Equations  U  23 24  tx 3.6  Bounds on Moments of- S p a t i a l D i f f e r e n c e s  31  3.7  Bounds on Moments o f Temporal D i f f e r e n c e s  33  3.8  S-Continuity  35  3.9  S o l u t i o n o f the SPDE  and the Standard P a r t  CHAPTER FOUR - The Dawson C r i t i c a l B r a n c h i n g D i f f u s i o n 4.1  Introduction  38 41 41  V  4.2  A H y p e r f i n i t e Difference Equation  43  4.3  The C o e f f i c i e n t s  45  4.4  The T o t a l Mass P r o c e s s  4.5  S-Continuity  4.6  C h a r a c t e r i z a t i o n by a M a r t i n g a l e  4.7  New R e s u l t s on the Dawson Measure V a l u e d D i f f u s i o n  Q M  52  o f the P r o c e s s  55 Problem  62  CHAPTER FIVE - The C r i t i c a l B r a n c h i n g D i f f u s i o n i n One Dimension  66 70  5.1  Introduction  70  5.2  The SPDE and the Measure D i f f u s i o n  71  CHAPTER SIX - The Support o f the F l e m i n g - V i o t  Process  73  6.1  I n t r o d u c t i o n and C o n s t r u c t i o n  73  6.2  The Dimension o f a P u t a t i v e Support S e t  75  6.3  A U s e f u l S t o c h a s t i c D i f f e r e n t i a l Equation  80  6.4  V e r i f i c a t i o n o f Support  83  APPENDIX A — Some I n e q u a l i t i e s Used i n Chapter 3  '  88  A.1  Purpose  88  A.2  Some I d e n t i t i e s  88  A.3  Some I n e q u a l i t i e s  91  APPENDIX B. - I n t e r n a l S o l u t i o n s t o SPDEs i n H i g h e r Dimensions REFERENCES  97 100  TABLE 1  of  Notations  Meaning I n f i n i t e s i m a l g r i d spacing i n  space  I n f i n i t e s i m a l g r i d spacing i n  time  At/Ax  2  *-countable g r i d representing *-finite  grid representing  R^ 1  [0,  t  ]  2 I n t e r n a l I.I.D. on T x X Borel  S-L  subsets of  random  variables  X  Coefficients for a discrete * - f i n i t e G r e e n ' s f o r m u l a ; a l s o d e n s i t y o f an i n f i n i t e s i m a l random w a l k I n t e r n a l s o l u t i o n to s t o c h a s t i c difference equations Internal  *-finite  analogue  Internal  *-finite  analogue of  of  I n t e r n a l s o l u t i o n to unforced analogue o f the h e a t e q u a t i o n L e b e s g u e m e a s u r e on  R  I n t e r n a l m e a s u r e on  X  mass  Ax  d  , which  a t each g r i d  Space o f p o s i t i v e f i n i t e  *-finite  A *-finite  assigns  point m e a s u r e s on  !Space o f p r o b a b i l i t y m e a s u r e s on Space o f functions  infinitely with  R^  differentiable  compact s u p p o r t i n  R  R  d  Space o f  C  f u n c t i o n s bounded on  Space o f  C,  f u n c t i o n s whose  b  d e r i v a t i v e s a r e bounded on  R  R  second d  P r e d i c t a b l e square (increasing) associated with a process x  process  3 CHAPTER ONE  Introduction  1.1  Why t h i s  The  Thesis  a i m o f t h i s work i s t o i n t r o d u c e nonstandard  particular  methods, and i n  t h e use o f h y p e r f i n i t e d i f f e r e n c e equations, t o t h e theory o f  multi-dimensional stochastic processes.  Non-standard  analysis  i s a  p a r t i c u l a r l y a p p r o p r i a t e t o o l when i n v e s t i g a t i n g r a n d o m p r o c e s s e s r e g i o n i n space, processes case.  which  over  a  e v o l v e i n t i m e , e s p e c i a l l y when t h e s u p p o r t o f t h e s e  i s c o n f i n e d t o a r e g i o n o f i n f i n i t e s i m a l volume, as i s f r e q u e n t l y t h e  The e v o l u t i o n t h r o u g h  time o f a l l t h e processes  may b e d e s c r i b e d b y t h e h e a t o p e r a t o r  g  discussed i n this  work  A , but i n the opinion o f the  ot author, nonstandard  t e c h n i q u e s may b e e q u a l l y f r u i t f u l  p r o c e s s e s whose d e v e l o p m e n t t h r o u g h as  well.  time  i n the analysis of  i s d e s c r i b e d by d i f f e r e n t  operators  4  1.2  SPDEs  The approach to the theory of SPDEs which we w i l l follow has i t s home i n the theory of multiparameter processes,  and i n p a r t i c u l a r i n the theory of multi-  parameter stochastic integration that has been developed i n recent years. Walsh (1986) contains a systematic  treatment of t h i s theory.  emphasizes sample path properties.  This approach  An a l t e r n a t i v e approach considers SPDEs as  stochastic evolutions on a space of functions, and emphasizes a n a l y t i c properties.  See Dawson (1975) and  information on t h i s approach.  (1985) and the references there for further  We w i l l not follow i t closely here.  The type of SPDE we w i l l be considering most often i n t h i s work, i s  Cl-1).  where  |r- = Au + f (u) W^ , dt tx t e R  +  and  x e R  .  d  W  i s "white noise" on  R  +  x R  d ;  that i s , the  tx derivative, i n the sense of d i s t r i b u t i o n s (or generalized functions) of a random process i)  E(W^)  iii).  if  = 0  W^  , indexed by sets  2 i i ) E(W^)  = X(A)  A n B = 0 , then  W^  A <= R  where  X  X  .R^  •>  such that  i s Lebesgue measure  i s independent of  on white noise see Walsh (1986).- Chapter 1. of the point values of  +  f (u)  Wg  .  For further information  i s a real-valued function  u .  Equation Cl-1). cannot possibly hold i n the c l a s s i c a l sense of an between the values of functions at every point i n the domain. far too rough.  term i s tx Rather we usually i n t e r p r e t (1-1) i n the weak sense. That i s , OO  i f we multiply <>j (~x) then  (.1-1) by a  c c  The  equation  W  £  ( R )  and integrate over a rectangle  (smooth, with compact support) function CO,T]  X A , where  A  contains  supp c>j ,  5  r  r  u  (1-2) A  Ox  <j> (x) dx =  Jo  (  u  JA  S  A<J>(x)dx)ds X  f(u  sx  )<j>(x)dW  sx  where the l a s t i n t e g r a l on t h e r i g h t i s the m u l t i p a r a m e t e r s t o c h a s t i c i n t e g r a l i n the sense o f I t o d i s c u s s e d i n Walsh be d e r i v e d  from  (1986)  Chapter 1.  Equation  (1-2) may  (1-1) taken i n the c l a s s i c a l sense, by i n t e g r a t i o n o f the middle  term by p a r t s . An e x i s t e n c e t h e o r y f o r (1-1) has been developed when  d = 1 ,  f : R -> R , i s a L i p s h i t z - continuous f u n c t i o n which grows a t most  linearly at infinity.  Dawson (1972) e s t a b l i s h e d e x i s t e n c e and uniqueness  these c o n d i t i o n s , u s i n g a H i l b e r t space Funaki  t  under  approach.  (1983), e s t a b l i s h e d the same r e s u l t , w i t h j o i n t c o n t i n u i t y o f  sample p a t h s i n in  i n the case  and i n x  t  and  x .  Walsh  (1981) e s t a b l i s h e d a modulus o f c o n t i n u i t y  f o r s o l u t i o n s o f an e q u a t i o n s i m i l a r t o (1-1) and i n v e s t i g a t e d  f i n e r sample p a t h p r o p e r t i e s , under the same c o n d i t i o n s on  f .  In Chapter Three o f t h i s t h e s i s an e x i s t e n c e r e s u l t i s - e s t a b l i s h e d f o r assuming o n l y c o n t i n u i t y and l i n e a r growth o f The  s i t u a t i o n f o r d >_ 2  f  for d = 1 .  i s entirely different.  may be regarded as a d e r i v a t i v e o f o r d e r  d + 1 I  The term  W  tx  i n (1-1)  o f a continuous f u n c t i o n o f cl  x R  unbounded v a r i a t i o n (the Brownian Sheet) on  R  no hope o f f i n d i n g a continuous f u n c t i o n  t o s a t i s f y the e q u a t i o n , even i n  the weak sense o f (1-2) . process  v , such t h a t  u  .  When  d >_ 2  there i s  The most we can hope f o r i s t o f i n d a continuous u  (1-1)  may be regarded as a d e r i v a t i v e , i n the sense o f  6  d i s t r i b u t i o n s , o f order  d - 1 , of  v .  In t h i s case  u  w i l l not, i n  g e n e r a l , have p o i n t v a l u e s , and i t i s d i f f i c u l t t o see what sense can be made of  the term  (1-2).  f(u ) sx  I n o r d e r f o r our t h e o r y o f s t o c h a s t i c i n t e g r a t i o n t o make sense o f  (1-2) we would need of  the q u e s t i o n i f  p o i n t values of The  o c c u r r i n g i n t h e s t o c h a s t i c i n t e g r a l on t h e r . h . s . o f  f(u )  t o be an adapted continuous  i s supposed t o be a r e a l f u n c t i o n o f the ( n o n - e x i s t e n t )  f  f o r which  u .  functions  (1-2) can be r e a s o n a b l y expected Walsh  (see Appendix B ) .  t o make  (1984) has shown e x i s t e n c e and  uniqueness o f s o l u t i o n s t o (1-2) i n t h i s case. results  This i s out  f  sense a r e the c o n s t a n t f u n c t i o n s .  his  process.  We were n o t a b l e t o extend  7  1.3  The  Dawson C r i t i c a l  Measure-Valued Jirina  Measure V a l u e d  Diffusion  Branching Processes  (MB P r o c e s s e s ) w e r e f i r s t  obtained  by  (1958) a s a l i m i t o f a b r a n c h i n g d i f f u s i o n o f a l a r g e number o f  particles.  These p r o c e s s e s were s t u d i e d e x t e n s i v e l y by Watanabe  co-workers,  and  Process".  We  and  l a t e l y many d e t a i l s o f t h e f i n e s t r u c t u r e h a v e b e e n o b t a i n e d  b y Dawson a n d H o c h b e r g w i t h t h e c a s e we  (1968)  (1979).  The  name o f Dawson i s p a r t i c u l a r l y a s s o c i a t e d  s h a l l s t u d y h e r e , h e n c e we  r e f e r t o i t o f t e n as the  w i l l h o w e v e r m o s t o f t e n make u s e o f a m a r t i n g a l e  "Dawson  characterization  o f t h i s p r o c e s s d e s c r i b e d i n R o e l l y - C o p p o l e t t a (1986). A s i m p l e c o n s t r u c t i o n , f o r the case o f a i n i t i a l follows.  L e t p a r t i c l e s be d i s t r i b u t e d  initially  on  according to a Poisson point process with intensity each p a r t i c l e  independently executes  t h a t each p a r t i c l e i.e.  expectation  p  1 a s s i g n a mass, o f ' — A  each  A  or a portion thereof, .  Suppose t h a t  on  R  .  Also  suppose u ,  t i m e s , w h o s e number i n a u n i t  I f the p a r t i c l e  i n t o two  splits,  both  and  time  particles, daughter  c a r e e r s from the p o i n t of b i f u r c a t i o n .  t o each p a r t i c l e ,  thereafter  branching with rate  , the p a r t i c l e d i e s or s p l i t s  outcome b e i n g e q u a l l y l i k e l y .  p a r t i c l e s begin independent  R^  motion  independently undergoes c r i t i c a l  a t f i x e d or e x p o n e n t i a l l y d i s t r i b u t e d  i n t e r v a l has each  a Brownian  L e b e s g u e measure i s as  We  o b t a i n a random measure on  now R  d  at  time. Now  suppose t h a t b o t h the i n i t i a l  r e c i p r o c a l of the weight  density of p a r t i c l e s  a s s i g n e d t o each p a r t i c l e )  a r e a l l o w e d t o go t o i n f i n i t y  i n s u c h a way  that  —•  and  A,  (which i s the  the branching r a t e  i s constant.  Then t h e r e  A  i s a l i m i t i n g p r o c e s s t a k i n g v a l u e s i n the space Dawson  (1972) i n d i c a t e d a c o n n e c t i o n i n  o b t a i n e d as a l i m i t o f a p a r t i c l e  system,  and  d  o f p o s i t i v e measures on  d = 1  u,  between t h i s  process  t h e s o l u t i o n t o t h e SPDE  R  8  d-3)  but f  |E  this  : u —>  connection /u  +  . 2 9x  ^  „  theory  t h r e e , we may now c l o s e  and hence  discussed  i n 1.2.  the  i n dimensions  This  since the function  (1-3) does n o t f a l l With the existence  d >_ 2 .  Lebesgue n u l l  process.  previously  Specifically  we  s e t i s a.s. never charged, i i ) the  Lebesgue s e t i s a . s . a continuous f u n c t i o n o f time,  iii).  convergence o f a sequence o f bounded f u n c t i o n s  a.s.  convergence o f t h e i n t e g r a l s o f those f u n c t i o n s w i t h  measure, u n i f o r m l y  like  t h e Dawson  to establish several  unknown r e s u l t s o n t h e s a m p l e p a t h s o f t h e Dawson P r o c e s s .  mass o n any g i v e n  the i d e n t i t y o f  H o w e v e r a n o n - s t a n d a r d a n a l o g u e o f ( 1 - 3 ) may b e  i s used i n Chapter Four  show t h a t i ) a n y g i v e n  theorem i n chapter  t o make s e n s e o f a n e q u a t i o n  a n d i t s s o l u t i o n may b e shown t o c o i n c i d e w i t h  construction  under t h e p u r v i e w  MB p r o c e s s i n o n e d i m e n s i o n .  A s m e n t i o n e d i n 1.2, i t i s d i f f i c u l t  constructed,  ,  t h e gap i n C h a p t e r Five,' and e s t a b l i s h  s o l u t i o n o f (1-3) w i t h  (1-3)  tx  h a s n o t y e t b e e n made r i g o r o u s ,  i s not Lipshitz,  of t h e existence  the  =  9t  on f i n i t e  time  intervals.  i n measure, i m p l i e s t h e respect  t o t h e random  9  1.4  The F l e m i n g - V i o t P r o c e s s  In F l e m i n g and V i o t  (1979) a measure-valued  p r o c e s s was  i n t r o d u c e d as a  l i m i t , under s u i t a b l e s c a l i n g s o f time and space, o f the Ohta-Kimura stepwise m u t a t i o n model.  F u r t h e r r e s u l t s on the s t r u c t u r e o f the sample paths have  been o b t a i n e d by Dawson and Hochberg  (1982).  Among these r e s u l t s i s the  f a c t t h a t f o r f i x e d times the s u p p o r t o f the random measure has H a u s d o r f f dimension not g r e a t e r than 2, almost s u r e l y .  In Chapter S i x we  establish  t h i s r e s u l t f o r a l l times s i m u l t a n e o u s l y u s i n g Non-standard methods t o a m p l i f y some o f the i d e a s o f Dawson and Hochberg  (1982).  For  further  i n f o r m a t i o n on the F l e m i n g - V i o t p r o c e s s , the r e a d e r i s r e f e r r e d t o s e c t i o n 6.1.  CHAPTER TWO Non-Standard A n a l y s i s and P r o b a b i l i t y 2.1  Some D e f i n i t i o n s and Notations from Non-Standard A n a l y s i s For  a r e a l i n t r o d u c t i o n the ideas of non-standard a n a l y s i s , with a minimum  of t e c h n i c a l apparatus, we r e f e r the reader to Cutland (1983). An i n t e r n a l object i n the non-standard universe i s one which may be referred to i n the non-standard language.  One of the consequences of the  t r a n s f e r p r i n c i p l e i s that i n t e r n a l objects described i n the non-standard language i n h e r i t a l l the q u a l i t i e s of standard objects that are described i n analoguous standard language. This i s u s e f u l when d e a l i n g with h y p e r - f i n i t e c o l l e c t i o n s , which may be treated as f i n i t e s e t s , though they are generally infinite. We w i l l u s u a l l y denote non-standard objects by c a p i t a l l e t t e r s or underlined l e t t e r s .  Unless otherwise noted, lower case roman l e t t e r s w i l l  stand f o r standard objects.  The embedding of a standard object i n t o the  non-standard universe w i l l be denoted by an a s t e r i s k (*) to the l e f t . We say that we denote t h i s if  x_ e *R  x ~ 0 ; x s y  |x| > n , f o r every  there i s a unique and c a l l  x  i s infinitesimal i f  x e R  means  n e N .  the standard p a r t of  k  n  and  F e  x_ « *x . We say that JC , denoted °x  *C(. R , R") , then k  ;  n e N ;  x e *R  is infinite,  i s f i n i t e (not i n f i n i t e ) then  or  may be extended, i n the obvious fashion, to any space. f e C[R ;R )  f o r every  x_ - y_ « 0 . We say  I f x e *R  such that  |x| < ^  x  i s near standard  s t ( x ) . These concepts In p a r t i c u l a r , i f  st(F) = f <=> ° F (x) =  f(°x)  for a l l nearstandard x e *R We s h a l l require t h e . f o l l o w i n g axiom of s a t u r a t i o n (see Cutland 1983, 1.9).  11  If  A n n neN a)  i s n o t empty.  I n f i n i t e s i m a l Underflow. a > 0 , x e S  x e S  2.1.1.  S-continuous i f f  Let F  s e t A , and e v e r y  F : *N -»- A  be i n t e r n a l ,  0 < °x <_ a .  (S  function  x ~ y => F (x)  function  which extends  f .  S £ *R , a n d s u p p o s e f o r  T h e n f o r some  e & 0 ,  f o r standard). F  : E c  *R  PS F (y_) . n s (* - R ) ' n  e  i s called  n  .. T h a t t h i s i s t h e a p p r o p r i a t e  t o l i n k s t a n d a r d a n d n o n - s t a n d a r d , - . i s shown b y ,  (Cutland  F : E <= *R i s  S  function  S-continuity  An i n t e r n a l  notion of continuity  T h e o r e m 2.1.2.  then  e <_ x_ <_ * a .  