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Hyper-finite methods for multi-dimensional stochastic processes 1986

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HYPER-FINITE METHODS FOR MULTI-DIMENSIONAL STOCHASTIC PROCESSES By MARK ALLAN REIMERS B.Sc, The University of Toronto, 1978 •Sc., The University of 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 i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics The University of B r i t i s h Columbia We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1986 © M a r k A l l a n Reimers, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying 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 thesis we introduce Non-Standard Methods, i n p a r t i c u l a r the use of h y p e r f i n i t e d i f f e r e n c e equations, to the study of space-time random processes. We obtain a new existence theorem i n the s p i r i t of K e i s l e r (1984) f o r the one dimensional heat equation forced non-linearly by white noise. We obtain several new r e s u l t s on the sample path properties of the C r i t i c a l Branching Measure D i f f u s i o n , and show that i n one dimension i t has a density which s a t i s f i e s a non-linearly forced heat equation. We also obtain r e s u l t s on the dimension of the support of the Fleming-Viot Process. Edwin Perkins ACKNOWLEDGEMENT I w o u l d l i k e t o thank 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 and 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 c o m p l a i n i n g . i v TABLE OF CONTENTS page ABSTRACT i i ACKNOWLEDGEMENT i i i TABLE OF CONTENTS i v TABLE 1 1 CHAPTER ONE - Introduction 3 1.1 Why t h i s Thesis 3 1.2 SPDEs 4 1.3 The Dawson C r i t i c a l Measure Valued D i f f u s i o n 7 1.4 The Fleming-Viot Process 9 CHAPTER TWO - Non-Standard Analysis and P r o b a b i l i t y 10 2.1 Some D e f i n i t i o n s and Notations from Non-Standard Analysis 10 2.2 Non Standard White Noise 14 2.3 Adapted Stochastic Integrals 16 CHAPTER THREE - The Heat Equation with Non-Linear Stochastic Forcing 20 3.1 Scope 20 3.2 White Noise on the space U 23 3.3 Hyper-Finite Difference Equations 24 3.4 Some Useful I n e q u a l i t i e s 26 3.5 Bounds on Moments of U 2 7 tx 3.6 Bounds on Moments of- S p a t i a l Differences 31 3.7 Bounds on Moments of Temporal Differences 33 3.8 S-Continuity and the Standard Part 35 3.9 Solution of the SPDE 38 CHAPTER FOUR - The Dawson C r i t i c a l Branching D i f f u s i o n 41 4.1 Introduction 41 V 4.2 A Hyperfinite Difference Equation 43 4.3 The C o e f f i c i e n t s Q 45 4.4 The Total Mass Process M 52 4.5 S-Continuity of the Process 55 4.6 Characterization by a Martingale Problem 62 4.7 New Results on the Dawson Measure Valued D i f f u s i o n 66 CHAPTER FIVE - The C r i t i c a l Branching D i f f u s i o n i n One Dimension 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 of the Fleming-Viot Process 73 6.1 Introduction and Construction 73 6.2 The Dimension of a Putative Support Set 75 6.3 A Useful Stochastic 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 of Support 83 APPENDIX A — Some In 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. - Internal Solutions to SPDEs i n Higher Dimensions 97 REFERENCES 100 TABLE 1 o f N o t a t i o n s Meaning I n f i n i t e s i m a l g r i d s p a c i n g i n space I n f i n i t e s i m a l g r i d s p a c i n g i n t i m e A t / A x 2 * - c o u n t a b l e g r i d r e p r e s e n t i n g R1^ * - f i n i t e g r i d r e p r e s e n t i n g [0, t ] 2 I n t e r n a l I . I . D . S-L random v a r i a b l e s on T x X B o r e l s u b s e t s o f X C o e f f i c i e n t s f o r a d i s c r e t e * - f i n i t e Green'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 t o s t o c h a s t i c * - 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 I n t e r n a l * - f i n i t e a n a logue o f I n t e r n a l * - f i n i t e a nalogue o f A I n t e r n a l s o l u t i o n t o u n f o r c e d * - f i n i t e a n a l ogue o f t h e h e a t e q u a t i o n Lebesgue measure on R I n t e r n a l measure on X , w h i c h a s s i g n s mass A x d a t each g r i d p o i n t Space o f p o s i t i v e f i n i t e measures on R d !Space o f p r o b a b i l i t y measures on R^ Space o f i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s w i t h compact s u p p o r t i n R Space o f C f u n c t i o n s bounded on R Space o f C, f u n c t i o n s whose second b d d e r i v a t i v e s a r e bounded on R P r e d i c t a b l e s q u a r e ( i n c r e a s i n g ) p r o c e s s a s s o c i a t e d w i t h a p r o c e s s x 3 CHAPTER ONE I n t r o d u c t i o n 1.1 Why t h i s T h e s i s The aim o f t h i s work i s t o i n t r o d u c e n o n s t a n d a r d methods, and i n p a r t i c u l a r 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 e q u a t i o n s , t o t h e t h e o r y o f m u l t i - d i m e n s i o n a l s t o c h a s t i c p r o c e s s e s . N o n - s t a n d a r d a n a l y s i s 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 random p r o c e s s e s o v e r a r e g i o n i n s p a c e , w h i c h 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 p r o c e s s e s 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 ca s e . The e v o l u t i o n t h r o u g h t i m e o f a l l t h e p r o c e s s e s d i s c u s s e d i n t h i s work g may be d e s c r i b e d by t h e h e a t o p e r a t o r A , b u t i n t h e o p i n i o n o f t h e o t a u t h o r , n o n s t a n d a r d t e c h n i q u e s may be e q u a l l y f r u i t f u l i n t h e a n a l y s i s o f p r o c e s s e s whose development t h r o u g h t i m e i s d e s c r i b e d by d i f f e r e n t o p e r a t o r s a s w e l l . 4 1.2 SPDEs The approach to the theory of SPDEs which we w i l l follow has i t s home in the theory of multiparameter processes, and in particular in the theory of multi- parameter stochastic integration that has been developed in recent years. Walsh (1986) contains a systematic treatment of this theory. This approach emphasizes sample path properties. An alternative approach considers SPDEs as stochastic evolutions on a space of functions, and emphasizes analytic properties. See Dawson (1975) and (1985) and the references there for further information on this approach. We w i l l not follow i t closely here. The type of SPDE we w i l l be considering most often in this work, is Cl-1). |r- = Au + f (u) Ŵ  , d t tx where t e R+ and x e Rd . W is "white noise" on R+ x Rd ; that i s , the tx derivative, in the sense of distributions (or generalized functions) of a random process Ŵ  , indexed by sets A <= R + X . R ^ •> such that 2 i) E(W^) = 0 i i ) E(W^) = X(A) where X is Lebesgue measure i i i ) . i f A n B = 0 , then Ŵ  i s independent of Wg . For further information on white noise see Walsh (1986).- Chapter 1. f (u) is a real-valued function of the point values of u . Equation Cl-1). cannot possibly hold in the classical sense of an equation between the values of functions at every point in the domain. The W term is tx far too rough. Rather we usually interpret (1-1) in the weak sense. That i s , OO £ i f we multiply (.1-1) by a c c ( R ) (smooth, with compact support) function <