whenever  Definition  Let  whenever  A useful notion i s  iff  For every i n t e r n a l  : N -> A , t h e r e i s a n i n t e r n a l  some  sets,  Two c o n s e q u e n c e s o f t h i s a r e  Denumerable Comprehension. f  b)  i s a d e c r e a s i n g f a m i l y o f non-empty i n t e r n a l  {A } n neN  (1983), Theorem 1 . 6 ) .  *R  be i n t e r n a l .  S-continuous on  Nonstandard P r o b a b i l i t y  Then  °F  exists  and i s continuous  E . T h e o r y r e a l l y came i n t o  i t s own a f t e r - t h e d e v e l o p m e n t  of Loeb Measure.  T h e o r e m 2.1.3.  (see C u t l a n d  Every i n t e r n a l t  o  a classical  *finitely  ( 1 9 8 3 ) , Theorem 3 . 1 ) . a d d i t i v e measure space  o - a d d i t i v e measure space  F c L ( F ) , and i f A e F , then L(y_) (A) = inf{°y_(B) is  B e F , with The  of  L (y_) (B A A) = 0  extension of  by  (Q,  L (y_))  If  , such  rise:  that  A e L(F) , then  F u r t h e r , i f L ( y ) (A) <  00  , then  there  . (f, L(F),  I n the case o f p r o b a b i l i t y  (ft, F, P)  L(F),  (A) = °y_(A) .  | A £ B , A e F} .  (complete) measure space  C f , F, y_) .  L  (|J,  (|~, F, y_) g i v e s  L(v_))  i s called  t h e Loeb e x t e n s i o n  m e a s u r e s , we w i l l  F, P) , c o n t r a r y t o o u r u s u a l  denote t h e Loeb convention.  12  T h e o r e m 2.1.4. If  E  i s an i n t e r n a l  measurable.  field  Conversely  a - f i n i t e with respect to internal  st  ° F ( x ) = f ( x ) L ( y ) - a.e. .  If  has f i n i t e F  t h e n we may  F : [ •+  i s an  is  called  support,  2.1.5.  S-integraBle, i f f  F dp  F(x)  infinitesimal  6  |F|^  is  is  which  of  - 1  : [~ -> *R^  such F  and  T £ *R^  is  S-continuous  is  there i s  i f  00  a lifting , then  f : T  f  R  of admits  i s continuous,  and f o r which  (the near-standard p o i n t s ) .  d  [  is  that  i s called  If  (R )  Such an  f .  function  i s finite,  dy +  I < 6}  {x: 1F(X)  T h e o r e m 2.1.6.  F  An i n t e r n a l  measurable,  i . e . L ( y ) ( { x j f (x) ^ 0 } ) <  , V 21 e r n s t  a uniform l i f t i n g  F  F  L(E)  u {°°}  the case h e r e ) , then  Such an i n t e r n a l  obtain a l i f t i n g  ° F ( x ) = f(°x)  If  [ , and  is  (as i s a l w a y s  w i t h t h e same p r o p e r t y .  st  Definition  f : [ •->  E-measureable f u n c t i o n  f  a lifting  p_ , i f  3.1, 3 . 5 ) .  of subsets of  i f  L(y_)  an  f  (1983) T h e o r e m s  E-measurable f u n c t i o n then the p r o j e c t i o n s t ° F : [ + R  internal L(E)  (see C u t l a n d  F  on  ( f , F, y )  i s called  and  | F ( x ) I dp & 0 , f o r a l l  { x : | F ( X ) | > H>  H .  and i n f i n i t e  F  i s said  t o be  S-IJ  with respect to  S - integrable.  (see C u t l a n d  (1983) T h e o r e m 3 . 9 ) .  S - i n t e g r a b l e on  F dy = A  (f,F,  p)  F d L(y) A  , then f o r a l l A e F  ,  F  D e f i n i t i o n 2.1.7.  ( T , F, y_)  An i n t e r n a l measure space  i scalled ( E , F, y )  h y p e r f i n i t e r e p r e s e n t a t i o n o f a t o p o l o g i c a l measure space (i) (ii) (iii)  T  i s a h y p e r f i n i t e i n t e r n a l subset o f  F  i s t h e i n t e r n a l power s e t o f  [  i s y - m e a s u r e a b l e , i f f s t "*"(B) n [  In t h a t case  y(B)  The  c a n o n i c a l example  -  1  (8)  Let  Ax. l  i = 1, 2,...,d  (see C u t l a n d  R  , w h i c h we w i l l  i = l,...,d.  °M A x . ^ 0 . l I  X = {k. Ax.,...,k.Ax.)Ik. e *Z, k.I < M.} 1 i j D ' l l ' l setting the value o f  . i s the discrete use f r e q u e n t l y .  (1983) T h e o r e m 4 . 1 ) .  be any i n f i n i t e s i m a l , such t h a t  n D  i s L (y) - m e a s u r e a b l e .  of a hyperfinite representation  r e p r e s e n t a t i o n o f Lebesgue measure on  T h e o r e m 2.1.8.  i f f  *E  A set B c E  = L (y_) ( s t  a  A_ o n e a c h p o i n t  L e t M. e *N\'N l  be  infinite,  Then l e t .  D e f i n e a measure  x_ e X  t o be  A —  d II Ax- • i=l  on  X  by  Then  (X, P(X) , (A_)) i s a h y p e r f i n i t e r e p r e s e n t a t i o n o f t h e r e c t a n g l e { x | |x. |  °M^Ax_^} c_ R^ , e q u i p p e d w i t h L e b e s g u e  I n t h i s t h e s i s we w i l l p r o v e r e s u l t s a b o u t from h y p e r f i n i t e g r i d s  X  to  *R .  measure.  S-continuity of functions  A l l t h e p r e c e d i n g d e f i n i t i o n s and * d  theorems a p p l y as i f t h e s e f u n c t i o n s were s t e p f u n c t i o n s on a r e c t a n g l e Theorems a b o u t  S - c o n t i n u i t y on a h y p e r f i n i t e g r i d  in  X , c a n be t r a n s l a t e d  i n t o t h e o r e m s a b o u t t h e weak c o n v e r g e n c e o f a s e q u e n c e o f p r o c e s s e s o n a sequence o f f i n i t e such a t r a n s l a t i o n .  grids to a continuous l i m i t .  We w i l l  R  not explicitly  make  14  2.2  Non  Standard White Noise  Anderson Motion,  n a m e l y an  /Kt  size  (1976) i n t r o d u c e d  a h y p e r f i n i t e representation of  i n f i n i t e s i m a l random w a l k w i t h s p a t i a l e x c u r s i o n s  i n a time step  At  .  Implicit in this  non-standard representation of white  n o i s e on  random v a r i a b l e s  0  £  , e a c h o f mean  time-line of spacing Recently  X  be  and  of white  t h e l i n e , as  (1985) h a s  as  a hyperfinite lattice  in  R^  as d e s c r i b e d  defined a family  P_CA). =  a p p l i c a t i o n s we  J  1} A — |ft| L  a n d we may We  . for  F  A e F  define  k  the  e  * ^ z  will  £  be  •  A n  the  a hyperfinite  {£  only  need  , o r on a l l o f above, and  }  let  of I.I.D.  R^ ft  . be  an  S-L  X£X  E(£  —  ) =0  and  be  shown by  21  may  ( f t , F, P)  .  Then  as  the coordinate using a  the  standard  i s an  example.  ft  analogue:  , and  define  internal probability  space,  lattice  i s then  *finite The  of  ft  Let  maps.  *countable  exemplary space  *a - i n t e r s e c t i o n s ) .  d e f i n e d as i n t h e  ft  family of i n t e r n a l subsets  * a - f i e l d g e n e r a t e d by  * a - u n i o n s and be  Let  w i l l m o s t o f t e n be  l'"*"' d be  IID  var(5 ) = 1 . For x. n e e d f i n i t e n e s s o f a l l t h e h i g h e r moments o f  e x i s t e n c e of such a space  ft = {-1,  to  of  well,  The  k  , on  F o r o u r p u r p o s e s we  i n t e r n a l random v a r i a b l e s , s u c h t h a t  x  At  a sum  a  g e n e r a l i z e d Anderson's c o n s t r u c t i o n  *  E,  c o n s t r u c t i o n was  variance  n o i s e on r e c t a n g l e s  i n t e r n a l space•on which are  most o f our  of  .  a - f i n i t e Radon s p a c e s .  representations Let  At  Andreas S t o l l  to.arbitrary  Brownian  X = { ( k ^ A x , •. - ,k_.Ax) |  {-1,  subsets  of  1} ft  .We  may  i s the unique  F  ( c l o s e d under  i n t e r n a l p r o b a b i l i t y measure P_  take  P_  may  * a - a d d i t i v e measure  15  on  F  : x e X}  (<4  such t h a t ,  P({co  The  transfer principle  x  a r e i n d e p e n d e n t and f o r any  = l}) = P ( {  w  x  = -l}) =  x e X ,  i.  2  guarantees t h a t the Kolmogorov  E x t e n s i o n Theorem  o v e r t o the non-standard s e t t i n g , and t h e r e f o r e t h a t such a measure In  t h i s c a s e a g a i n we w i l l  Loeb e x t e n s i o n o f  take t h e p r o b a b i l i t y space  A c X ,  W(A) =  —  d-dimensional  r } xeA  If  t o be t h e  , define f o r internal x ~ W : A -»- W(A) i s called  A  E, IT Ax ; t h e map x . . i — i=l X .  S-white n o i s e on  Lemma 2.2.2.  P)  exists.  ( f i , F, P_) .  G i v e n s u c h a s p a c e fi , a n d r a n d o m v a r i a b l e s sets  ( f i , F,  P_  carries  °A_(A) < °°  and  Stoll  (1986)  £  shows  _A(A A B) = 0 , t h e n  °W(A) = °W(B)  P-a.s.  T h u s we may make  Definition  2.2.3.  F o r each Loeb measureable  a s t a n d a r d random v a r i a b l e  w(A)  w(A) whenever  A  i s i n t e r n a l and  T h e o r e m 2.2.4. The of  family  ( X , X)  (Stoll  set A c X , with  i s well defined  = °W(A)  (up t o a n u l l  L (A_) (A)  < °° ,  s e t ) by  P-a.s.  L(X_) ( A A A ) = 0  .  (1986) T h e o r e m 2 . 5 ) .  (W]('A'I .|,.;L/'CA)••".<: ~ } /  i s a w h i t e n o i s e on t h e Loeb e x t e n s i o n  w i t h r e s p e c t t o t h e Loeb p r o b a b i l i t y space  ( f i , F,  P) .  16  2.3  Adapted S t o c h a s t i c I n t e g r a l s  Let  At  and  Ax  be i n f i n i t e s i m a l s , and l e t T  be a l a t t i c e o f  spacing  At  r e p r e s e n t i n g a l i n e segment i n R  spacing  Ax  i n each d i r e c t i o n r e p r e s e n t i n g a r e c t a n g l e i n R  E, tx  an i n t e r n a l space s u p p o r t i n g a c o l l e c t i o n S-L  2  random v a r i a b l e s , as d e s c r i b e d  .  d  {W  =  n  °t>t  t  F  we d e f i n e a f i l t r a t i o n  a(F ) v W , where  ±  F , t e T t —  W i s the c o l l e c t i o n o f  We say t h a t an i n t e r n a l p r o c e s s on  t <_ n}) dX  R  x R  +  dX  d  u  x fl , i f f o r H e *NXN  x, u) | °U  , 0 < s < t , y e X , i s —  We s h a l l c a l l a p r o c e s s t e R  +  (u) ? u  (1983) §3.  n e N ,  F t_ u^_  Since our f i l t r a t i o n  integrate discontinuous  (w)}) = 0 o.  —  U  2  I({|x|  > H ;  u t  x  ^  .  o  _t, o  n  X  ^  x  ^  i s F^-adapted, i f  measureable.  on  , u (jo). , (s e i-0, t ] , x e R sx  measureable.  P-null sets. For  l i f t i n g o f a process  a  and  t X  —  We say t h a t an i n t e r n a l p r o c e s s  sy  On the Loeb  « 0 , and  LC*. x X x P ) C ( C t , '-t - X — — -  TJ  ^ ^ w  t x  on ft t o be the  , t e s t ( T ) , by  p r o p e r t i e s o f t h i s f i l t r a t i o n see Hoover and P e r k i n s  u^Cw)  L e t 9, be  to represent  tx  0 < s < t , x e X} . — —  sx  F  dW  We d e f i n e an i n t e r n a l f i l t r a t i o n  Cft, F, P)  .  D  above.  a l g e b r a o f i n t e r n a l s e t s g e n e r a t e d by space  be a l a t t i c e o f  , (t,x) e T x X , o f I.I.D. - -  In t h i s s e c t i o n we w i l l use the n o t a t i o n  F v^At A x tx  , and l e t X  +  D  F  R  x R  , F^_-adapted, i f , f o r each  , a) e ft), i s B ( [ 0 , t ] ) x B( R ) x f t D  i s continuous  and we do n o t wish t o  i n t e g r a n d s , we s h a l l n o t make d i s t i n c t i o n s between  adapted, p r o g r e s s i v e l y measureable, o p t i o n a l , and p r e d i c t a b l e  processes.  We s h a l l u s u a l l y b e w o r k i n g w i t h  F^adapted  liftings  o f F^-adapted  processes.  Theorem 2 . 3 . 1 .  Suppose  and t h a t a n i n t e r n a l S-L (T x X x n 2  ;  C2-1)  u  ^- ^ ^  ;  s a  n  F^_-adapted p r o c e s s o n  L  U (to) l i f t s tx  x . p ) ) . Then f o r a n y t e R  t  a.s.  dw sx s x  y 0  where t h e i n t e g r a l on t h e l e f t  Proof:  "'  F^ a d a p t e d p r o c e s s —  L ( A ' *'A  u  w  x  y  <s<  t  x  R  x R  x  u , and i s a n d t ss " t  +  u dw  'x  S5 sx  i s i n the sense o f I t o .  We f i r s t e s t a b l i s h a n i s o m e t r y p r o p e r t y :  I  I  0<s<t  xeX  U  dW -  §  -  §  J I II  V I  U dW  +  2X sx s*j \o£s.<t ( x e Xsx §S  (since  = E/ y SA\0<s<t "  y  v  d  w s  [u  xeX  s  2 x  —  i  dw  2  sx —  +  1 112  |u||  L (TxXxfi)  0  <3  < £  d  sx s<s'<t\xeX  y  d  u  u  sxsx x'/x — — x:JteX  I E ( U )Ax At + 0 xeX - 2  U  i s(conditionally)  x  X  I 0<s<t  I {I O4 I s'  (using  ,aw  sx  -  IxeX  independent  dw  E(dW  U  2  1 +2 x 0  sx'  ) = Ax At)  2  d  ^  -  of F ) g  x  dW s  Hence t h e m a p p i n g  space o f  U  '  >  F -adapted'  a r e a n y two  S-L  y 0<s<t  I 0<s<t  S-L  2  £ ^ ^ 0<s<t 2  £ U dW x e Xv -s  F^-adapted l i f t i n g s  —  I E|U. tx xeX  —  - U' | tx 1  £ U xeX  2  t  , <  s<t  y us x dws x tv  xeX  acts  x  isometrically  of  from the  Hence, i f U  and  u ,  xeX  —  —  Ax At d  a .s.  dW  s  yudw  0<s<t  o  I  I  0<s<t  xeX  A s i m i l a r a r g u m e n t shows t h a t i f t  y  x  2 p r o c e s s e s t o *L (fi) .  (TxXxo,)  I D dw - y  xeX  °y 0<s<t  Hence  U  1  u' dW §  §  x  x  RJ t , t h e n  a.s. 0  —  , s o t h a t t h e r . h . s . o f (2-1) i s w e l l  defined  up t o a n u l l s e t . We m u s t show t h a t t h e r . h . s . o f (2-1) c o i n c i d e s w i t h Consider a process  u  o f the form  u. (o>) = I _ " tx It r j  where  R  i s an  F  . . ( t ) • I . (x) A  ,t )  • I_(u>.) , K  m e a s u r e a b l e s e t i n fi , a n d 1  the I t o i n t e g r a l .  A  i s Lebesgue measureable  in  R  .  LU )(A x  'f i n d  %2 ~  Pick  ^2  '  A s t ( A ) ) .= .0; .  P (R A R)  £  t-  =0  Let  n  P^ ^-  d  c  internal  A  By T h e o r e m 3.2  1  '  a  and an i n t e r n a l  in  X  such  that  i n H o o v e r and P e r k i n s  subset  R  o f fi - , s u c h t h a t  ( 1 9 8 3 ) , we R e  may  and  .  _ 1  u\ (to) = I tx  u  Then  s  x  d  w  sx  =  K  dw s x  l (w) v  R  [  a .s.  1 T  )  ^  l '  t  '  t  A  t  2  )  >  <  A  I I t <s<tAt  d  w  2-  » (by. d e f i n i t i o n noise  xeA  I  0<s<t  I  U  xeX  sx  dW  u  dw. ) tx  sx  T h u s t h e l e f t h a n d s i d e a n d r h s o f T h e o r e m 2.3.1 functions  o f the white  coincide  o f t h e f o r m i n d i c a t e d , and hence f o r l i n e a r  f o r simple  combinations  thereof.  2 Such l i n e a r  combinations are c l e a r l y  dense i n the H i l b e r t  space o f  F^-adapted  p r o c e s s e s on,  [0,..:t]-, as i n t h e c a s e - o f t h e o r d i n a r y  integral.  H e n c e t h e two  s i d e s must c o i n c i d e  the  isometry property  f o r a l l such  o f the s t o c h a s t i c i n t e g r a l ,  u  and  L  stochastic U  by  and o f t h e h y p e r f i n i t e  sum.  CHAPTER THREE  The  3.1  Heat E q u a t i o n w i t h Non-Linear S t o c h a s t i c  Forcing  Scope  In t h i s chapter I w i l l use h y p e r f i n i t e methods to prove a weak theorem f o r s o l u t i o n s o f e q u a t i o n s o f the  (3-1)  on f  R  |» dt +  R  x  t o be  such  .  Here  W  *  2  *  f ( u )  tx  form  ,  i s "white n o i s e " on  tx  c o n t i n u o u s , and  R  +  x R  .  We  to s a t i s f y a growth c o n d i t i o n :  suppose the there  function  is a real  K ,  that  (3-2)  We  dx  existence  f (u) £K(l+u ) 2  solve  C3-1)  C3-3J.  where  subject  to an i n i t i a l  u(0,x) = u  u^(x)  for a l l  2  (x)  condition  ,  i s a bounded continuous  With minor changes i n n o t a t i o n as w e l l as  u  , and  u^(x)  c o u l d be  function. f  c o u l d be made to depend on  taken as a random f u n c t i o n .  s i g n i f i c a n t changes, the same c o n s t r u c t i o n modified or by  the  by  u .  t  is  + g ( t , x, u)  i n t r o d u c t i o n o f a bounded, non-zero, c o n t i n u o u s n o n - l i n e a r  ,  function  2  of  u  m u l t i p l y i n g the term  t o those i n Appendix A,  .  With a p p r o p r i a t e  inequalities  analogous  a treatment v e r y s i m i l a r t o the remainder o f  c h a p t e r can be done f o r e q u a t i o n  (3-1)  on a s t r i p  R  +  x [ a , b]  x  With more  can be made to work i f (3-1)  the i n t r o d u c t i o n o f an a d d i t i o n a l f o r c i n g term  9 u —— 3x  and  with  this  D i r i c h l e t o r Neumann t y p e b o u n d a r y c o n d i t i o n s random i n i t i a l  condition  handled by e n l a r g i n g As  discussed  t h a t i s , i f <|> e C  C3-4)  u  u  specified at  independent o f the white noise  0  and W  tx  b .  A  may b e  0, .  t h e p r o b a b i l i t y space  i n s e c t i o n 1.2, ( 3 - 1 ) i s o n l y oo  a  solvable  i n a 'weak' s e n s e ,  ( R) ,  u  <$> (x) d x -  tx  Ox  d> (x) d x =  u  sy  Ad)(y)dyds  ft f(u 0  This this  i s c a l l e d t h e "weak" f o r m o f ( 3 - 1 ) .  chapter asserts  a stochastic process  that there u  9, , f o r w h i c h Section  (3-4) h o l d s f o r a n y  3.2 i n t r o d u c e s  may b e s o l v e d  00 c  on  t  and  x , may b e  f , defined  (R) .  t h e p r o b a b i l i t y space  construction of a white noise Section  (f> e C  (3.9.2) o f  Q , such t h a t , f o r any  continuous i n  tx on  )<j>(y)dw sy  The m a i n t h e o r e m  e x i s t s a space  , jointly  sy  ft  and d i s c u s s e s t h e  0, .  3.3 e x h i b i t s a h y p e r f i n i t e a n a l o g u e o f ( 3 - 1 ) a n d shows how i t internally,  fora solution  U tx  Section Proofs  U  tx  3.4 c o n t a i n s  s e v e r a l i n e q u a l i t i e s which a r e used  subsequently.  a r e d e l a y e d u n t i l A p p e n d i x A.  S e c t i o n 3.5 c o n t a i n s e s t i m a t e s o n t h e moments o f t h e i n t e r n a l s o l u t i o n , w h i c h a r e n e c e s s a r y f o r 3.6 a n d 3.7. Section  U. - U . tx ty  3.6, o b t a i n s  b o u n d s o n t h e moments o f s p a t i a l  differences  I n s e c t i o n 3.7-we o b t a i n b o u n d s o n t h e moments o f t e m p o r a l  differences that  U  U\ - U .We tx rx  u s e t h e r e s u l t s o f 3.6 a n d 3.7 i n 3.8 t o show  i s ,with probability  1, a l i f t i n g  of a jointly  continuous  process  tx I n 3.9 we v e r i f y t h a t t h e p r o c e s s i s a weak s o l u t i o n  of  (3-1).  u  actually  satisfies  (3-4),  hence  23  3.2  W h i t e N o i s e on  the  space  ft  Let a positive i n f i n i t e s i m a l { k A x | k e *Z>  .  Now  t^ > 0  suppose  {kAt,  k e *N}  Now  p i c k aa i s given  such t h a t  Ax  be  such t h a t and  given.  0 < °a  let  t  be  Let  X  be  < - j , and any  let  the  set  At = a Ax  number o f t h e  .  2  form  t_ ~ t . Let T be { k A t l k e *N, k < t , / A t } - f t — -f h a v e m e n t i o n e d i n s e c t i o n 2.2 t h a t i f ft i s a * c o u n t a b l e s p a c e  .  c  1  We  r  (such  as  variables and  {-1,  {E,  1} tx-  t h e random  )  t 6 T,  5^  such t h a t  -,TxX  on w h i c h a r e x e X} -  defined  such t h a t  possesses f i n i t e  (internal)  set function  higher  {w^(oj) |A e B(['0, t ] !  adapted to the  filtration  F  t  this white noise in  spirit  x  f  T E /AtAx t—x (, t ,• x )\e A•»  R) } .  Further,  d e r i v e d as p e r  g e n e r a t e d by t h a t we  moments o f a l l o r d e r s ,  >  A  "white noise"  filtration  a f a m i l y o f I . I . D . random 2 E(E ) = 0 & E (£ ) = 1 V(t,x) — tx — tx --  the v a l u e s  shall solve  to t h a t employed i n K e i s l e r  {E,  (3-1).  s < t} sx _  The  .  then  induces  t h i s white noise  s e c t i o n 2.3  from the  I t i s with  m e t h o d we  ,  a  is  internal respect  s h a l l use  (1983) f o r s t o c h a s t i c O.D.E.'s,/  is  to  similar  24  3.3  Hyper-Finite Difference  Equations  Let  l i f t i n g of  F(u)  condition  be a u n i f o r m  (3-2).  Let  u n i f o r m l y bounded on  TT  +U  a  t+At,x  (3-5)  f ( u ) , s u b j e c t a l s o t o the growth  be a u n i f o r m l i f t i n g o f X .  which i s a l s o  Then c o n s i d e r the h y p e r f i n i t e analogue o f (3-1) ,  U  t,x  u^  . - 2 U +U t,x+Ax t,x t,x-Ax  At  F(U  . 2 Ax  tx  ) £  tx  /At A J  or e q u i v a l e n t l y ,  U,.,.., = U + A t Ax [ u . - 2 U +U t+At,x t,x t,x+Ax t,x 2  We may and  E  U.. At,.-  Q  ^  solve for  and  . ] + F(U ) E t,x-A tx *tx  (3-5) i n p r i n c i p l e i n d u c t i v e l y .  x e X  The s p e c i f i c a t i o n o f  g i v e s us enough i n f o r m a t i o n t o f i n d  £, , we may At,1  solve f o r  U . 2At,-,  F(U  ) £ tx  U  , then '  t+At,x  values of  U  t+jAt,  U  . . t+2At,x-A  difference equation:  , U x  , and so on.  Knowing  U  (u) .  We  t+2At,x  e n t e r s i n t o the d e f i n i t i o n tx  , U  . , and then t+2At,x+Ax  The c o e f f i c i e n t  Q ~Y  n A t  w i t h which  X  e n t e r s i n t o the d e f i n i t i o n o f  , (3-6  U / X  f i n d a c l o s e d form e x p r e s s i o n f o r t h i s i n d u c t i v e d e f i n i t i o n as f o l l o w s .  From (3-5) i t i s c l e a r t h a t the v a l u e of  .  U  , and so on.  C o n t i n u i n g i n t h i s manner, we d e f i n e an i n t e r n a l p r o c e s s may  — Ax  x  U • „ ,may be found .trHiAt,y = 0 Vx  = 1 ;  x  F(U ) E tx tx  as the s o l u t i o n o f a  ^ 0 ;  (n+l)At nAt „ ^nAt ^ ^nAt Q = a Q . + (l-2a)Q + a Q . *x • *x-Ax x x+Ax L  five  But density  we may r e c o g n i z e  o f a Markov Process,  Lemma 3.3.1.  step  i n f a c t , a random w a l k . t Q-  The c o e f f i c i e n t s  r a n d o m w a l k on^ t h e one  (3-6) a s t h e d i f f e r e n c e e q u a t i o n  lattice  are the "density"  X , starting at  t o the r i g h t (or l e f t )  governing the  Thus we h a v e  o f an i n f i n i t e s i m a l  x = 0 , a t time  i n each'time p e r i o d  At  A t , and t a k i n g  with probability  a .  W i t h t h i s n o t a t i o n , we may w r i t e  C3-7L  y  u tx  L  0<s<t  +  I  Q  l  Q- ^x-y yeX - i L  v  t + A t  y,X The  S  y -I  ) ? s y •* 1^ Ax  u ° l  second term on t h e r . h . s .  the h y p e r f i n i t e h e a t equation  F(U  o f (3-7) i s t h e d e t e r m i n i s t i c s o l u t i o n t o  (3-5), w h i c h we w i l l  designate  hereafter  U tx  3.4  Some U s e f u l I n e q u a l i t i e s  We w i l l sections.  find  Proofs  the p a r t i c u l a r  Lemma 3 . 4 . 1 .  Lemma 3.4.2.  Lemma 3.4.3.  I  I -xeX  t  eT  chapter.  K , depending on a  constant  , depending on  a , such t h a t f o r  I  I x€X  A  )  CQ-j S  < K  2  ~  x  a , such  that  2  0<s<t  CQ- - Q-  , and  do n o t i l l u m i n a t e  constant  < K - a  a  /t/At . -  K a  There i s a c o n s t a n t  Lemma 3.4.4.  three  2 i (Q-) < K ^ x — aJ t  There i s a f i n i t e  t , "  0<s<t  t o A p p e n d i x A, s i n c e t h e y  There i s a f i n i t e  v  a l l  are deferred  subject matter o f this  I xeX  t/A?t c *N ,  for  the following i n e q u a l i t i e s h e l p f u l i n the next  a , such t h a t V  , depending on  z e X  | z | / Ax . "  K  There i s a constant  , depending on  a  a , such t h a t , f o r a l l  r < t  I 0<s<r  We s h a l l  I xeX  CQ-"") - Q- " )  2  < K  /(t-r)/At  a l s o r e q u i r e t h e f o l l o w i n g theorem o f Burkholder  (1973,  Theorem  2.1.1., s p e c i a l i z e d s l i g h t l y ) .  T h e o r e m 3.4.5.  Let M n  , n e  N  be a h y p e r m a r t i n g a l e  a n d l e t <M> n e n  be t h e a s s o c i a t e d p r e d i c t a b l e square f u n c t i o n , a n d l e t p > 1 Then t h e r e  i sa finite  E(M ) < K - n - p P  EC<M> n  P / / 2  K P  constant )  + K E(max P  Q  <  k  <  n  depending o n l y on |M k+1  - M | ) k P  .  be  p , such  finite. that  N  27  B o u n d s o n Moments o f U  3.5  q > 1 , and l e t R (t,x) = E | U | q - - tx  Pick  Consider  tx  any f i x e d  2  .  q  L e t H (t) = sup R (t,x) q q - -  ( t , x ) / a n d l e t ji b e t h e m e a s u r e o n  [0, t ]x X  2 defined by  y ( { ( s , y ) } ) = (Q-" )  — . Ax  §  -  X~Y  -  L e t l y l denote 1  i n what f o l l o w s , a f i n i t e  y ( [ 0 , t ] x X).  1  Let  c  denote,  constant, depending only on  and  t  , w h i c h may c h a n g e i t s e x a c t v a l u e f r o m  line  q  to line.  Now  R  Ct,x). < c E q - - -  I  I  0<s<t  y X  1/2 ) Q±~ - E ( f ) "X sx  S  Y  X  £  A  apply Burkholder's  R  I  1  |2q  tx  F s  .  (3.4.5) t o o b t a i n  I  F  2  ( U ) y(s,y)]  q  (3-8) + c E | s u p | F(u ) |E /y(s,y) | — sy sy - 0<s<t —  2  q  +  c  1  lu  1  | tx  1  2  q  .  yeX The s e c o n d  t e r m o n t h e r . h s . o f (3-8) i s c l e a r l y e q u a l t o  c E [ max <s<t  2 2 F (U ) E ?y  c E[ I " 0<s<t  I veX  0  R ( t , x ) < c E |  F (u  y(s,y)] - -  ) E  2  a  I  I  0<s<t eX Y  2  S  F  2  ^  which  y(s,y)] "  ( U  tl  )(1  q  i s bounded by  .  +E  2  51  Thus  )  Hil^-| | |g q  |  y  y  1  t o ( s , y ) may b e r e g a r d e d  with respect t o thef i l t r a t i o n  inequality  (t,x). < c E [  + c  x  I n t h e summation above, t h e term c o r r e s p o n d i n g as a m a r t i n g a l e i n c r e m e n t  rU  2<5  S  ECU  +  |u  c  I  2q  H e n c e we may  28  A p p l y i n g Jensen's I n e q u a l i t y t o t h e p r o b a b i l i t y measure we  y(-)/|y| ,  find  Ct,x) < C E [ I " " ~ ~0<s<t  R q  I  |F(U ) |  for  q (  1  +  y X £  Now we u s e t h e f a c t s t h a t uniformly  2  °t <_ t  \^\) ?y  Hitfl]|y| \V\  |y| /|y| = | y | q  q 1  , that  5  sy  b y Lemma 3.4.2, t h a t t h e moments o f  i s independent o f  Ct,x) , s i n c e  in  R ( t , x ) < c U + - -  U  q  0  Q  I <s<t  < ca  Now u s i n g  sy  , and t h a t  y e  I x  I  E |Fltr;>| ^  K  2 q  (l  2 q  ( l + E\K  2  i s bounded  £ sy  i s bounded  are f i n i t e ,  uniformly  I 0<s<t  I R ( s , y ) CQ- - ) q - x-y yeX  l  H  0<§<t  q  Ax = / A t / a  (s) c  I yeX  —fpA  2q  (Q^) *x-y  |2q  yeX  | ) ( Q - " - ) 7^) *-y * 2  A  +E|U ) -' s y  0<s<t  +  U tx  \ * . 't x  i s bounded, t o o b t a i n  < cCl  <_ c C l +  U  C  +  |y|  1  that  |S  q  2  Ax  ;  T^) Ax  At) .  X  , a n d lemma 3.4.1 we  find  Lemma 3.5.1.  (3-9)  r -1/2 H (t) = sup R ( t , y ) < _ c ( l + I H(s)(t-s) At) , f o r a l lt i n y °fs<t q  [0,  t ^ l , where  c  Now, i t e r a t i n g  i s a c o n s t a n t depending on  a , q  (3-9) a n d i n t e g r a t i n g b y p a r t s ,  and  t  f  .  H  I 0<s<t  (t) < c  <  c(^tj+  I (1 + H 0<u<s  (1 + c  (u))(s-u)  I (1 + H ( ) ) ( I (s-u) 0<u<t '* u<s<t ~  At)(t-s)"  1 / 2  1  /  2  U  (t-s)  1  /  2  1 / 2  At  At)  At '  (3.10) I  <.c(l+  1  where  c(l +  c We  t  now  Proof.  f  I 0<u<t  i s another  (u))  (u)  H  (  I  s  2  (t -s)  At) A t  1 / 2  f  At)  c o n s t a n t d e p e n d i n g o n l y on  There i s a c o n s t a n t  , such  that  E_|  We  take  c  may  V  q  r e q u i r e a type of Gronwall's  Lemma 3.5.2. and  (1 + H  |  c  d e p e n d i n g o n l y on , for  t h e maximum o f  H  q, m a x l u | y 0y  t <_ t  q  , and  (0)(= maxlu 1  y  * of equation  (3-10).  We  proceed  t h e lemma h o l d s f o r t = n A t . S u p p o s e now t h a t f o r n e  H CkAt) <_ c C l + c A t ) Then  H C n A t ) <_  c  (.1 + c  = c ci  k  by  i n d u c t i o n , on  k = 0,  £ (1 + c A t ) 0<k<n  + c ^—r- — 1  k  At)  At  cAt (summing a g e o m e t r i c = cCl + c A t )  n  n e  Z  series)  1, . . . , n-1 (by  Oy  1  2 q  f o r any  | )  2  q  and  . a x e  the  + .  N  , for  t^ .  lemma.  <_ c e x p ( c t )  t o be  a, q, and  (3-10))  .  For  n = 0 ,  This i s the i n d u c t i o n By verifies Then we close).  step.  t h e t r a n s f e r p r i n c i p l e , we H ( n A t ) ' <_ c (1+cAt) notice  that  ,  n  (1+cAt)  n  may  conclude that t h i s  f o r a l l n e *N  £ exp(cnAt)  internal  such t h a t  , (in fact  argument  n A t <_ t  .  they are i n f i n i t e s i m a l l y •  31  3.6  B o u n d s o n Moments o f S p a t i a l  Let  U  = U 1—X  + V L.X  random f o r c i n g .  Differences  , where  V  represents  "CX  U  i s S-continuous  interested i n the continuity o f  (infact  V .  o n t h e moments o f t h e d i f f e r e n c e s  V  a  , and  There i s a constant  -  | v  V  v  -  tx  12q  - V ty  respect  to the internal f i l t r a t i o n  I  I  2q  0 <_ t <_ t  ,  q  ± °l - zl  •  a s a sum o f m a r t i n g a l e F  q > 1 , max|u | ,  i n X , and f o r x  We may w r i t e  estimates  - V ty  x, y  Proof:  tx  s e c t i o n we o b t a i n  c , depending o n l y on  t ^ , such t h a t , f o r a l l i  S - s m o o t h ) , a n d we a r e  In this  tx Lemma 3.6.1.  t h e c o n t r i b u t i o n from the  increments  with  ; b y (3-7), s  V  - V £  = a.  We w i l l  l  y 0  <s<t  estimate  (Q-~- - Q ^ ) F ( U  zex  the  V*  V«'  ) £  ( — )  V  2  H  2 q - t h moment o f t h i s u s i n g B u r k h o l d e r ' s  Inequality  CTheorem 3.4.5). We w i l l  designate  by  yCs,z)  t h e measure on t-s t-s (Q- - - Q- -) x-z y-z  2  to  each p o i n t  Hlv^  " V  (s,z) the weight - -  t y  |  2 q  F (u 2  < c EC  sz  [0, t ] x X  At — Ax  )dy(s,z)) - -  q  [0,t)xX  2  2  + c E ( max F (u ) £ " 0<s<t ^ -S  Z  y(s,z)) " -  q  which  assigns  32  < c E(  F  2  --  [0,t]xX  [ F (U 2  +  <  C  C  E  (  E  sz  CF (U  )  ) ]  2  , dy_(s,z)  ( u  K sz 2  q  [ i+  ) q i i j I  q  y  ^ S ' 5 ) ,  |5|  sz  2 q  ]  sz  q  | y |  T ^ -  (  §  '  q  5  )  -|y|  q  II  (Jensen's I n e q u a l i t y a p p l i e d t o - — r )  < c  K  2 q  H q  (using independence o f  <_ c  Ix  (s) - ^ " y  U sz  and  { s  -'*  E  ]  sz  • |y|  q  , a n d a l s o f i n i t e moments o f  E ) sz  - y|  ( u s i n g lemmas 3.5.2 a n d 3.4.3, n o t i n g t h a t  At — = ctAx )  •  33  3.7  B o u n d s o n Moments o f T e m p o r a l  Differences  I n t h i s s e c t i o n we o b t a i n e s t i m a t e s V  tx  - V rx  Lemma 3.7.1. max  o n t h e moments o f t h e d i f f e r e n c e s  |TJ | uy  There i s a constant , a , and  t  I E V —' t x Proof:  S  , such t h a t , f o r a l l  r  I2q  |  V < c rx' —  t '-  V  - V  tx  to the i n t e r n a l  £  ^ ^. r<s<t  ~V Z£A  u  designate  to each p o i n t  by  F  x-z - -  differences  with  s  ^  sz  y(s,z)  x e X , -  r < t .  IS  zeX  0  t , r <_ t , and a l l — -r  —  a s a sum o f m a r t i n g a l e  filtration  <s<r  x  We w i l l  rx  q > 1 ,  ,q/2  r  We may s u p p o s e w.1.o.g. t h a t  We may w r i t e respect  -  c , d e p e n d i n g o n l y on  25  ^  s z Ax  t h e measure on  [0, t) * X  which  assigns  t-s r-s ^ At (Q- - - Q- -) - — , i f s < r , a n d t h e x-z x-z Ax -  (s,z) the weight - 2  weight  ^2 _|^ X  Ax  ' ^  - - -  *~ '  <  T  ^  e n  u s  i- 9" Burkholder's n  CTheorem 3.4.5) I l E V - V — tx rx  2 c  *  < c E( — —  F (U 2  J  sz  )dy(s,z))  q  [0,t)xX  + c E ( max - <s<t 0  F (U 2  ) £  2  y(s,z)) ~ ~  q  Inequality  34  < C E  (F (U 2  )) (i q  +  (using Jensen's  <c  E  K  2  Q  -^'5>.|y| y  | )  \  2q  K  q  Inequality)  H (s) ^  •  |yl  q  (using independence of U , and E, , and the sz sz f initeness of E I E, I ) sz 2q  —  < c(t-r)  1  q/2  (using lemma 3.5.2, and lemmas  3.4.2  and 3.4.4, with  35  3.8  S-Continuity and the Standard  The  Part  main r e s u l t i n t h i s s e c t i o n i s t h a t  h y p e r f i n i t e d i f f e r e n c e equations  , the s o l u t i o n t o the  (3-5) i s a . s . S - c o n t i n u o u s .  We s h a l l  o b t a i n t h i s by a p p l y i n g a non-standard v e r s i o n o f Kolmogorov's C o n t i n u i t y Criterion:  * T h e o r e m 3.8.1. lattice  F  : fi x r -»-  Let  which represents  k U  u  _ x  U I  I  i  h-  1  , d  y  rectangle i n R  .  on a h y p e r f i n i t e  I f there  ,K  such t h a t f o r  x, y e T  a r e such t h a t  ,...,y  1  exist  k = l,...,d  d  k  <_ K I x - y |  l i e s a l o n g t h e k*^  be an i n t e r n a l process  a finite  3 ,...,3  p o s i t i v e r e a l numbers  R  , whenever  coordinate  axis,  then  i f  x - y  6, < f, /$, , k = l , . . . , d k k k  there  o  is  a s e t fi' c fi o f L o e b P r o b a b i l i t y  1, a f u n c t i o n  6 (w)  , <5 (w) > 0 o n fi' , , k 6  and  a constant  c , such t h a t f o r  k=l,...,d  U  - U  ' x whenever  x , y e T, | x - y |  < 6(w)  coordinate axis.  Inparticular  Proof:  (1984)  See S t o l l  a s 3.8.1, b u t h i s p r o o f T h e o r e m 3.8.2. of  and x - y  U  l i e s along  '-  -  1  the k^ T  <  4"  a  n  d  The r e s u l t he s t a t e s i s n o t a s d e t a i l e d  i s sufficient.  The h y p e r f i n i t e p r o c e s s  ^2  <  \  u  -(- ^ ^  constructed by the s o l u t i o n  a )  x  a  n  d  A  c  X  i s a r e c t a n g l e whose s i d e s h a v e  l e n g t h , t h e r e i s a s e t fi' <= fi o f p r o b a b i l i t y 6(a)).  o n fi' , a n d a c o n s t a n t  1, a p o s i t i v e r e a l  c , depending on  3  , 3 1  , such t h a t  V  1  .  C3-5), i s a . s . S - c o n t i n u o u s o n n e a r s t a n d a r d p o i n t s i n T x X .  if  t  y' —  i s a.s. S-continuous on  Lemma 3.2.  t  < c x - y  wefi'  , x,  y e A , t , r < t  2  max U  yeX  °l  Moreover,  finite function , a , and  36  1  l tx ' r y ' U  Proof:  U  Pick  c  q e R  ' t ~ ^1  (  y  + |x - y|  such t h a t  +  3.6.1 a n d 3 . 7 . 1 , t h e r e  8  ) , i f | t - r | + |x-y| < 6 .  <  1  are constants  -—— 2q  and  such  that  c  8  2  <  q  2  2q  .  B y lemmas  I |2q I 1 2 + ( q / 2 - 2) E V - V < c t - r ^ — tx rx — '-' ;  1  1  I2q  I —  1  tx  ty  Hence by Theorem of  U  on any s e t  U  1  I -  i2+(q-2) -  1  3 . 8 . 1 , t h e s t a t e m e n t o f t h e lemma i s t r u e w i t h T x A  but the near standard and  —  , where  part of  X  A  i s an i n t e r n a l  i s a  a-union o f such  i s a l i f t i n g o f a smooth f u n c t i o n , w h i c h  equation.  finite  An e x a m i n a t i o n o f t h e e x p l i c i t  form  A  V  i n place  rectangle .  Now  in  X ,-  U = U + V  i s - a - s o l u t i o n - t o the heat  (3-7) f o r U  yields quickly  that  U. tx  - U  Hence t h e t h e o r e m  In general  < c ( t - r — — —  ry  +  x —  y ) —  i s true f o r U = U + V  the exponents  j  and  ^  .  •  are best possible  (see Walsh  (1986),  Corollary 3.4). We may a l l o w  slightly  to r e l a x the conclusion  more g e n e r a l  slightly.  depend o n l y on t h e boundedness o f  enough t o e n s u r e t h e  itself  will  n o t be  discontinuous  c o n d i t i o n s , i f we a r e p r e p a r e d  The a r g u m e n t s i n lemmas 3.6.1 a n d 3.7.1 max  |u  y X  '  £  is  initial  S-continuity of  I  .  Thus t h e b o u n d e d n e s s o f  °T  0  U - U .  However, i n t h i s  S - c o n t i n u o u s i n t h e monad o f z e r o .  the conclusion  o f t h e theorem w i l l  U  I f  case  U  TJ i s bounded b u t 0 h a v e t o be r e s t r i c t e d t o  37  °t,  °r > 0  .  R e t u r n i n g t o t h e c a s e when  i s c o n t i n u o u s , we  o f p h r a s i n g t h e c o n c l u s i o n o f Theorem 3 . 8 . 2 i s t h a t almost surely  in  *C([0,  t ] x R f  as t h e s t a n d a r d p a r t o f  t,  x  , a.s.  : R)  I t i s clear that that set , 1 It  -  the i n t e r n a l  =  i  C([0, t ] x R  are of course r e a l l y  the h y p e r f i n i t e time l i n e  , i f  :  R)  time  t  .  Corollary  infinitesimal C3-5I i s We  o  - U  Q  up  o  =  o  x  U  | < —} n  e N,V  x, y £ X ,  i s i n the  Loeb-measureable. (t,x) K  (t,x) , i f  co e ft'  1  for a l l t  t o some i n f i n i t e number T x X  .  L  .  , and  Construct the  Thus t a k e take  ft  solution any  finite  have  The  solution  grid representing  S-continuous  in  note a l s o t h a t  u  hence by d e f i n i t i o n  U(OJ)  fora l l  tx  e N, 3k  £ Z  interested i n solutions  T  define a process  t,  f o r any  co i  i s nearstandard  A l l t h e o r e m s p r o v e d p r e v i o u s l y h o l d t r u e up u n t i l  T h u s we  3.8.3.  u  way  .  l a r g e r enough t o s u p p o r t a w h i t e n o i s e on as b e f o r e .  may  s e t s , hence i s  0  We  tx  U  t x  U  = {co :\/n  ° tx^oO" u (o»  in  1  — k  Thus t h e p r o c e s s  sample p a t h s  ft  - r l < — =>|u  -  1  a - a l g e b r a g e n e r a t e d by  H e n c e we  U(OJ) , o r e q u i v a l e n t l y  t, r < t , | x - y | < — - — -f - ' k  has  .  f i n d that another  (°|x| t  is -  Cin s e c t i o n  on R  +  T x X  , where  T  now  i s an  , c o n s t r u c t e d from the d i f f e r e n c e  equations  < «>} n {°'t < °°} , a . s . 0CF s 2.3)  ) measureable u  is  f o r any  F^_-adapted.  s -  with  o  s > -  t  38  3.9  S o l u t i o n o f t h e SPDE  We of  now  u  of section  tx  We We  must check need f i r s t  Definition X £  R  if  °6  a new  3.9.1.  An  , i s called  d  x. *1  (. . . (6  the f i n i t e  x. \  condition  0,  $  internal  a lifting  $ ) . . . ) (x) =  function  $  We  to order  k  of a  function  <J) (° )  2.2  and  ...3 . X  \  x.: i  (6  cj> : R  x.  .  -> R ,  d  Here  6  x.  ($)(x) =  l  A consequence o f t h i s  x , °[$(x+Ax) - 23>(x) +$ (x-Ax) ] / A x  C  lattice  for a l l x e X -  x  3x, \  to order 2 of  note that every  solution  i n sections  o n an i n f i n i t e s i m a l  d i f f e r e n c e o p e r a t o r i n the d i r e c t i o n  is a lifting  a  definition.  that f o r nearstandard  if  i s i n fact  (3-4).  [§(x,,...,x.+Ax,...,x ,) - $ ( x , , . . . , x ) ] / A x . -± ~ i -d - l -d is  3.8  (3-1) w i t h r e s p e c t t o t h e w h i t e n o i s e d e f i n e d on  3.2.  is  show t h a t t h e p r o c e s s  2  definition  = - ^ r - <j>(°x) 8x  (j) .  function  cj>  has  a canonical l i f t i n g  to order  k  ,  * namely  <f>  Now  restricted  f i x any  to the  e C  oo  ( R)  lattice. , and  let  $  be  a lifting  to order 2 which i s  o  exactly  0  on  values of  (j) , ( t o a v o i d ( u n n e c e s s a r y ) Then  U  |  tx  • $ (x) -  u <Hx)dx t x  is a  x  whose s t a n d a r d p a r t s  concern over the convergence  (uniform) l i f t i n g  I u «Mx)dx = * a  0 x  x l i e o u t s i d e the c l o s e d  S  °l  of  (U  u  t x  of  d> (x) a . s . tx  - U ) 0 x  «(;,  *  - c o u n t a b l e sums) .  f o r any  Ax  support  t ss t  .  Thus  39  u xeX  0<s<t  U  '  I  Now  U  I  sx  .  2 U  SyX+Ax  s  + u  I  I  0<s<t  xeX  F(U  g x  Now  sx  —  —TIZIZ  ) ?  -  $  (  X  Ax A t  )  --  0(x+Ax) - 2 $ ( x ) +'$(x-Ax) Ax A t Ax  ) $(x) /AtAx E  sx  i s a uniform l i f t i n g  t e r m o n t h e r . h . s . o f (3-11) i s  F ( U ) $ (x)  sx  / A t Ax  C$ (x+Ax) - 2 $ ( x ) + $ ( x - A x ) ] / A x  Hence t h e f i r s t  +  Ax  U _ sx  xeX  F(U  s,x-Ax  x  $(x)  xeX  I  0<s<t  + °  s , x •At • $ ( x ) - A x  1  0<s<t  (3-11)  - u  4  s+At,x At  i s a (uniform)  H e n c e b y T h e o r e m 2.3.2, t h e s e c o n d  F -adapted  a.s.  lifting  s  u <f>" (x) sx  a.s. o f  u c(>"(x)dx • R sx of  f ( u )4>(x)  sx  t e r m o n t h e r . h . s . o f (3-11) i s a . s . e q u a l  't f (u  to 0  R  u  (3-12) R  sx  tx  T h e o r e m 3.9.2.  )<j)(x)dw . sx  <i> (x) d x - u R  Thus we h a v e ,  Ox  c|> (x) d x  a  jointly  S  '  u R  There i s a Loeb space  (3-1) h a s a s o l u t i o n  =  d) e C ( R) c  sx  , on which  continuous i n t  t e R  (j) (x) +  f(u 0  R  S ,X  )<Mx)dw  any e q u a t i o n o f the form  and x  with respect t o the  c a n o n i c a l w h i t e n o i s e o n fi . NOTE:  We b e l i e v e i t i s p o s s i b l e t o e x t e n d K e i s l e r ' s  principle  t o t h i s Loeb space  fi  (see K e i s l e r  internal transformation  (1984)),. I n t h i s c a s e , t h e  SX  equation  (3-1) h a s a s o l u t i o n w i t h r e s p e c t t o a n y w h i t e  supported Walsh  by  R  0, .  (1986) h a s e s t a b l i s h e d u n i q u e n e s s f o r t h e c a s e when  L i p s h i t z . Presumably convenient  n o i s e on  this  counterexample.  i s false  f  i  i n g e n e r a l b u t we do n o t know a  CHAPTER FOUR  The  4.1  Dawson C r i t i c a l  Chapter well.  use  o f h y p e r f i n i t e d i f f e r e n c e e q u a t i o n s , w h i c h was  3 f o r one  s p a t i a l dimension  It  Branching  attempted  explained  branching.  grid point  is- taken  I f we  e a c h e x e c u t i n g an let  x e X , a t time  u t  = A U +  where  Z  i s an i n t e r n a l n o i s e w i t h  Hence  Z  may  t h e r e f o r e we  this  4.4  be w r i t t e n as  i n 4.5  R  by  , d->"-l ,  s t a n d f o r t h e d e n s i t y _of„particles  , then,  i f the i n i t i a l , d e n s i t y . ;  U  satisfies  a  hyperfinite  This noise  /u w W  Z  E(Z. ) = 0 tx'  where  W  , E (Z ' tx —  is like  i s however, a l i t t l e  an  tx | U  u tx' —  e x a m i n e t h e t o t a l mass o f t h e p r o c e s s to e s t a b l i s h  _ d AtAx  awkward t o work w i t h , set out i n  c o n s t r u c t e d i n 4.2,  some c o n t i n u i t y r e s u l t s , w h i c h y i e l d  the standard p a r t i s w e l l - d e f i n e d .  I n 4.6  we  verify  *  S-white noise i n  a d o p t t h e s i m p l e r scheme o f d i f f e r e n c e e q u a t i o n s we  with  form  °*  many r e s p e c t s .  one  i n f i n i t e s i m a l random w a l k ,  l a r g e e n o u g h , i t i s p o s s i b l e t o show t h a t  d i f f e r e n c e equation o f the  In  t  U^  as  here.  non-standardly,  X , representing a portion of  number o f p a r t i c l e s ,  undergoing  i t does i n  However i t does s u c c e e d  i s p o s s i b l e t o r e p r e s e n t t h e Dawson p r o c e s s  an i n f i n i t e  successful i n  f o r higher dimensions  t o t h e same e x t e n t as  D i f f u s i o n , a s w i l l be  p l a c i n g , on a h y p e r f i n i t e g r i d  any  be  as i s s p e l l e d o u t i n A p p e n d i x B.  t h e Dawson C r i t i c a l  use  may  T h i s approach does not succeed  dimension,  at  Diffusion  Introduction  The  and  Branching  that this  easily  standard  and 4.2. and that  part  does i n d e e d c o i n c i d e w i t h  t h e Dawson p r o c e s s .  I n 4.7 we o b t a i n  new r e s u l t s a b o u t t h e p a t h w i s e r e g u l a r i t y o f t h e Dawson p r o c e s s , nonstandard  construction.  several using  43  4.2  A Hyperfinite Difference  Let {xix = - -  be any i n f i n i t e s i m a l ,  (k A x , . . . , k Ax) , k. e 1 d I  W(: w i l l of  Ax  R  and l e t  Z , d > 1}  .  A very  be  so t h a t  X  s i m i l a r treatment i s possible  represents  i freflecting  a r e imposed on s e v e r a l h y p e r - p l a n e s i n  of a rectangle  in  R  R  d  R  .  on t h e whole  boundary  , or along  t h e edges  , b u t t h e i n e q u a l i t i e s a r e m e s s i e r , and indeed  depend  t h o s e f o r t h e unbounded domain. Let  At  be an i n f i n i t e s i m a l ,  such t h a t  At/Ax  I n case  d = 1  p a r t s o f t h e t r e a t m e n t much e a s i e r . 1 At  X  t r e a t h e r e t h e c o n s t r u c t i o n o f t h e Dawson P r o c e s s o n l y  conditions  on  Equation  d  £ 0 . we  This  makes some  require  2  < j &  , a s i n C h a p t e r 3.  Let  T  be a h y p e r f i n i t e t i m e l i n e o f  spacing  At :  T We w i l l It and  -  suppose  =  tx The  C4-11  0 < ° (MAt)  < °° .  , k £ M} .  +  Let  t  = MAt .  TxX  i s e a s i e r , a n d i t s u f f i c e s f o r o u r p u r p o s e s , t o t a k e fi = {-1, 1}  to l e t  PCE  = {t : t = kAt , k e *Z  E t x (co) b e t h e c o o r d i-n a t e  11 = P { E  tx  = -1} = i  2  analogue o f equation  C6.U) = t tx  or  .  (3-5) i n h i g h e r  CAU) + - t x \ ./  equivalently,  map, •. . a s - . o u t l i n e d i n s e c t i o n 2.2.  d  A — -  u  IE  2At I  tx  ,  (r: •)  U , = U + AtCAu) +1 / t+At,x t x tx % A U  dimensions i s :  A  2  /  1 E tx  ,  Thus  where  6  (6 U) t  the  t x  I s a f i n i t e d i f f e r e n c e analogue of  t = ^  u t +  ^  t  Laplacian i n  C4-2)  u  x  t  R  x  ^  / A t , and  x  +  The  i s the f i n i t e d i f f e r e n c e analogue  of  :  CAU)  /u  A  — : 9t  At  U  2  /  j_i=l  U  t ( x ,...,x.+Ax,...,x )  - -1  -  l  A \ t ( x ,...,x.-Ax,...,x ) - -1 - i -d  4 - /  u  -d  2d U  . t ( x , , .. .,x ) - -1 -d  /  U  tx is  term Ax -  (  n  substituted  for simply  -  *  i n order to ensure, that, a non-negative value at  tx  At in  A x  (4-1),  -  ( t , x ) , a l l o f whose  n e a r e s t n e i g h b o u r s a r e n o n - n e g a t i v e , w i l l n o t become n e g a t i v e a t t h e n e x t time step. 0 < U <  The v a l u e s o f  U  f o r which the l i n e a r term i s taken are  4At d '  Ax For function  an i n i t i a l on  c o n d i t i o n f o r (4-1) we may X  , requiring only  use any n o n - n e g a t i v e def  that  r . d I u Ax , the Ox xeX  i n i t i a l m a s s , be f i n i t e a n d t h a t t h e mass o n p o i n t s o f n e a r - ^ s t a n d a r d be i n f i n i t e s i m a l i n sum. B o r e l measure on  R  d  b y such, a n i n t e r n a l preceeding remarks).  (we w i l l d e n o t e function  U  Q  We  may  X  internal total  which are not  r e p r e s e n t any f i n i t e  positive  the space o f a l l such measures  ( s e e C u t l a n d (1983) T h e o r e m 4.7  and  d ( R ))  45  4.3  The C o e f f i c i e n t s  Q "u  From  (4-1) a n d (4-2) we o b s e r v e  enters into  of  t h a t each v a l u e o f  t h e d e f i n i t i o n o f subsequent  the former  term  U  tx  's .  /  SY  At  U sy A — — 2  . d Ax  £  sy --  We d e n o t e t h e c o e f f i c i e n t  i n t h e d e f i n i t i o n o f t h e l a t t e r by  t-s 0~ ~ , o b s e r v i n g t h a t x-y  these  coefficients  a r e homogeneous  i n space and t i m e .  analogue o f a Green's f u n c t i o n formula: U At sy t  x  Y  0<s<t  Lemma 4 . 3 . 1 .  K £ /  ~y  ^ - -  The c o e f f i c i e n t s  \ * AAx  i n f i n i t e s i m a l random w a l k  is  d-dimensional  Proof:  Qx  B^_  the  starts  B  X .  on  t  The s t a n d a r d p a r t o f  2d  (4-1) a n d ( 4 - 2 ) .  We o b s e r v e  t  t h a t these  f o r the difference  X , w i t h parameter  , on  0 e X , and a t each time  nearest neighbours  B  o f r a t e 2.  t o a Markov p r o c e s s , B  at  Oy  a r e t h e i n t e r n a l d e n s i t y f o r an  Brownian motion  from  equations correspond  of  y x-y yeX - -  A s i n Lemma 3.3.1 we may c o n s t r u c t d i f f e r e n c e e q u a t i o n s  coefficients  where  I sy / --  2  D  internal a.s.  U £  v  t+At Q-  We may t h e n w r i t e t h e  step  B  o f i t s current position,  t e T  t a k e s a s t e p t o one with infinitesimal  2 probability  a = At/Ax  put with p r o b a b i l i t y Let  B  i t  f o r each o f t h e 1 - 2da ~ 1 .  denote t h e  th i  (Bt+A-t  " B^).  2  | B  s  +  = P(B  coordinate o f B  <  B  l  = t  t  2  2  t  = 2At . •  stays  ~  .  Since steps i n opposite  i s an i n t e r n a l m a r t i n g a l e .  1  >  B  The p r o c e s s  = x) .  t  ; 0 <_ s <_ t ) = 2 a A x  predictable quadratic variation  t  possibilities.  X  d i r e c t i o n s have equal p r o b a b i l i t y , E(:  Q- ^  2d  Also  Hence t h e ( i n t e r n a l )  Using Burkholder's  Inequality  on  i  t h e h i g h e r moments o f  t  .  the  Now a s t e p i n o n e d i r e c t i o n e x c l u d e s  same t i m e s t e p .  H e n c e E [ (B- .  3  < B , B^>  steps  to a s s e r t that  „  1  - B ) ] = 0 1  t+At  B  Now we i n v o k e  Hoover and P e r k i n s  has a.s. a standard  S-integrable  part  b  during  i f i ^ j .  t  a r e independent, so t h a t t h e i n t e r n a l  = 26..t .  1  t  i s  a s t e p i n any o t h e r d i r e c t i o n  - B ) (B  1  t+At Successive  ±2 (B^_)  , we may c o n c l u d e t h a t  process (1983) T h e o r e m  8.5  , a n d we o b s e r v e t h a t  b  satisfies i) ii)  b^  i s a martingale  <b , b > 1  3  t  = 26 i  (again w i t h reference characterize  t  i =  l,...,d  i ,j =  l,...,d  :  t o Hoover and P e r k i n s  d-dimensional  (1983)).  Brownian Motion.  •  We n e e d a l i t t l e m o r e i n f o r m a t i o n a b o u t t h e o  Lemma 4.3.2: :  is  If  — ^ — '  Now i ) a n d i i ) a b o v e  °t > 0 , ~  Q's . 2  Q-  x" — = i d Ax  II  P  (°x) , w h e r e t -  —  t h e d e n s i t y f o r B r o w n i a n m o t i o n o f r a t e 2.  ||  P (x) = exp{ } t r— 4t » 4TT t Qx Hence i s S-continuous i n Ax 1  d  t  for  Proof:  °t > 0 .  We know f r o m 4.3.1 t h a t t h e d i s t r i b u t i o n  t h a t f o r b ^_ , w h i c h h a s d e n s i t y  P ^_ , f o r a n y  Q  the  lemma w i l l  f o l l o w then,  0  i f Q-/Ax x  d  i s  We o b s e r v e t h a t t h e f o l l o w i n g e q u a t i o n  C4-4I  (6 Q ) t  Now i t i s c l e a r  x  =  (A Q ^ )  for  B  °t > 0 .  i s t h e same a s The s t a t e m e n t o f  s-continuous f o r  °t > 0 .  holds:  x  f r o m t h e d e f i n i t i o n o f t h e random w a l k  B  that the  coefficients the  d  Q -  indices  Claim:  t, x ,Q - > Q - ._ for - x — x+Axe, . k  a u n i t vector i n the  We p r o v e t h i s b y * - f i n i t e Q  At x = P ( B = x) = 6 - . x 0 0  (4-4)  and (4-2),  C4-5)  Q x  (4-6)  t+At Q _ .  Now  x  k = l , . . . , d , where  i n d u c t i o n on t .  + a ( x  e k  d i r e c t i o n , w h i c h p o i n t s away f r o m 0 .. I t i s clearly true f o r  Suppose t h e c l a i m h o l d s  = (l-2da)Q-  t + t  t o change o f s i g n o n any o f  x  F o r each  represents  are symmetric w i t h respect  t e T . -  f o r some  t = At  Then b y *  a £ Q. ) , and . . x.+ Axe j=l d , r + a )  t = (l-2da)Q- _ .  t n-  i f x 5^.0 t h e n a l l t e r m s a p p e a r i n g -k  _  i n (4-5) a n d (4-6) a b o v e l i e  i n t h e same h a l f - p l a n e ; e a c h t e r m i n (4-6) i s s h i f t e d b y e A x r e l a t i v e t o , k and h e n c e , b y t h e i n d u c t i o n a s s u m p t i o n , i s n o t g r e a t e r t h a n , t h e c o r r e s p o n d i n g term i n (4-5).  Now  Hence  suppose  possibilities  Qx  > Q- , - . - x+Axe, k  (w.l.o.g.  for $  k = 1) x  = 0 . We p i c k o n e o f t h e t w o  and s t i c k w i t h i t .  Lety  be a  By o u r a s s u m p t i o n s o n A t , i n a n y d i m e n s i o n  Cl-(2d l)a)Q§  (4-7)  +  Now t Q-Ax,y  adding  x  a O-  > (l-(2d l)a)Q^ +  +  a  07 + Ax,y  t , = Q, we o b t a i n ' Ax,y  a  x  (k-1) t u p l e .  d , (2d+l)a  > (1-(2d 2)a) | ^  Q~ -Ax,y  +  t  o  Q  x  +  <_ 1 .  a  Hence  p * ^  b o t h s i d e s o f (4-7) a n d u s i n g  (4-8)  (l-2dcO  + a(Qj  Oy  Ax,y  Now b y t h e i n d u c t i o n  d  C4-9)  a  Adding  1 j=  t Q5 u y  . j  > «  I  j=2  t 2Zx,y Axe j +  '  ( 4 - 8 ) a n d ( 4 - 9 ) we o b t a i n  t+At 2  This  assumption  d  y + A x e  2  + Q) >_ (l-2da)Cj£ + a (Q^ + Q | ) -Ax,y Ax,y Oy 2Ax,y  5y  establishes  t+At  ^Zx.y  *  t h e c l a i m f o r { Q ^ ^ | x e X} . +  t  Now we i n v o k e t h e  p r i n c i p l e o f i n d u c t i o n under t h e t r a n s f e r p r i n c i p l e t o e s t a b l i s h t h e c l a i m for a l l  t e l .  Now b y Lemma 4 . 3 . 1 , f o r a n y i n t e r n a l r e c t a n g l e  X  (4-10)  I  —=- A x d xeA Ax  d  P  c  In l i g h t o f the monotonicity  °(Q-/Ax ) = P d  Q  (°x)  Therefore also is  (x)dx , f o r  A  °t > 0  claim just established, this  means  i f °t > 0 , a n d x e n s ( X ) .  . d Ax  i s S-continuous i n t -  for  t > 0 , since -  P (x t  continuous i n t , i n t > 0 .  We p r o v e t h e f o l l o w i n g i n e q u a l i t y i n o r d e r t o o b t a i n moment b o u n d s i n s e c t i o n 4.5.  A :  Y-  xeA  I t h a s no i n d e p e n d e n t i n t e r e s t .  We i n t r o d u c e  the notation  V Y  Lemma 4.3.3.  There i s a constant  K < °° a n d a p o s i t i v e i n f i n i t e s i m a l  At'  49  such t h a t f o r a l l i n t e r n a l and  sets  y e X  t,r, s e T  , whenever  r-s > t - r > At' ,  (t-r) < K- (r-s)  t-s r-s Q-A:y - - QA:y  Proof:  Suppose f i r s t  Now c l e a r l y  max  °(t-r) > 0  f o rfixed  A:y  t, r  and  s ,  -  A  ^  - Q- -  Vt,r,s:y  \'t,r,s:y  A = { x e X|Q§ < Q-} y,t,r,s x-y x-y  where  The  that  IQ- - - Q- -1 = Q- -  A,y  is  A , and a l l  standard part of this  s e t i s easy t o i d e n t i f y ,  using  Lemma 4 . 3 . 2 .  I t  the b a l l  A  ^ = {x y,t,r,s  where ball  e R | d  1  Mx-y|| 2 ~ 2  1  1  J  n  y = °y , t = °t , r = °r i s computed by s o l v i n g  i.e. [4Tr(r-s)]  d/2  < 2d ^ - s ) (r-s) (t-r) and  s = °s . -  P^ ^(x)  exp[-  = P  4(r-s)  (x)  4  r-s  <=>  Now  (P y,t,r,s  (r-s) (t-s)  .2 (t-s) r-s . t - s , 6 = 2d logC ] t-s r-s o  r-s  J  r  (x-y) - P (x-y))dx t-s J  The r a d i u s  '  6 t,r,s  of this  -  -] =  d/2  ^ r r-s  c  [47T(t-s)]  d/2  exp[-  -]  4(t-s)  50  a-l' (4tr)  •  J  d/2  d-1  to  '  S  ,  1  2  ,«<  1  d-l  i n thef i r s t  dp ,  , 2,., d-1 exp(-p / 4 ( t - s ) ) p dp  d/2  i s thearea o f the surface,  d-l  „  d/2 e x p ( - p / 4 ( r - s ) ) p  1  (t-s)  }  p = p^/tTs"  Now l e t  r  t.r.s ' '  .d/2  S  ,  (r-s)  1  TT) (4TT)  where  t  S, , , o f t h e u n i t b a l l d-1  integral,  and  p' =  p/2/r-s  i n R  i n the second,  obtain [ (4TT)  d/2 j  t,r,s/2/r-s  ^ P  p  t,r,s/2/t-s -p  d-1  Since the function bound t h i s  _ 2 e  f(p)  = p  e  i s b o u n d e d , we may f i n d  i n t e g r a l by K  6  t, r, s  ( ~ i / r— vr-s v t - s  K / l o g (1+ ) ( * r-s  /t-s  - /r-s,  /t-  Now i f r - s >_ t - r we may b o u n d t h i s  pEE ( ! 2 " ' (  K  r  < K  S  }  (r  s)  f u r t h e r , by  _)  /t-r(/t^s~ + /r-s)  t-r r-s  T h u s , i f °(t-r) > 0 (t-r) (4-11)  r - s > t - r = >  s u p \c£ A:y AcX  Q£ .I A:y -s  —  (<rK --s7 ^ ) T-  ,  Vy e X  K  to  51  Since real for  e > 0 t  the  internal  , t h e n by  - r > A t ' ss o  We internal  statement  (4-11) i s t r u e f o r  the p r i n c i p l e  of infinitesimal  overflow  for a l l  i t must  hold  .  •  b e l i e v e t h a t t h i s Lemma i s t r u e proof  t - r > e  of this  i s not  easy.  for  t - r  down t o  At  but  an  52  4.4  The T o t a l Mass P r o c e s s  Now  l e t M  whose d e n s i t y  =  t  ) xeX  U Ax tx —  L  v  U  M  b e t h e t o t a l mass o f t h e i n t e r n a l m e a s u r e  i s obtained by s o l v i n g  (4-1) i n d u c t i v e l y .  We h a v e b y  definition,  r 0<s<t In what f o l l o w s q  and  t  v )  v  xeX  t-s  yeX  eX  i~l  x  \jt  There i s a constant <_ t  - i  2 q  ) <  U d  x  Ax  y  c o n s t a n t whose v a l u e s d e p e n d o n l y  change from l i n e  c  d e p e n d i n g on  e  c  C  2 q  1  q > 1 , M  , and  (Burkholder's I n e q u a l i t y ) , the f a c t  + c E( "  I I 0<s<t yeX ^  max <s<t  0  L  a, b, a A b < a —  1 • U  U AtAx ^  sy  A t A x1 | d  )  -£  d  ,2 • * £ ) 5*  yeX + c  Now c E(  M 0  2 q  t h e s e c o n d t e r m i n (4-12) a b o v e may be b o u n d e d b y  max | M 0<s<t -  A t | ) <_ c E| q  \ M At| 0<s<t -  q  .  Hence  on  to line.  t  and t h e f a c t t h a t f o r any  E(M ) < c E(| t — -  (4-12)  At  s  ,  U s i n g T h e o r e m 3.4.5  t-s Q- - = 1 x-y  /U /  denotes a f l e x i b l e  E(M  Proof:  r  , a n d w h i c h v a l u e s may  Lemma 4 . 4 . 1 . such t h a t  c  r  (4-12) b e c o m e s  that  t ^ ,  E ( M ) < c EI T M At t — - „_ s 0<s<t 2 q  r  0  < c E| I M A t l ^+ ~ ' 0<s<t 5  (4-13)  c(M +D 2 q  0  < c y E(M )At + c(M +l) — « . s 0 0<s<t 2 q  for  a l l  t <_ t  f  2 q  .  We c o m p l e t e t h e p r o o f w i t h a n a p p e a l  t o the appropriate version o f Gronwall'  Lemma (Lemma 3.5.2).  We may now e s t i m a t e t h e d i f f e r e n c e s  (4-14)  •  E|M t  (4-12)  to  i norder t o  Then  (using Burkholder's from  as f o l l o w s ,  L e t q > 2 , and w.l.o.g. take  o b t a i n c o n t i n u i t y o f t h e mass p r o c e s s . 0 <_ r < t <_ t  M, - M t r  M I r'  2 5  < c E — -  1 U AtAx | sy y e X --  I J i_ r<s<t  d  q  1  L  v  I n e q u a l i t y a n d a p p l y i n g t h e same r e a s o n i n g a s i n g o i n g  (4-13))  < c E | I M r<s<t ^ ( t - r )  q " 1  (t-r) - -  q  < c E ( I M - ^ ) ( t - r )q s sx. r<s<t -t -- rq  s  (applying Jensen's I n e q u a l i t y — — t-r  ) on t h e i n t e r v a l  to theprobability  r < s < t) - - -  measure  54  < ~  (4-15)  c  I Ellfl) r<s<t " S  (t-r)  <_ c e x p ( c t ) ( t - r )  H  f  (t-r) - -  q  ,  b y Lemma 4 . 4 . 1 .  T h u s we h a v e  Lemma 4.4.2. on  The t o t a l mass p r o c e s s M  0 < t < t  Proof: In  / d e f i n e d above, i s a.s.  .  Apply t h e Kolmogorov C o n t i n u i t y C r i t e r i o n fact  S-continuous  M  i s S-Holder  continuous o f order  (3.8.1) t o ( 4 - 1 5 ) . — - e  f o r any  • E > 0 .  55  4.5  and  S-Coritinuity o f the Process  For  internal  X*" t  stand  sets  for T x£A  and  d  —  functions  C  °(AF)(x)= Af(°x)  Lemma 4 . 5 . 1 .  F  lifts  , Vx  F  If  Ax  L  l e t the notations .  d  We  introduce the  2  such  d  R e c a l l f r o m s e c t i o n 3.9 t h a t i f a n  to order  2 a function  f e C, „ , t h e n b, 2  i s an i n t e r n a l e X  f u n c t i o n such  that  , then the process  X  F  i s  S-continuous on  a.s.  Proof:  r < t e T  For  ,  7* - X = t r F  x  Y F ( x ) (U - U )Ax y tx rx eX  d  l I L ————J F(x)  r<s<t  =  (4-16)  i  ^ ^ r<s<t  x  I  I  =  sx-i  s+At,x  r  AtAx  x  y  ,  r<s<t  F(X)  r  r/  U x s  (Au  )  L . - S - X — —  (AF)(x)  U  SX  Ax  -  V d V A+Ax A  d  Usx A  A  i  2At  ;  ,  sx  At  x  u +  I  ^ ^ r<s<t  L  that  e ns(X) .  | A F ( X ) | <_ K , Vx  |F(X) | +  J F(x) U tx xeX  F  = { f e C ( R ) | 3K < °°  2  .  d  function  functions  v  I A f (x) | <_ K \ / x e R }  +  internal  T  U Ax tx  u  c l a s s o f (standard)  |f(x)|  A , and i n t e r n a l  [ F ( x ) ( ^ sx x  sx A _p 2  . d At  g sx  /AtAx  d  AtAx  d  X^  56  The  Now  f i r s t t e r m i n (4-16) a b o v e we  AF X- < K M . s s  Hence i f  . -p j?. 2q EX - X < c E( -' t r 1  + c E(  I X r<s<t -  At  q > 1 -  _ )  M At) s r<s<t L  r e c o g n i z e as  I r<s t <  2q  I xeX  (using Burkholder's Inequality  F (x) u  I E(M ) s r<s<t  + c  I E( r<s<t  2 q  —  d  q  f o r t h e s e c o n d t e r m on t h e r i g h t  i n c o r p o r a t i n g the term i n v o l v i n g  < c  Ax At)  2  "max"  a s we  (t-r) t - r - -'  I F (x) \/ — xeX 2  U  d i d i n (4-12))  2q  Ax ) At d  SX  q  r | F ,^ l E |X | r<s<t 2  (we now  r e c o g n i z e t h e s e c o n d terms as  (4-17)  < c  if  Now  we  t  a p p l y t h e N.S.  (Theorem  3.8.1).  (t-r)  t  2 q  + c  (t-r)  q  , u s i n g Jensen's  and  At  and  F X -  2  2 <_ K M ) -  ,  I n e q u a l i t y a second time.  v e r s i o n o f the Kolmogorov  Continuity  Criterion  •  57 A Um sup °X n-*>° t e [ 0 , t ] "  Lemma 4.5.2.  n  °  =  *  a - S  where  f  A t  n  = { x e X|  |x| >_ n} ; t h a t i s , X  t  i snearstandard i n  M ( R ) F d  for a l l  , a.s.  Proof:  Let H  be i n f i n i t e .  0 < _ F < _ 1 , F = 1  on  Let F  A , F = 0 H  Q-  t+At  AF —  ) U  i s bounded.  Ax  d  that Then  .  y e ns(X)  -t-s Q s 1 , so t h a t  U  and  —  1  Now b y Lemma 4 . 3 . 1 , f o r e a c h  By a s s u m p t i o n  A^ H 1  < ECX/" ) - 1 ( 1  E(<>  r )  on  be an i n t e r n a l f u n c t i o n s u c h  e  =  r )  „t+At Qz 0 .  i s nearstandardly concentrated, so t h a t  Q  lim n  I  °(  U  y£A n  -X»  Ax ) = 0 . d  -  Hence,  (4-18)  I  Etxf) <  "yeX  t  e U ~ 0 X °Z  xl  =>  s 0 a.s.  t  F Now b y Lemma 4.5.1  X  i s a.s.  S-continuous.  T h e r e i s a c o u n t a b l e S-dense s u b s e t o f  T  f o r which  (4-18) h o l d s .  Hence  A °X = O U t e T , a . s . => sup °X = 0 a.s. ~ " teC0,t ] £ A ° n P Therefore sup X — > 0 a s n -> , teCO,t ] £ A n l , but since X > X we m u s t h a v e c o n v e r g e n c e a . s . F  H  f  00  f  A  n  +  n U  58  X : T  Theorem 4.5.3. nearstandard  M  F  ( R )  *C( R ;  in  i s a.s. S-continuous. (°X)  ( R ) ) , and d  = L(X)  q  Thus 0  s t  -  X 1  i s  for a l l  t t e [ 0 , t ] a.s.  Proof:  R e c a l l t h e weak t o p o l o g y o n f , x  continuous  n  ( f ) -»- x ( f )  2 cl C _( R ) b, 2  f u n c t i o n s which  M  Let  T  •t  ( R ) .  {$ } k  $  X^_  4.5.1 e a c h  .  :  Mp( R ) d  Let  {(j) } k T  + x <=>  be a c o u n t a b l e  f o r a l l bounded collection of  c o n s t i t u t e a convergence determining c l a s s f o r  be l i f t i n g s  to order 2 o f the  i s a.s. S-continuous.  {cj> } . k  Now b y t h e L o e b  Then b y Theorem  construction  ( s e e s e c t i o n 2.2)  <h  (4-19)  Therefore  d L(X ) o s t  = °X  t h e l . h . s . o f (4-19) i s c o n t i n u o u s  convergence determining c l a s s . for  k  Hence, almost  a . s . f o r each  surely  <j)  L(X ) ° s t t  -  1  i n the = L(X ) s  s t  -  1  t s s e T .  a l l  *  F r o m Lemmas 4.4.1 a n d 4.5.2, for  0  a l l t  , a.s., so t h a t  X  ° (X^)  i s i n fact nearstandard e x i s t s and equals  L(X ) f c  °  d  i n  st~J-\^  ( R ) t  a.s.  •  Now we show a s t r o n g e r f o r m o f c o n t i n u i t y .  T h e o r e m 4.5.4. '' end  Let X  o f s e c t i o n 4.2).  0  Let A  °t > 0  on a c o a r s e r g r i d  Proof:  We t a k e  Let  Y = t  At'  1  L  T v  TEA  be n e a r s t a n d a r d  T'  be i n t e r n a l . c T  f r o m Lemma  QA:y  + A t  -  U Ax Oy -  i n  d  *  M  F  Then  d ( R ) X  of infinitesimal  4.3.3. and l e t  t  (as d i s c u s s e d a t t h e i s S-continuous f o r  spacing  A t ' independent  of  A  59  U Y  Then  X  A  t  =  2  t  •^Ax )  I I Q j . - ( Jv AtAx' sy 0<s<t yeX Y  £  d  2  A :  sy  = vJ; + Y t t 2  F r o m Lemma 4 . 3 . 3 , t h e r e i s a r < t < 2r -  K  such  that, i f  i n 7" , |Q- QI < K(t-r)/r A:y A:y - -  .  —  I y  -Of  |0j A:  eX < — — r  x  A : y  |U °y  Ax  Hence i f t ~ r -  and  °r > 0  d  (t-r) M - 0  ~ 0 . Thus  Y^  i s S-continuous  £  Now s u p p o s e using Burkholder's  <  t e T '  i n T'  n st  - 1  (t>0)  .  , t - r < l , 0 < y  iY  q >_ 2 .  2 2i 2 I - Y < c E t r —  and (4-13)),  2c  Y ^  1  u  0<s<r -  +  C  E  + c E  <  C  E  Then  I n e q u a l i t y a n d a b s o r b i n g t h e t e r m s i n v o l v i n g a maximum  t h e s u m m a t i o n s , a s we d i d i n s e c t i o n 4.4, ( ( 4 - 1 2 )  E —  l f and  <  ( t - rY)  I  v r-(t-r)'<s<r  I  < ^. r<s<t  I  I  y£A  ye  (Q- A:y  yeX  -  -) A:y  Q-  X  i  Y  U AtA: sy  d q  sy  (Q|.5) Us y A t A A:y 2  u  Ct-r)  -±  2-2Y  M  At  0<s< r - ( t - r )  (we i n v o k e Lemma 4.3.3 a n d n o t e  (r-s)> (t-r)  .)  into  60  q  1  + C E  1 • M  r-(t-r) <s<  At -  Y  |pjf - A:y  (since  T  At q  M  r<s<t  1  (4-20)  -| < 1) A:y' —  1  + c E  < c ( ( t - r )  (  2  -  r  -  2  Y  )  ( t - r )  q +  Y  (t-r) )  q  q  +  ( f o l l o w i n g the u s u a l Jensen's i n e q u a l i t y t r a i n o f development, b y now Our f a i t h f u l  o  M  '  ' -f '  01  so  familiar).  servant the constant  a n d  ' k  q  ut  n o t  o n  t  c  h a s now  a c q u i r e d dependence  on  - •  a n d  2 y  The b e s t c h o i c e o f criterion  f o r (4-20) seems t o b e —  ( T h e o r e m 3.8.1) we  get  .  Using the Kolmogorov  S - c o n t i n u i t y w i t h a modulus  -j- e  f o r any  e > 0 . Remark:  • Of c o u r s e  failure of  i f  p u t s a l l i t s mass o n a n u l l { t « o}  S - c o n t i n u i t y i n t h e monad  C o r o l l a r y 4.5.5.  Let  F  X^_  Proof:  K  Let  is  S-continuous  on  i  T  |F| .  be a bound f o r  zft  = 0<s<t I yeX I • (xeX Iv w  Let  X  F  y  =  yeJ  Then  F  "F  X* = X  ~F  + X  fc  .  Now  . „.  i f  •  -r<  t > r e T  X  Then  y  v  F(x)  .  v  ~  o~  yeX xeX  QI~J x-y sy) ( / u _ A t A 2x _ £ -£ T  get  —1 n s t (t>0) , a . s .  U  F(x)  , we  .  t let  A  be any bounded i n t e r n a l f u n c t i o n on  F  the process  set  z  d  A  syA x ) ^ °£ d  +At  x-y - *•  Ax ,  U  d  Oy -  and  61  |x 1  - x | < K | X t r — t F  F  1  Following T'  n  {  f  >  0  }  1  - x r  {  F  >  0  }  | + K | X ' t  {  f  <  0  -  }  x r  1  the f i r s t part  o f Lemma 4.5.4,  {  F  <  X  0  }  is  F  S-continuous i n  {°t > 0} . Now i f q > 1 ,  E  |X  F  - X | F  2 q  < c E|  + c E| "  I 1 ( 1 0<s<r yeX  I r<s<t  F ( x ) ( Q ^ - Q :|)) - £ - -  2  x  x e  I K yeX  2  < 2 K  2  X  u  AtAx | d  U  AtAx | d  2 q  -£  2 q  Now  C  F ( x ) CQ-  £  -  -  b y Lemma 4 . 3 . 3 . obtain  Q|~|)  )  Hence f o l l o w i n g  I~F  and t h u s  any  (Q|  -  -  <g~-)  2  the second p a r t  < c  —  o f t h e a r g u m e n t i n 4.5.4  we  ~FI2q  ~X  is  ,  ^ < c(t-r) — - -  S - c o n t i n u o u s on  We b e l i e v e  that  X  ,2q/3  T  a.s.  i s Holder  •  S-continuous o f index  2 ~  ^  z  o r  z > 0 .  Remark:  I f we t o o k  4.5.5 w i l l will  sup  the s i m i l a r r e s u l t  E X - X t r  Remark:  2  hold  t  f  t o b e h y p e r f i n i t e , °t  a.s. u n t i l  a l lfinite  = °° , t h e n 4 . 5 . 3 , 4.5.4  t i m e s , and hence a . s . on  u s e t h i s f a c t w i t h o u t f u r t h e r ado i n s e c t i o n  4.7.  ns(T)  .  and We  62  4.6  C h a r a c t e r i z a t i o n by  We in  the  show now  a Martingale  t h a t the measure-valued process  l a s t s e c t i o n as  s t u d i e d by  Problem  Dawson a n d  o  L(X ) t  others.  t  s  We  _  will  use  one  of the  Dawson a n d  The  Kurtz  (1982) a n d  continuous  x  i s the  t  paths,  and  m  t  X  =  We  b  Ms.  refinements  C R^)  t  0  b r a n c h i n g measure d i f f u s i o n  We  will  (R ) ) may  be  ( R )  valued process  d  F  c o n d i t i o n such  A$  such  D(A)  <f>  x  <{>  a)  a n d b)  martingale.  above f o r f u n c t i o n s  argument.  $  be  a lifting  <M°x)d L ( X )  r  (recall is ° r ) $ X  ( R^) b /2  <|> e of  only.  R )  in  I t i s i n f a c t unnecessary  to  ,  C, „( b, 2  in  t o go b e y o n d suppose  C  2  d ( R ) .  b, 2 |$| <_ K  .  a.s.  (X)  hs  o  A  t o o r d e r 2, a n d  , t e T'  t  =  D(A)  i s weakly continuous  i s the c l o s u r e under  a limiting  let  =  °t  $ e  .  (which  done b y  a  w i t h weakly  that for a l l  ds i s a c o n t i n u o u s  R o e l l y - C o p p o l e t t a ' s a r g u m e n t i n h e r T h e o r e m 1.3 For  follows:  s  (Lemma 4.5.3) t h a t o u r p r o c e s s  verify  extension to  c o n d i t i o n i i i ) ) i s as  x^  s  know a l r e a d y •  x 0  ,2 x • ds  Stroock  A  x  r  M  the g i v e n i n i t i a l  t " o t  <m >  C  (unique)  domain o f the L a p l a c i a n <j> d e f „ a )  The  martingale  others.  c h a r a c t e r i z a t i o n o f the c r i t i c a l  Theorem 4.6.1.  F  diffusion  (1986), which are  g i v e n i n R o e l l y - C o p p o l e t t a ( 1 9 8 6 ; T h e o r e m 1.3,  M  constructed  l a r g e r c l a s s o f m a r t i n g a l e p r o b l e m s i n v e s t i g a t e d by H o l l e y and  (1978),  (the  , w h i c h we  i s i n f a c t the measure  1  characterizations given i n Roelly Coppoletta of the  x  t  X  $(x) U Ax — tx  f r o m 4.5.2  t h a t t h e mass o n  always i n f i n i t e s i m a l d  , by  Loeb L i f t i n g  X \  ns(X)  a.s.) Theorem  ( T h e o r e m 2.1.6)  Then  Hence, <j> m  is  4>  t  =  t  x  <j) "  x  A*  t  x  o  0  ds  5  i n d i s t i n g u i s h a b l e from the $  M  def  $  =  x  -  $  x  =  I xeX  -  - U  )Ax  (  =  l xeX  $ ( x ) (A U  $  Hence  irr  =  M -  T h e o r e m 5.2  <m^>  ^  y  -  xeX Y  Vt e T "  m^  Ax At d  d  d  the  A  d  t  with  respect  Hs  2—  integration-by-parts)  d X  A x  ]  ?  sy 1  to the  implies that  irr  internal i s an  fc  Perkins i s a.s.  t  same p a p e r ,  AtAx  A Ax  sx  , a.s.  coincides  part of  (A$)(x) sx -  -  *(x)[7u  [st(M )]  standard  ) -  a.s.  a n o t h e r r e s u l t o f Hoover and  the  AtAx  i n f i n i t e s i m a l analogue of  L  since  u  A t  o f H o o v e r and  Further, function  rr  i s a martingale  °t  I  ~ U ) sx  A  Y 0<s<t  I  -  d  , i s+Atx  *(£)  « . 0<s<t  Now  U  I xeX  y  (4-21)  Ox  I 0<s<t  (using the  of  x f A t  0<s<t $(x)(U -  part  A*  r  J  -  standard  (1983).  This  continuous i n with  Perkins  = st[M  the  verifies t  (Theorem  F^_  F -martingale, t a)  . by  o f T h e o r e m 4.6.1.  , the p r e d i c t a b l e  square f u n c t i o n  (1983)  filtration  [m^]  t  .  square Now  by  6.7)  ] ,  i n t e r n a l s q u a r e f u n c t i o n , and  by  T h e o r e m 8.5  of  the  64  CM  provided  $2  (M^)  (with respect  ]  ~ <M  T  > vt  , a.s.,  i s S-integrable,  t o F^_) s q u a r e  $  <M >  where  i sthe internal  predictable  process.  Now b y B u r k h o l d e r ' s I n e q u a l i t y ,  i f p > 2  P/2 sup ( M ) )  E(  "  $  t  <t  <  P  £  E  c  ~  "  P  T 0<s<t  l $ ( x )U Ax At •• sx xeX 2  u  d  p/2 < c K - p M s  (where  M At  \  E  0<s<t  i s t h e t o t a l mass p r o c e s s )  w h i c h i s f i n i t e b y Lemma 4.4.1.  $ 2  Now a p p l y C h e b y c h e v ' s I n e q u a l i t y as  t o conclude that  (M )  i s  required. Now  (4-22)  <M > $  0<s<t  xeX  \ 0<s<t  I $ (x) U xeX  AtA?  tr I  I  0<s<t ~  xeX  $ (x)  v  sx  I  2  -  r  {u < sx  A t , ( 4— —-i d Ax  4  A —t  d  - Us x ) A t A x  A  Now t h e s e c o n d t e r m a b o v e may b e b o u n d e d b y  r, j. 0<s<t -  sx —  ~v xeX r I I { X  i  < L}  , 4Atl {u < TJ sx d A  -AX  AtAx  d  S-integrable  Ml  I  I  2  0<s<t  < K  xeX — { x > L>  • t • (2L) •  2  By 0  K  +  I X^'0<s<t -  2  L -»•  00  .  Hence, f o r each  <k <nr>  —  < —-r} Ax  d  At  }  The f i r s t  0  t  term i s i n f i n i t e s i m a l  t -  a.s. =  °  $ ) X „ s 0<s<t -  2 2  v  L  A  At  r e q u i r e d b y c o n d i t i o n b) o f T h e o r e m 4 . 6 . 1 .  T h e o r e m 4.6.2. constructed  The s t a n d a r d p a r t  others.  x  f c  L  a.s  f-S  finite.  x 0  2  ds , s  Hence  o f the h y p e r f i n i t e process  by s o l v i n g t h e d i f f e r e n c e e q u a t i o n s  U , i s the c r i t i c a l  for a l l  t  a.s. ,$ = <M >  11  as  {u  u  Lemma 4 . 5 . 2 , t h e s t a n d a r d p a r t o f t h e s e c o n d t e r m i n (4-23) g o e s t o  a.s. as  ,» .\ (4-24)  L  Ax At d  /  ^  d  Ax  (4-23)  i  u  X  (4-1) f o r an i n t e r n a l  density  b r a n c h i n g m e a s u r e d i f f u s i o n c o n s i d e r e d b y Dawson a n d  4.7  New R e s u l t s  This  o n t h e Dawson M e a s u r e V a l u e d D i f f u s i o n  section represents  j o i n t w o r k w i t h E d P e r k i n s , my t h e s i s  Now t h a t we know t h a t t h e m e a s u r e v a l u e d via  process  s o l u t i o n o f the h y p e r f i n i t e difference equations  " d e n s i t y " TJ  x^_  supervisor.  w h i c h we  constructed  (4-1) f o r a n i n t e r n a l  , i s i n f a c t a c o n s t r u c t i o n o f t h e Dawson P r o c e s s ,  we may u s e  t-X  this  construction  t o a s c e r t a i n some r e g u l a r i t y p r o p e r t i e s  o f the process.  F i r s t we n e e d Let A c X  Lemma 4 . 7 . 1 . Then on a s e t  Proof.  fi  of probability  L e t {A } „ n neN  f o r each  be a Loeb-measureable s e t o f 1  ,V  L ( X )(A) = 0  measure  t e T* n s t  _ 1  zero.  (t>0)  be a n e s t e d sequence o f i n t e r n a l s e t s such t h a t  .  A c A - n  n e N , and  t h a t t h e A 's n  ° (y (A ) ) ->• 0 . E x t e n d {A } t o *N i n s u c h a way x n n a r e s t i l l n e s t e d ( t h a t t h i s may b e d o n e , f o l l o w s f r o m  w^-saturation). Let with  H  u (A ) ss o . S u p p o s e x H Now b y Lemma 4 . 3 . 2 , i f K i s a b o u n d f o r p  be i n f i n i t e  °t > 0 .  i n *N .  Q-  H i  E C X ^ ) = EC H  *€A„ — rl  I xeA  Thus f o r e a c h Let  Ax  —  t  t e T' , — (x) ,  °-  t+At  Ax ) = d  H  Then  there  I yeX  U  Ax  H —  d  < Ky (A )M x  H  0  =  0 .  -  t i s a s e t fi- o f p r o b a b i l i t y  A  1  on w h i c h  X  H  be a c o u n t a b l e S-dense s e t i n T' . L e t fi' b e t h e s e t o f , H A H i p r o b a b i l i t y one on w h i c h X i s S-continuous f o r t e T . Let fi  TT  { t }, -k k e N  =  n k=l  fi n n %c  .  Then on  fi H  ,  (X t  ) = °  lim V -k. -  t  t  (x. ^  ) = 0 l  f o r some  ~ 0  sequence  {t } . .  Thus  —  X  1  for  each  H t  \J x 0 Vt  -  ~~  e T'  , hence by i n f i n i t e s i m a l  to e ft , t h e r e i s a n i n f i n i t e s i m a l  e(co) s u c h  rl  underflow  that  A  For  n e *N , l e t Y  = sup  teT'  n  the sequence H e *N . a set  {Y } n ne N  Hence  ft  X  A 1 .  n  t  i sinternal.  ° (E (Y ) ) 4- 0 n  of probability  as  n  =°  Y  i s S-L  f o r each  J  N , and thus  through  A L ( X ) ( A ) <^L(X )(A )= °X  °Y  n  4-0  on  < • °Y  n  Hence  n , and  E ( Y ) x o , f o r any i n f i n i t e H  Now  o n e . Now  V/t e T' , V/n e N •  Then  L ( X ) ( A ) = 0 , \ / t e T' t -  o n ft . A  •  H e n c e , we may d r a w  C o r o l l a r y 4.7.2. x  (A) =  Proof:  Vt  0  I f A c R  d  i s a Lebesgue n u l l  _ 1  i s a Loeb n u l l  We i n t r o d u c e t h e n o t a t i o n s  is  If  f : R  d  •+ R  Let F : X  let  be  *R  •  x^ = t  f  dx.  In particular  and  x  x^_  = x  (A) x  f  i s a.s. continuous i n  L ( X ) CA) = 0 \/t  (since  M  t  be a l i f t i n g w i t h r e s p e c t t o  { x : x e X °F(x) ^ f(°x)}  4.7.1,  a.s.  set.  f o r any Lebesgue s e t A .  Proof: A  = 0) , t h e n  i s a bounded measureable f u n c t i o n , t h e n  a.s. continuous i n t > 0 .  t > 0  (X(A)  > 0 , a.s.  stCA)  T h e o r e m 4.7.3.  set  i sfinite  By Lemma 4.5.5  e T'  Vt -  v  X  F t  which  a . s . Now  a.s.) .  Hence  L(y ) x  i s a Loeb n u l l  F  i s S-L x  o  f f c  i s S-continuous a.s.  1  of  of  f .  Then  s e t . B y Lemma (X,X ) V t a . s . fc  = f °F(x)dL(X ) = ° X Vt j t t F  e  T'  68  Theorem  4.7.4.  Let  <j>  be a bounded m e a s u r e a b l e f u n c t i o n on  R  , and  suppose a sequence converges t o  {A } o f u n i f o r m l y bounded measureable f u n c t i o n s k keM i n Lebesgue measure. Then f o r any e > 0  6  sup He,t ]  Ix  - x  t  t  |  —>  0 ,  f  Proof:  Let  $  and  respectively. Then i f  'E(X  $  JC  Extend  k e N  , be l i f t i n g s  {$, }, k keN  of  <)>  and  <j>  JC  t o an i n t e r n a l s e q u e n c e  , k e N  {$ } k e *N . k  °t > 0  k  t  )  ° J u" (*(X) tx xeX  =  - $  v  R  d  u  -i  -> 0  °  as  tx  (x))Ax k —  (<)>(x) - <j> ( x ) ) d x x  k -»•  00  , since  u  tx  d  where  u = st(U) .  i s bounded and  L  1  ( R)  for  t > 0 .  Hence  i f He  *  N  i s infinite,  °E(X  (4-25)  Now  $ X^  Lemma 5.5.2.  H  )  = 0 $  and  T  ,  hence  -are b o t h  a . s . Vt  ~ Q Vt  def =  „ max X teT' £  0  as  k ->-  00  e T .  S - c o n t i n u o u s on  T  i  n st  (4-25) h o l d s f o r a c o u n t a b l e  " H  H  Y^ — >  ~ 0  H  $  X  Y„  Therefore  X  Therefore since $  of  H  X^_  =>  e T'  H  a . s . , and  ^  0  through  a.s.  N .  -1  (t>0)  S-dense  a . s . by subset  69 Now  we  k st(X  $ st(X ))  ) (resp.  t  sup t [e,t j €  f  (Lemma 4.7.3)  have seen  are  t  I k | x^ t  4> - x Y  t  I  a.s. =  k £ N  that for  (resp.  indistinguishable processes.  o max teT'  i  X *  ki  =  °  Y k  P —>  „  0  x^_)  and  Hence  n  •  70  CHAPTER 5  The  5.1  C r i t i c a l Branching Diffusion  i n One  Dimension  Introduction  In Chapter  T h r e e we o b t a i n e d a n e x i s t e n c e  3u — = Au + f ( u ) 3t  where  f  a Lipshitz  t h e o r e m f o r SPDEs o f t h e f o r m  w tx t  grows a t most l i n e a r l y a t i n f i n i t y , w i t h o u t t h e n e c e s s i t y c o n d i t i o n on  Dawson c r i t i c a l  f .  As d i s c u s s e d  branching diffusion  ,,. (.5-1)  l —  d t  i n Chapter  has been b e l i e v e d  of  imposing  One, t h e o n e d i m e n s i o n a l to satisfy  t h e SPDE  i 3 u /- • = — - + Zu w „ 2 tx 2  A  dX  However t h e t h e o r y o f such the f u n c t i o n of Chapter indeed  i s not Lipshitz.  In this  T h r e e t o show t h a t t h e c r i t i c a l  satisfy  continuous  u ->  an e q u a t i o n has n o t been well-known  (5-1) i n one s p a t i a l  density.  s e c t i o n we u s e t h e  branching diffusion  dimension,  and thus  since results  o f Dawson d o e s  i t has a.s. a  jointly  5.2  The SPDE a n d t h e M e a s u r e  Let  Diffusion  d = 1 , a n d l e t fi be t h e s p a c e  s e c t i o n 4.2. equations  Let  U  tx  X  t  U. = A U + tx - tx  (/rT A tx  tx  — -  -  2  be t h e m e a s u r e - v a l u e d  c r i t i c a l branching diffusion.  -*-)  At  v  ^  " x  S - c o n t i n u o u s on  of  u  TxX  ,  -  p r o c e s s , whose i n t e r n a l d e n s i t y i s  I n the case  additional information that i f U  satisfies  difference  £ tx  .  I n s e c t i o n 4.6 we v e r i f i e d t h a t t h e p r o c e s s  is  as d e s c r i b e d i n  (4-1):  6  and l e t  TxX  be t h e s o l u t i o n t o t h e h y p e r f i n i t e  U (5-2)  {-1, 1}  0  is  d = 1  x ^ = °(X^)  was i n f a c t t h e  h o w e v e r , we h a v e t h e  S-continuous and  a.s. by C o r o l l a r y  U .  3.8.3.  S-L''" , t h e n  Furthermore  U  tx  u = °U  ( 3 - 1 2 ) , t h e weak f o r m o f an SPDE; t o c o m p l e t e t h e i d e n t i f i c a t i o n  a s t h e s o l u t i o n t o ( 5 - 1 ) , we n e e d o n l y n o t e t h a t , s i n c e  t h e n f o r u e ns T  x 0 ,  R)  On t h e o t h e r h a n d , s t a r t i n g  f r o m t h e SPDE  (5-1) we n o t e t h a t t h e 2  martingale problem i s e a s i l y s a t i s f i e d  since f o r  t Ao 0  R  sx  <Kx)dw  sx  i s a m a r t i n g a l e , whose i n c r e a s i n g p r o c e s s i s  ft u R I f we a l s o assume t h a t  u  Q  is  sx  2 A (x)dsdx  L^ , t h e n  <> f e C  c  ( R)  rt  u  o and  a dominated  satisfied T h u s we  convergence  f o r <J> e C  have  Theorem 5.2.1: —  2 b , 2.  Let  dsdx <  sx  argument ensures  ( R)  R  +  x R .  that the martingale problem i s  (which c o i n c i d e s w i t h  u_ (x) 0  be c o n t i n u o u s and  white noise constructed i n section onto  a.s.  00  There i s a j o i n t l y  L  D(A)  ( R)  .  i n this  Let  w^ tx  case) .  be the  TxX {-1, 1}  2.2 f r o m t h e L o e b s p a c e  of  continuous non-negative  process  u  such  that  U  and  fora l l 6 e  0x  =  U  0  (  X  '  )  „( R) , a n d a l l t e  b, z  R  +  3 d) 2  C5-3)  u <t)(x)dx = tx  U  0x  ^^  d  x  u  +  R  R  /u 0  R  sx  — - (x) d x d s sx „ 2  * (x) dw sx  M o r e o v e r a n y s o l u t i o n t o (.5-3) i s t h e j o i n t l y unique measure-valued  dx  continuous  s o l u t i o n t o the martingale problem  density o f the  d e s c r i b e d i n Theorem  4.6.1.  Remark:  The e x i s t e n c e o f a j o i n t l y  realizations one  of the one-dimensional  constructed here.  jointly  continuous  t h i s process  This follows  continuous density holds f o r a l l Dawson b r a n c h i n g d i f f u s i o n , from  not only the  the f a c t t h a t the existence o f a  d e n s i t y i s a measureable p r o p e r t y o f the sample p a t h s o f  Csee C u t l e r  (1985)).  CHAPTER  The  6.1  Support of the Fleming-Viot Process  I n t r o d u c t i o n and C o n s t r u c t i o n  As m e n t i o n e d  i n C h a p t e r One,  case o f a model used d  SIX  quantitative  individuals "drift", "Drift"  the F l e m i n g - V i o t p r o c e s s i s the  i n theoretical genetics:  characters.  i s c o n s e r v e d , and  and m u t a t i o n . i s m o d e l l e d by  Briefly,  the Ohta-Kimura model f o r  i n t h i s m o d e l , t h e t o t a l number o f  the dynamics i n v o l v e  M u t a t i o n i s m o d e l l e d by  two p r o c e s s e s , g e n e t i c  a random w a l k  r e p l a c i n g i n d i v i d u a l s a t random by  whose g e n e t i c t y p e m a t c h e s a n o t h e r i n d i v i d u a l c h o s e n of the p o p u l a t i o n . denote  by  t e R  , d i v i d e d by  +  pCt,-)  I f we  denote  the t y p e s by p o i n t s  p ( t , k ) , t h e number o f i n d i v i d u a l s o f t y p e  forms  N  limiting  space  Z  individuals  a t random f r o m t h e k e Z k  , t h e t o t a l number o f i n d i v i d u a l s  a continuous-time countable state  new  on  , t h e n we  d  alive  at  I  Lffp). =  (conserved), then  M a r k o v jump p r o c e s s w i t h  l^je  p" 1  i  1  (k)  + i- , i f  { p(k) P(k)  f e  1 D  )  Z p(k)  where  + D p(i) 9 .](f(p  [y p ( i ) p ( j )  , i f  -2d, i f 0,  k =  j  k =  i  otherwise,  =  1, i f  1  = 3 otherwise,  may  time  generator:  C6-1)  rest  and  and  - f (p))  74  y  and  D  mutation  are p o s i t i v e  d e s c r i b i n g the r a t e s o f " d r i f t "  and  respectively.  Each o f the 2d  constants  N  individuals  t a k e s one  possible directions at Poisson  i n d i v i d u a l d i e s and  N-1  ( " m u t a t e s " ) i n one  t i m e s whose r a t e i s  i s r e p l a c e d by  a g i v e n o t h e r o f the  step  another  individuals  whose t y p e  ( i . e . an  D/N  .  of  the  Similarily  each  coincides with that of  "offspring" of  that  9  individual) To 1/N  1/N  respectively:  1 / 2  (6-2)  X  A  T  = —  takes values The  in  r e - s c a l e time R )  M^(  R )  and  Viot  by  let  d  (1979) i s t h a t i n t h e  d e f i n e d by  r e f e r to Fleming  and  Viot  (1979) a n d  (6-2)  l i m i t as  R  .  d  N •+  converges i n the space o f  measure converges i n Dawson a n d  Hochberg  M  ( R ) d  ,  00  (R ) .  We  (1982) f o r more  this.  However, a n o n - s t a n d a r d immediate from  (6-2).  We  c o n s t r u c t i o n of the Fleming-Viot process  simply  Stated i n non-standard  take  N  infinite  language the  6.1.1. with  For  N  infinite,  S-continuous  paths  and  in  t e n.s.(  ns(  and  ( R ))  is  l e t the s t a t e space  r e s u l t of Fleming *  process  space  , t h e s p a c e o f p r o b a b i l i t y measures on  d  p r o v i d e d the i n i t i a l  Theorem  and  eA  valued processes  .  Y/N  k)  2  r e s u l t of Fleming  d e t a i l s on  A e B(  for  p(N t,  the measure-valued process  *Z  with rate  c o n s t r u c t t h e F l e m i n g - V i o t p r o c e s s , we  and  2  according to a Poisson process  and  be  Viot i s  + R ) (6-2)  d e f i n e s a.s.  , i f initially  a l lbut  a an  * d ri n f i n i t e s i m a l f r a c t i o n of the N ' p a r t i c l e s a r e on n e a r s t a n d a r d p o i n t s i n Z /vN . N o t e t h a t t h i s r e s u l t a s s e r t s t h a t a l t h o u g h t h e p a r a m e t e r s p a c e i n (6-2) i s technically a.s.  * + R , that  Hence X  process  called  has  X  ~ X"_  i n t h e weak t o p o l o g y w h e n e v e r  a standard p a r t , x  the Fleming-Viot  t ~ r e st  , which i s a measure-valued Strong  Process.  -1  + (R ) ,  Markov  75  6.2  The  We their  Dimension  use here  results.  o f a P u t a t i v e Support  Set  t h e t e r m i n o l o g y o f Dawson a n d H o c h b e r g What we  a i m t o show i n t h i s  chapter i s that a  c o n s t r u c t i o n makes much o f t h e i r w o r k more n a t u r a l as w e l l as e x t e n d i n g t h e i r  results.  The  (1982) a n d  some o f  non-standard  (and a good d e a l easier'.)  main r e s u l t  ( T h e o r e m 6.4.4) o f  this  chapter a s s e r t s t h a t the dimension o f the support o f the F l e m i n g - V i o t process is  a t most 2 f o r a l l times s i m u l t a n e o u s l y , a.s.  able  t o show t h i s o n l y f o r f i x e d  times).  d i m e n s i o n o f a s e t , t h a t i n s e c t i o n 6.4  (Dawson a n d H o c h b e r g w e r e  In t h i s  s e c t i o n we  w i l l be  shown t o be  alive  a t some t i m e  d e r i v e the the s u p p o r t i n g  set. Consider a p a r t i c l e p r o c e s s e v o l v e s and may  disappear  i t may  time  t h e p a r t i c l e w a n d e r s , a t some t i m e  ( t o be  r e p l a c e d by a n o t h e r p a r t i c l e  s e r v e as t h e  disappears at time r  are  (or i n d i v i d u a l )  f o r the replacement  r  l a t t e r c a s e we  In this  'descendants'  t e r m i n o l o g y we  will  time of disappearance of t h a t p a r t i c l e , time  t  particle  .  a t time  t  has  the paths of p a r t i c l e s split. at  time We  t i m e , by  a t any  Two  particles  a unique  a n c e s t o r a t any  r > t  time  time  whose d e s c e n d a n t s  comprise  a l l of the  or  else  t  .  which  at  For ease o f  s > t , up u n t i l  the  of that particle Furthermore  at  every  s < t ; i f we  follow  converge, but they w i l l  of a given p a r t i c l e  c o n s t r u c t a s u p p o r t i n g s e t f o r t h e mass o f  looking for a small (finite)  particle  never  a r e s a i d t o h a v e a common a n c e s t o r  t , i f they are both descendants will  a t time  relation.  b a c k w a r d i n t i m e , t h e y may a t a time  this  the  o f some o t h e r p a r t i c l e  i s the descendant  Note t h a t a n c e s t r y i s a t r a n s i t i v e  As  say t h a t b o t h p a r t i c l e s  of the o r i g i n a l p a r t i c l e  say t h a t the p a r t i c l e  r > t  .  somewhere e l s e ) ,  'type-model' .  t  N  a t time  particles  s e t o f a n c e s t o r s , a t an e a r l i e r N  particles  a t the l a t e r  time.  t . at  any  time  76  C o n s i d e r any {a^}  and  ^A^}  finite be  time  strictly  interval  [ 0 , T]  .  Let  e > 0  , and l e t  d e c r e a s i n g s e q u e n c e s o f p o s i t i v e numbers  such  a/A 2  that  A  e  n  a /A n n We  -> oo  n  0  2+E  the  n  as  and  n -*  sets  {t  .  We  let  t JC  N n,k  N  particles  radius  a  n  the system  alive  N  a t time  t  particles  smaller balls of radius  , n,k  .  a t time  ——  Let  A  N  , n,k  t  when lies.  I will  necessary.) L e t k ( t ) = Ct/A n  A.  is a  the union o f b a l l s  ancestors a t time .  Let  A  be  n,k  t  ]  of the  of  , , n,k-l  the union  of of  (Technically  make t h e d i s t i n c t i o n b e t w e e n s t a n d a r d a n d  n  ,  ( i n t e r n a l ) o b j e c t s , but s i n c e they are a l l  identify  i n which  interval  t  =  u m=l  i  n  st(A  n=m  n,k  n  (t)  Ct  internal  only  ,, t , ,), t n,k n,k+l  )  (standard) B o r e l s e t f o r each standard  t u r n o u t t o be  our supporting s e t .  number o f a n c e s t o r s a t t i m e hence o f a l l the p a r t i c l e s Dawson a n d an  be  n  Then l e t  (6-3)  which  n,k  . n,k  , c e n t e r e d a t t h e same p o i n t s .  o f the above a r e n o n - s t a n d a r d  near-standard  = A ( 4 + e ) / (4+2e) ^ ^ n  n  = k A , 0 < k < T/A } form a p a r t i t i o n n / Jc n — — n be t h e number o f a n c e s t o r s a t t i m e t of n,k-l  a.  all  a  {T/A^} a r e a l l i n t e g e r s s o t h a t f o r e a c h  , centered a t each o f the of  c h o i c e w o u l d be  |t  n T]  (A c o n v e n i e n t  suppose w . l . o . g . t h a t  (finite)  CO,  .  00  The  dimension  system  in  Co,  of this  t  , , of the N particles n,k * a l i v e between times t , and n,k  Hochberg t r e a t e d t h i s problem  infinite particle  t  for fixed  T]  .  This  s e t depends on a t the times t  t  will the , n,k  , ) . n,k  t i m e s by c o n s t r u c t i n g  to d e s c r i b e the Fleming-Viot process  at that  (and  particular N the  time.  particles  U s i n g o u r h y p e r f i n i t e m o d e l , we c a n u s e t h e  a t a l l times.  distribution  o f the  number o f a n c e s t o r s o f  Dawson a n d H o c h b e r g  N  E  taken  particles  -ST  (6-4)  N T m  time  (1982;  ( i n r e v e r s e time)  to m  (e  = H? . (1 + -f-T,)  (6-5)  >_£.) A n  n,k  E(T ) = c/A / A  / a  k=  n  }  I  =J^-  yk(k+l) A  scheme.)  for integral  N  L  •  yk(k-l)  = P(°T > °A ) c/A n n  N  transform  ,  N  ( T h e i r a r g u m e n t a p p l i e s verbatim t o o u r h y p e r f i n i t e P (N  showed t h a t  t o reduce t h e  only, had Laplace  k=m+l  to estimate  (6.23))  same s y s t e m o f  + 1  A  n  We may u s e  -2- . A n  From  (6-4)  (6-3)  4~^,and  Y  Y  n  c  n  N  N  H  C /  A  _  n  ~  , k=  c  +  [ rY . k. ,(. k„ -. l ) ],  1  n  2  A A  n  V k-  Y  1 2 -£+ 1 A n  3  c  3 ~ )  l - i  3  3'  ( "D  Y  k  1  2 ,  1  r  N  3  J  ~  k  2 3 3y c oA  Now  by Chebychev's I n e q u a l i t y ,  P (T^^  > n  hence, tak i n g  (6-6)  + Y  °AA 3/2 h ——j—) f3yc  h = °/5c (yc-1)/SE~ , n P ( N , > T -) n,k A 5  n  <  A  n  (for -  3C(YC-1)  c > Y" ) . 1  < — h  ,  78  Let  6 > 0 .  Let  y (A)  denote t h e Hausdorff  £  x  2+  E  measure o f a s e t  Then  PC  max N , > 6/a k<T/A ' — n n  <  A  P(N > n,l  n  2 + £  ]  k  6/a ' n  2 + e  ) '  A < f-  2 _ _  n  n  a  (6-7)  ,  ~2+7  3  =  ~2+F " a  n  A 3  }  ( Y 6  n  (  F  R  O  M  (  6  _  6  )  )  . >  n  A  6  a  _  n  ( Y  2  ~  ~2+F a n  1 }  A 0  as  n ->  since  00  a 2+e Hence  P(  max a t£[0,T]  N  n  necessary,  n  , ,,> ' n k  (  t  n 2+e n > <  5)" "0» >  a n c  j by t a k i n g a subsequence, i f  )  we c a n e n s u r e  0+c  (6-8)  max a t [0,T]  N  n  n  £  , _, -> 0 ' n #J  k  (  t  a.s.  )  oo  Now f o r e a c h balls of radius  a  y  and  hence  y  ( e  m  z  00  u m=l  , A .  m  , m,k  m  Hence  ... (t)  i s a covering of  (6-8) e n s u r e s  ( n A , ,,,) = 0 n,k ( t )  n A , . by n,k ( t ) n=m n 1  N , _. m,k ( t ) m fJ  that  uniformly  in  t , a.s.  m  00  n A , ,,..)= 0, f o r a l l m n-k (t) n  t e [ 0 , T]  a.s.  This  i s true  79  f o r any  E >  Lemma 6 . 2 . 1 : Hausdorff  0 .  The r a n d o m s e t - v a l u e d  frunction  A  d i m e n s i o n a t most 2 f o r a l l time, a.s.  d e f i n e d by  (6-3) h a s  6.3  A Useful Stochastic Differential  We A t any  Equation  s t u d y t h e numbers o f d e s c e n d a n t s o f a s p e c i f i e d t i m e , we  of the  N  , and  stop the F l e m i n g - V i o t p r o c e s s , designate restart  suppose the process associated with  this  (Markov) p r o c e s s .  i s r e - s t a r t e d a t time  the  designated  descendants a t times  t > 0  .  Y  n  t  this  i n the o r i g i n a l  i)  = — N  ii)  c h a n g e s o n l y when  one  particle  Y  n)  .  out  s e c t i o n w.l.o.g. d e n o t e t h e mass  a t time  0  and  with  a l i v e a t time  R e c a l l t h a t we  NY^  make no Now  out of the  NY  out o f the  d i s a p p e a r s , and  fc  i n c l u d e d i n the  one  particle  and  i s r e p l a c e d a c c o r d i n g t o t h e t y p e o f one  Disappearances  are  their t  including time  t , in  change t o  N-NY^_  i s r e p l a c e d by NY^_  a  , or  excluded p a r t i c l e s  disappears  of the  NY^_  particles. and  replacements  designated p a r t i c l e s Y^_  happening e n t i r e l y  o r amongst t h e  N-NY  w i t h i n the context  excluded  of  particles,  .  the replacement  other given p a r t i c l e  o f any  given p a r t i c l e  according to the  happens a c c o r d i n g t o a P o i s s o n p r o c e s s  type model  of  with rate  y  (6-1)). There are  NY  • (N-NY^_)  p o s s i b l e w a y s f o r an e v e n t  each happening a c c o r d i n g to a P o i s s o n process a decrease an  Let  (# o f p a r t i c l e s  of a type-model not  descendant  (see  this  particles  t h a t have remained unchanged u n t i l  Y^_  particle  any  .  For  n  particles.  count. Now  the  n  0  particles  which are descended from the o r i g i n a l particles  group of  event  of size  of type  with rate  of type y  , and  i ) to each  occur  causing  — i n Y. . T h e r e a r e (N-NY ) • NY p o s s i b l e ways f o r N t t t i i ) to occur, each w i t h the e f f e c t o f i n c r e a s i n g Y by  y  a n d o c c u r r i n g a t t h e same r a t e an  (internal) martingale.  as an event o f type i ) .  The a s s o c i a t e d p r e d i c t a b l e  i s easy t o compute, s i n c e t h e change t o 2NY (N-NY^)  Y  p r e d i c t a b l e q u a d r a t i c v a r i a t i o n s a r e each  <Y>  =  t  Y  i s  increasing process  a t a n y t i m e i s t h e sum o f  independent P o i s s o n p r o c e s s e s , each o f r a t e  t  Thus  — j . N  y  , whose  Thus  2y Y ( 1 - Y ) d t  0  *  (6-9) d<Y>  t  = 2y Y ( l - Y t  Hence b y H o o v e r a n d P e r k i n s Y  (6-9)  d  <  Y  >  t  =  2  Y Y (l-y t  )dt  (1983)"'" we may c o n c l u d e t h a t t h e p a t h s o f *  a r e a.s. n e a r s t a n d a r d i n t h e space  from  f c  C((0,°°); R) . ,t  ) d t and hence  enlarging our probability  (6-10)  dy  (up t i l l  t  F u r t h e r m o r e we h a v e t h e f o l l o w i n g  Lemma 6 . 3 . 1 :  Suppose  y  /2 y d-y ) Y  t  the time o f e x t i n c t i o n o f  = /2y y ( l - y > d b  0  = e  where  1  t  Y .  Then  dy  s p a c e we may f i n d a B r o w n i a n m o t i o n  t  o  b^_ =  o a standard Brownian motion  Let y =  i s t  t  y) . b  Thus b y  such  that  t  lemma.  0 < e < — . 2  Then t h e r e a r e f i n i t e  * Note. H o o v e r a n d P e r k i n s (1983) T h e o r e m 8.5 r e f e r s t o - d i s c r e t e t i m e processes X , s u c h t h a t i f Ax i s t h e c h a n g e i n X o v e r a n i n f i n i t e s i m a l time step A t , then s u p °|Axj = 0 a . s . By l o o k i n g a t o u r p r o c e s s Y o n l y a t i n t e r v a l s o f A t = 1 / N , we may e n s u r e s u p °~|AY| = 0 a . s . a n d bring Y i n t o t h e framework o f t h e theorem q u o t e d . 2  82  K^,  constants  P ( 3s  K  2  independent o f  e [0, t ]  e  such t h a t  , such  y  < ^ e) < K. s  Proof: Y  fc  Write  stays  y  fc  ^—^—  —  4  —  exp  C- r ^ - ] . K^t  1  as t h e time change o f a Brownian m o t i o n  i n the range  at least —  that  •  3  5  [- e, - E ] Now  use the  .  As l o n g  as  t h e d e r i v a t i v e o f t h e t i m e c h a n g e m u s t be estimate  - yeU- e) 2  P(  4  3 s e [0, t ]  such t h a t  l b I > c ) < 2 P ( | b I > c) s — ' t 1  —  c = 4 P(b  >_ c) < K  e  2 t  2 •  83  6.4  Verification  of  We  t h a t t h e random s e t  now  check x  the measure  o f T h e o r e m 6.1.1  fc  Lemma 6 . 4 . 1 :  Consider a t time  the path of the Then t h i s p a t h  (unique)  Brownian  t  motion  o f s e c t i o n 6.2  fc  any p a r t i c l e  ancestor of  p  does i n d e e d  support  i t takes a step of size  p  a t each  .  Let  time  r  s < t , for  , and  trace  s <_ r < t .  random w a l k whose s t a n d a r d p a r t i s a  of rate  Between the appearance and  particle,  A  (the F l e m i n g - V i o t P r o c e s s ) .  i s an i n f i n i t e s i m a l  d-dimensional  Proof:  Support  2D  .  ultimate disappearance —  o f any  given  from i t s c u r r e n t p o s i t i o n  t o any  one  /N  of  the  DN  .  2d  neighbouring positions according to a Poisson process with  Replacement  (which i s b i f u r c a t i o n  according to a Poisson process which particle. process  I f we B^_  imagine  , and  i s independent  such a motion  i t s coordinates  B^  martingale, since steps to the r i g h t 1 left.  Now  <B >  = 2DNt( —  1  o f an a n c e s t o r p a r t i c l e )  occurs  o f the motion  of  c o n t i n u i n g i n d e f i n i t e l y and , then c l e a r l y  each  B^  rate  the  call  i s an  this  internal  o c c u r a t t h e same r a t e a s s t e p s t o  the  2  )  = 2Dt  , since  B  1  i s t h e sum  of 2  independent  /N  1  Poisson processes, of rate  2DN  , and o f a m p l i t u d e  ——  .  The  motions  B  and  1  /N  ~B?  happen a c c o r d i n g t o independent  if  i ^ j .  standard part  Thus by H o o v e r and P e r k i n s b^_  a.s.,  d-dimensional Brownian  2  A g a i n we  P o i s s o n p r o c e s s e s , hence  and  this  motion:  b^  (1983)  (once a g a i n l )  <B , 1  B^  B  "  5 >  has  s a t i s f i e s the c h a r a c t e r i z a t i o n  t  =  0  a of  E ( b b 0 < r < s ) = b ; <b> = 2DIt. t' r — — s t  must l o o k a t the p r o c e s s B  t  at discrete  t o p u t i t i n t h e f r a m e w o r k o f T h e o r e m 8.5  of this  intervals paper.  (of s i z e  •  At =  1/N)  R e f e r r i n g t o 6.2 f o r t h e d e f i n i t i o n s o f t h e s e t s c l ~ n ((°A , ) ) > £ ) < — °e n,k — e K  Lemma 6.4.2:  P(x °  a  / K  2  A  , n,k  and  A , . n,k  A  for finite  n  constants  n,k  V  K  2 •  Proof: at  time  Consider the f a m i l y o f descendants t . . n,k-l  used  i ndefining  A n,k  3  o f any one o f t h e N  particles n / JC back from time t , n,k  I f we t r a c e  t h e movements o f a n y o n e o f t h e p a r t i c l e s  i nthis  f a m i l y , and i t s ' p r o g e n i t o r s ,  u n t i l we come t o t h e p o s i t i o n o f t h e o n e a n c e s t r a l p a r t i c l e we f i n d a m o t i o n a Brownian  Motion.  p  t , , , n,k-l  o f t h e k i n d d e s c r i b e d i n Lemma 6 . 4 . 1 , w h o s e s t a n d a r d p a r t i s Hence t h e d i s p l a c e m e n t o f any g i v e n p a r t i c l e  from i t s ' a n c e s t o r a t time particle  a t time  a t time  t ' , n,k-l  i sdistributed  a t time  N ( 0 , 2DA i ) . n  t n,k F o r anv J  t , , n,k  a„ E(I  (P)) £ P ( B  A C  A  . n,k  -a /K A 2  > - )  D A  < K  e  n  Hence N -a /K A I EI ( p . ) < °K e . - ;o I — 1 1=1 A , n,k 2  E(x  °, t  U°A . n,k  n,k  f)\<  ECX^ C A ) ) = t . n,k n,k C  —  \ N  n  and  PCx  (C°A ) n,k V  o  c C  -, „ - a / K A - i °e . e K  2  n  ) > e) <  n  2  n  n,k  fc  c K ( ( A , ) ) > £ ) < — e n,k e o  Lemma 6.4.3:  P(x  o  -VVn  n,k+l Proof: N(0,  As above, w i t h l a r g e r  4D A I ) . n  K  2  , since displacements are d i s t r i b u t e d  2  n  E > 0  Let  .  F r o m Lemmas 6.4.2  {x  P ( max  T  (°A  n,k  fc  '  6.4.3  ,) v  C  n  and  we  (°A°  Si.k+l  k  n  '  may  deduce  )} > k  2  n K T —  <  -  Cas  EA  -a /K  A  2  e  n  z  - v O  n  as  n  <*> .  n  per u s u a l , our t r u s t y s e r v a n t s , the constants  and  K  2  are  changing  v a l u e s when n e c e s s a r y ) . Hence  a  fortiori  PC max  (x  (°A  ) v  C  x+  .  (°A  ) } > f ) -> 0 .  C  n Now  c o n s i d e r the p o s s i b i l i t y ,  t h e mass o f t h e p r o c e s s E t  o f mass l i e s , , n,k+l  Let  s  , n,k  be  a)  Cs  t  n,k  the f i r s t  x^ t  configuration at  interval  a n d we s  , n,k  . ,.) n,k+l  a t l e a s t one  t h e mass  e  Cthat i s t h e time  .  may One  O  A  t  , n,k  , n,k  A  , n,k  t  , , , b u t t h a t more n,k+l  and  contain  1 - — 2  a t some t i m e i n b e t w e e n  such time,  s  n,k  t  of  , n,k  and  i s a s t o p p i n g time f o r  c o n s i d e r i t r e s t a r t e d a t time  s  , from n,k o f the f o l l o w i n g must o c c u r d u r i n g the  f o u r t h o f t h e mass a —  2  that l i e s outside  Cto r e - e n t e r  that lies outside e — N  e  A  , n,k  A  A  , n ,k  initially  ,) n,k  d e c r e a s e s by  p a r t i c l e s have a t most  3  e N  a t l e a s t one descendants  t n,k+l)  Consider case a ) . For each p a r t i c l e  than  •  travels a distance b)  at times  t  o u t s i d e the sets  the Markov p r o c e s s its  x  t h a t the sets  p  a t time  t  , , , the d i s p l a c e m e n t from i t s n,k+l  at  fourth  ancestor a t time  s  ,  i sdistributed  N(0, D(t  n,k  , , - s  n,k+l  )I) .  Recall  n,k  f r o m s e c t i o n 6.3 t h a t t h e number o f d e s c e n d a n t s a t a f u t u r e t i m e o f a n y g i v e n s u b s e t o f p a r t i c l e s , f o r m s a m a r t i n g a l e . Hence t h e e x p e c t e d mass a t time t • , o f descendants o f t h e e N p a r t i c l e s a t time s , , which a r e n,k+l n n,k  3.  outside b a l l s o f radius initial  e  mass  —  c e n t e r e d on t h e i r a n c e s t o r s , i s e q u a l t o t h e  , times the p r o b a b i l i t y  t h a t any one p a r t i c l e -a /K A  i soutside  such  2  2  a b a l l , w h i c h p r o d u c t i s bounded by Hence u s i n g C h e b y c h e v ' s  e K  e  ( a s i n Lemma  and  k, c a s e a) o c c u r s )  n  6.4.2).  Inequality,  " n a  P  Hence  P ( f o r some  (for a fixed  n  T k < -— , s , < t , A n,k n,k+l n  w h i c h g o e s t o z e r o a s n -*• . Now by Lemma 6.3.1 T P ( f o r some k < — , s , < t , A n,k n,k+l n  < K^e  K  a n d c a s e a) o c c u r s ) <  / K  < — — A  P( n  3  s < t , - n,k+l  (where T - A~ 1 n K  y  - s ,k n  <A n  s.t. s y  T  a  :  occurs)  y =  i s t h e process mentioned  4  -  1  E Y  0  I v  b  '  = ) e  in.6.3.1)  -e/K.A 2 n 6  w h i c h a l s o goes t o z e r o .  Hence  P ( 3k  < T^- such t h a t A n  and  x^ (°A °, ) < ^- a n d t n,k 2 n,k  3 s e (t ., t ) n,k n,k+l  such t h a t  (A ,) < n,k C  t  n,k+l  2  x (°A , ) > e) ->- 0 . s n,k C  A  1 " n — — e A n  00  a n d c a s e b)  2 n  / K  2 n A  87  »  Thus  C  P  sup x (A , te[0,T] ' t  n  k  ) —> (  t  0  as  n +  )  n  By  t a k i n g a s u b s e q u e n c e , i f n e c e s s a r y , we may e n s u r e ° c a •s x (A . .) —'—$ n,K m  sup te[0,T]  A g a i n b y t a k i n g a s u b s e q u e n c e we may a s s e r t  n  V nil  Then  0 .  r  sup  x (°A tcCO.T] ' n  sup x ( u te[0,T] n=m Z  sup te[0,T]  <  y  I n=m  sup  ~TL  -> 0  k  (  t  a.s.  00  )  t  o  <_  n  x (A^)  sup te[0,T]  <_  ) <  C  t  n  n  x (°A t  £  n  *  f o r any m e N ,  K  )  C  ' n k  x (°A  t [0,T]  as  c  A . , .) ' ^>  n  (  t  )  C  ' n k  (  t  )  m ->  00  T h u s we h a v e  Theorem 6 . 4 . 4 :  The r a n d o m m e a s u r e s  Fleming-Viot process, a r esupported dimension  a t most  2 , a.s.  x  f c  , which  are t h er e a l i z a t i o n s  f o r a l l times  t  on a s e t  A  of the of  APPENDIX A  Some I n e q u a l i t i e s  A.l  Used i n C h a p t e r 3  Purpose  I n A p p e n d i x A . 3 , we d e r i v e some i n e q u a l i t i e s  involving  the coefficients  g Q-  w h i c h w e r e p r e s e n t e d i n C h a p t e r 3.4.  I n s e c t i o n A.2 we d e r i v e  i d e n t i t i e s which provide a neat route t o the i n e q u a l i t i e s A.2  i n A.3.  Some I d e n t i t i e s s y A s we saw i n Lemma 3 . 3 . 1 , Q= p(B- , = x x-y - s-At  s i m a l random w a l k s t a r t i n g a t  y  size  with probability  At  some  Ax  t o the right or l e f t  , where  2  a = At/Ax  w i l l become c l e a r  1 <_ — .  a t time  where  y B-  i s an i n f i n i t e -  0 , and t h e r e a f t e r t a k i n g steps o f a , a t each time  The r e a s o n f o r n o t c o n s i d e r i n g  interval  1 1 — < a <_ —  later.  As we saw i n Lemma 4.3.1 t h e a b o v e r e m a r k s a r e t r u e a l s o i n where t h e random w a l k t a k e s s t e p s i n any o f  2d  directions,  each  d-dimensions, with  2 probability  a = At/A^ .  d-dimensions The  Lemma  first  although the  d-dimensional versions w i l l  identity i s t r i v i a l .  7  A.2.1.  Q- = l , i f x  L  x Ax  Lemma  T h e i d e n t i t i e s i n A.2 a r e t r u e i n  2  (Q|) = Q^-~ *  x  Ax  N  e *Z  £  A.2.2.  s —- £ At  £  ~ Z  At  > i f s/At £  *N  n o t be used  89  Proof:  I x  (Q|) -  ( I Q|) x  =  2  I x  -  2  x  I Q 9? y^x x  -  y^x  (by A . l )  =  - ^  1  -  1  (  B  x  s - A t  I ^ _  A  ^  =  ' ^ s - A t ^  = x)  t  - ^ B ^ ^ - x )  X  (by s y m m e t r y ; t h i s  step  fails  i f we c o n s i d e r a  reflecting  random w a l k . )  -  1  I £<B°_ x  = x) P ( B ° .  A t  s  2 A t  ^0|B°_  A t  =x)  1 - P(B° . f< 0) 2s-2At 28-At  0  y  (by d e f i n i t i o n ) .  •  * Lemma A . 2 . 3 .  I f  x/Axe  7  Proof: , .  Z  x/Axe  2  = ^I (Q-) x x  -  X  Z  * Z  -  (Q*  L  z/Ax e  2  , and  s/At e  - -  Q) x+z - -  y^ Q-x Q-x+z + I (Q ) x+z §  L  x  -  - -  x  - -  2  N  2  -  2 l  P(B°.  - ,,P(B°.  4 T  T T  . ;  •  (by Lemma A.2.2)  X  (by  =  symmetry)  Q S~ *0 2  2  =  2  - 2 P(B° = z) 2s-2At  A t  O A  (  2  ^  - 2 z  u  s , —  r  Lemma A. 2 . 4 .  J  I f —  ( £ S-  f )  +  Q  x/Axe*Z  Proof:  x  Y  Q  e  =  2  +  —  2  £  + 2  r t  2s-At _  A  +  0  Q-)  x  N ,  Q  x  (Q- -  v  ^  2s-At  f +  0  0  2  x  Ax  x  -  x  2r 2,-At _ 0 +  2  l  p  -  (  B  0  -  x  -  -  0  =  r+s-At  - -  =  r+2s-2At  (by s y m m e t r y a n d A.2.2) 2r+2s-At 2  "  "  -  ,„0  2  ^ r 2s-2At = C B  +  „. 0 )  „2s-At +  x  V  0 | B  -  0 £ 2 ^ +  -  A.3  Some  Inequalities  In t h i s s e c t i o n I prove  the four i n e q u a l i t i e s  3.4.1 t h r o u g h 3.4.4 o f  C h a p t e r 3.  Lemma A . 3 . 1 .  (3.4.1).  (Q) < K A t / t  I  K  There i s a c o n s t a n t , i f  X  t ^  xeX Proof:  such  that  * e  N  Clearly  xeX  x/Axe Z  x  By d e f i n i t i o n , o f 0 t h e sum o f  k  probabilities Since  k  +  1  *  x  A  t  0  = P (IS, I < zr) k — 2  ,  where  S  k  i s  I.I.D. random v a r i a b l e s t a k i n g t h e v a l u e s a , l - 2 a, a  respectively.  °a > 0 , t h e n b y C o r o l l a r y  - 1 , 0, +1  ) = 2ka . k 2.2.3 o f B h a t t a c h a r y a  with  Var(S  a n d Rao ( 1 9 7 6 ) ,  (2 ' /2kct  (k+l)At -0  e 1  dz  <  —  as  k ->  /2TT  "2 / 2 k a  2 '  /2ka  Now  e  1  -z  /2, dz  i s asymptotically  •/2ka  Thus t h e r e i s a f i n i t e  K  (k l)At +  0  such <  JC  " A  that  1  , 1, + o(—) 2/2irka /k x  as  k •>  00  By  the transfer principle  t = At  t h i s must h o l d  ft  for a l l of  N .  Thus  (checking  separately) _ J ± _ 2t-2At -  2t-At 0 <  y  At  <  t  * Lemma A.3.2 ( 3 . 4 . 2 ) .  I 0<s<t  There i s a constant  I xeX  K  such t h a t f o r  t/At e  N  (Q-) 1 K/t/At . 2  1 £ — f _ C Sn k = l /k n  Proof:  Follows  f r o m A.3.1 a n d t h e f a c t t h a t  n e  hence f o r a l l  * N  n e N ,  fora l l  .  •  5 Lemma A . 3 . 3 ( 3 . 4 . 3 ) . t  and  - e At  Proof:  There i s a f i n i t e  *  r  N , then  Suppose  ) 0<s<t  t s s ^ ) (Q- - Q) < K. X £ A - -  w.l.o.g. that  w a l k whose d e n s i t y  i s Q- ^ x +A  constant  z ^_ 0 .  Let B  g  J(t,x) =  I I(B 0<s<t  Let  L(t,x) =  I I (B = x ) A x . 0<s<t -  Let  L(t,x) - -  I KB = x ; B = x + Ax)Ax „ . S S+At 0<s<t  Then with  L  =  Now  > x) ( B  i s the true occupation  'pauses', w h i l e E(L(t,x)) = - - -  L  *  such t h a t i f - — e Ax  Z ,  I I?I I  be t h e i n f i n i t e s i m a l  random  •  Let  s  K  s + A  " B ) . -  density  ("local time"),  f o r o u r random w a l k  i s t h e d i s c r e t e analogue o f Brownian L o c a l Time. 7 P(B = x ; B . - B = Ax)Ax ^ " - s s+At s 0<s<t -  93  Y P(B = ) a s 0<s<t  = a E(L(t,x))  In Perkins induction  on  • Ax  x  u  .  (1982) Lemma 3 . 1 , T a n a k a ' s f o r m u l a i s e s t a b l i s h e d b y i n t e r n a l  t :  (B  fc  - x )  +  = J(t,x) + L(t,x)  Thus  L(t,0)  Now  J(t,0)  - L(t,x)  and  J(t,x)  E(L(t,0)  =  - (B  - x )  +  - J(t,0)  + J(t,x)  areboth i n t e r n a l martingales,  - L(t,x)) = E [ B p - E [ ( B - x )  +  t  hence  ]  (A.3.1)  Now s i n c e i n Lemma 4.3.2.  I  _ . 0<s<t  0 < °a < | , Q§ = V ( B _ s  = 0) >_ P (  A t  B s  _  A t  = x ) = Q| , a s shown  Now b y A.2.3  1  v xeX  (Q- - Q~ x  -  x+z - -  <* = 2  )  2  < 2  —  I  Q 2" 2  „ . 0 0<s<t u  <er  4 t  i  At<s<2t  U  I 0<s<2t-2At  1 2 E(L(2t-At,  At  n  - Q  2 s  z  "  A t  -c » 4t  -  P ( B = 0) - P ( B = z ) -  0) - L ( 2 t - A t , z ) )  |- E ( L ( 2 t - A t , 0 )  - L(2t-At,z)  , b y (A.3.1)  < ~~ a A x  Lemma A.3.4 ( 3 . 4 . 4 ) . r/At < t/At  There i s a f i n i t e  are i n  constant  K  such t h a t i f  *N , 2  I 0<s<r Proof:  I xeX  (Q!~- - Qr~~>  The l . h . s . above i s e q u a l t o  V L 0<s<r  I  (A-2)  r . s+(t-r) s. L (Q~ ~ ~ ~ Q~' xeX -  First,  ( Q  2s+2(t-r)-At  suppose  +  Q  2s-At _  2  Q  (t-r) 2s-A +  t-r< r  t )  0  and (t-r)/At  i s even.  T h e n many o f t h e t e r m s  (A-2) c a n c e l l e a v i n g  I  Since  °  t - r 2  Q -" 2  t-r r<s<r+—— - - - 2  AT  °  increases.  (t-r)/At  +  <s<t-r -  I t - r r + — <s<t - 2  < 1 a — - , the c o e f f i c i e n t s 3 Hence  Q  2 S _ A T  0  s Q0  a r e monotone  t h e sum o f t h e s e c o n d , t h i r d ,  above i s bounded above b y  KV  i w h i c h b y A.2.4 i s b o u n d e d b y  0  t-r 0<s<—— --2  s  2  x  0<s<r  in  <K/(t-r)/At  0 .  The f i r s t  b y A.3.2, a s r e q u i r e d .  decreasing as  a n d f o u r t h t e r m s i n (A-3)  t e r m i n (A-3) i s b o u n d e d b y  t-r Now i f t - r < r Y  r n  and  2s+2(t-r)-At  n  I LQ 0<s<r  ~~7~7~  2s-At -  + Q -  - 2 Q  I x  y ~ 0<s<r  — < -  I  2  L  0<s<r  AT  K  K  -  < < ^ >  eX  Q|)  x  v 2s~At \ Q < K t-r 0 < s < - — +At — 2  i /t-r/it  2  x  2s+2(t-r)-At 0  (by  <  A p p l y i n g t h e same  xeX  0<s<r  —  . .  ( i . e . r < t/2) , then  I  0<s<r  ,  1  - J ..  c a n c e l l a t i o n argument, t h i s i s bounded by  Now i f t - r > r  i s o d d t h e sum i n (A-2) i s b o u n d e d b y  (t-r)+2s  +  Q  2s-At 0~  A.2.2)  2s-At Qo  ( B Y  A,3  -  2)  t-r — — . At 1  Remark:  If  a = —  t h e n Lemmas A.3.3 a n d A.3.4 a r e f a l s e .  non-zero c o e f f i c i e n t s checkerboard pattern. for which  t-s x-y —-— + ~ At Ax  i n the array Thus  U  {Q-}  , s / A t e *N  i s independent o f £  x  i s an odd h y p e r - i n t e g e r .  The p a t t e r n o f  , x / A x e *Z , i s a U, , t-At,x  and o f any  T h u s t h e moment  J  U s;  inequalities  96  on s p a t i a l a n d using  temporal d i f f e r e n c e s w i l l  the s i m p l e s t  finite  difference  fail.  This  i s the reason f o r not  scheme i n C h a p t e r  3.  APPENDIX B I n t e r n a l S o l u t i o n s t o SPDEs i n H i g h e r  The s u c c e s s  Dimensions  o f the hyperfinite difference equation  approach i n Chapters  3 a n d 4 t o t h e e x i s t e n c e o f s o l u t i o n s t o SPDEs i n o n e d i m e n s i o n , Dawson m e a s u r e d i f f u s i o n use  i n higher dimensions,  o f h y p e r f i n i t e d i f f e r e n c e equations  SPDEs i n h i g h e r  the analogy  The k i n d o f  o f those  equations  i n Chapters  3  4, w o u l d b e o f t h e f o r m  (B-l)  ! ^ = Au + f ( u ) d W  the corresponding  t e T  internal  (6 U ) = t -x t  t'B-2)  and Now  x e X  fcx  , t e R  +  , x e  d  (A U. ) + F ( U ) 5 / t- x tx tx  , hyperfinite grids representing  i t i s easy t o s e e t h a t an i n t e r n a l  internal solution  R  equation  usual inductive construction. U  AtAx  R  d  , for  and  R  respectively.  s o l u t i o n t o (B-2) e x i s t s , b y t h e  What i s n o t s o c l e a r i s w h e t h e r o r n o t t h i s  has a n o n - t r i v i a l  some s p a c e o f d i s t r i b u t i o n s . such  might l e a d t o a general theory o f  t h i s hope has n o t borne f r u i t .  t h a t we w o u l d b e l e d t o c o n s i d e r a f t e r  and  l e a d s one t o wonder i f t h e  dimensions.  So f a r a t l e a s t ,  and  and t o the  standard part  u , presumeably i n  I t i s a l s o n o t c l e a r w h a t i t w o u l d mean f o r  a d i s t r i b u t i o n - v a l u e d process  u  t o be a s o l u t i o n o f ( B - l ) , i n  g e n e r a l , s i n c e n o n - l i n e a r o p e r a t i o n s on p o i n t v a l u e s o f d i s t r i b u t i o n s a r e undefined  or discontinuous a t best.  if  i s an o p e r a t o r v a l u e d  f(u)  I t seems p o s s i b l e t o make s e n s e o f ( B - l )  function of  u  with values  i n a class of  o p e r a t o r s on a space o f d i s t r i b u t i o n s , b u t t o pursue t h i s p o s s i b i l i t y  would  9 8  t a k e us  too f a r a f i e l d  I f we  from the ideas of t h i s  r e s t r i c t o u r s e l v e s t o the case where  o f a r e a l v a r i a b l e , and t h e n we  may  thesis.  still  that  F  f  i s a real valued function  i s some n a t u r a l l i f t i n g  ask whether o r not the i n t e r n a l  of  f , such  solution to  (B-2)  as  has  f , an  i n t e r e s t i n g standard p a r t . Sadly, The  i n s e v e r a l cases  first  c a s e we  compact s u p p o r t , Then s i n c e  (and  A t <<  infinitesimal  Ax  F  i s an  i s when  other set)  set  up  f  lifting  , changes t o the i n t e r n a l  U  U  to  (B-2)  a n e a s y i n d u c t i o n a r g u m e n t shows t h a t  i s bounded by  a finite  constant.  I f we  seek  are  the to  £ U Ax* f o r some f i n i t e i n t e r n a l r e c t a n g l e xeA — , t h e n we a r e n a t u r a l l y l e d t o e x a m i n e E ( F (U ) ) . We sy  A  3  a difference equation  f o r t h i s q u a n t i t y , and  may  show t h a t i t i s e v e r y w h e r e  infinitesimal.  infinitesimal. e i t h e r end  Thus t h o u g h t h e v a l u e s  o f the  (connected)  p r e c i s e l y o n e a c h monad. value  The  The  0 < p < i , o r to f i n d  E(M - M ) t u  2 q  n  a finite  examination  ^  t a k e n by F  <  P ^ _ 1 -  the t o t a l mass. is finite,  U  l i e almost  balance  point i s infinitesimally  to the heat  m a  Y  U  , is  always  , these values balance  at  very  close to  equation w i t h the given  the initial  t o t h e monad i n q u e s t i o n .  e x a m i n e d w e r e when W e  region of  of  standard point corresponding  o t h e r c a s e s w h i c h we  finite  support  o f the d e t e r m i n i s t i c s o l u t i o n  c o n d i t i o n , a t the  on  function of  o f compact s u p p o r t ) .  solution  T h u s t h e v a r i a n c e o f t h e i n t e g r a l o v e r any  4.4  'no'.  i s a continuous  S-continuous  tx the v a r i a n c e o f  (or  a.s.  considered  a t e a c h s t e p and  internal solution estimate  t h e a n s w e r seems t o be  F(U)  =  , when  either  t r y to f o l l o w the development o f s e c t i o n  I n the case  0 < p < - , we f i n d t h a t 2 a t l e a s t i n the case of r e f l e c t i n g boundary c o n d i t i o n s  r e c t a n g l e , s i m p l y by b o u n d i n g of the d i f f e r e n c e equations  U  by  (B-2), w i l l  1 + U  .  However, a  show t h a t , .  closer  the  values  U  are almost always i n f i n i t e o r  infinitesimal.  Thus t h e q u a n t i t y  £  U  2  i s actually  p  infinitesimally  __  smaller than  where E|M  U  , s i n c e a l m o s t a l l t h e mass comes f r o m p o i n t s  i s infinite.  - M |  t  £  xeX  << E (  2 q  0  T h u s we e n d u p w i t h , i n f a c t  1 M^) , a n d t h u s t h e t o t a l mass i s i n f i n i t e s i m a l l y 0<s<t -  c l o s e t o MQ a . s . I f we e x a m i n e t h e p r e d i c t a b l e associated with  it  t obe  to  0 .  £ (U  - U  "cx  X  ux  )$(x) -  2  p  £ x . s 0<s<t =  —  J U $ (x)Ax xeX - -  £ 0<s<t  x  2  d  A $  ^  , which  increasing process  A t , a s i n s e c t i o n 4.5 we f i n d  i s likewise infinitesimally  (B-2) f o r F ( U ) = iP  Thus t h e p r o c e s s we g e t f r o m s o l v i n g  close  , 0 < p <  t u r n s o u t t o have a d e t e r m i n i s t i c s t a n d a r d p a r t . If  F(U) = U  infinite  , i < p <_ 1 , t h e n , a g a i n  P  or infinitesimal.  t o t a l mass  M  Whenever  , we f i n d  U  U  must a l m o s t always be  I f we o b t a i n a n e s t i m a t e o f t h e v a r i a n c e o f t h e E(M - M )  i s infinite  =  2  U << U  2  p  Y 0<s<t  Y U AtAx xeX — 2  , and hence  p  Y  L  v  xeA  The moment b o u n d s o n  for  U  ,* - X* X* t 0  i s a.s. i n f i n i t e ,  y A £ X$ ^ s 0<s<t -S X  a)  M  At  t o be n e a r s t a n d a r d a s a measure-valued  F  i s essentially  (1986), o r b)  Ax  d  >>  Y U Ax sx xeA —  d  L  v  a s i n s e c t i o n 4 . 5 , shows  like  F ( x ) = c/x  ( B - l ) appears  F(x) = C , i n which  essentially,  i s simply too larg  stochastic process.  which  t oy i e l d anything are  c a s e we r e c o v e r t h e  t h e o r y o f SPDEs i n h i g h e r d i m e n s i o n s w h i c h was o r i g i n a l l y Walsh  p  —  andthus the quadratic v a r i a t i o n  The o n l y c a s e s w h e r e t h e f o r m u l a t i o n if  2  sx  .  cannot be o b t a i n e d . An e x a m i n a t i o n o f t h e p r e d i c t a b l e  increasing process f o r that this  U  d  linear  developed by  i s treated  i n C h a p t e r 4.  100  REFERENCES  R. M. A n d e r s o n , "A N o n - S t a n d a r d R e p r e s e n t a t i o n f o r B r o w n i a n I n t e g r a t i o n " , I s r a e l J . Math., 2 5 , 15-46 (1976).  Motion  and I t o  K. B. A t h r e y a a n d P. E . N e y , B r a n c h i n g P r o c e s s e s , S p r i n g e r V e r l a g , H e i d e l b e r g , 1972. R. N. B h a t t a c h a r y a a n d R. R. R a o , N o r m a l A p p r o x i m a t i o n s Expansions. J o h n W i l e y & S o n s , New Y o r k , 1 9 7 6 . D. L . B u r k h o l d e r , " D i s t r i b u t i o n F u n c t i o n I n e q u a l i t i e s A n n . P r o b . , 1, 19-42 ( 1 9 7 3 ) .  and A s y m p t o t i c  for Martingales",  C. 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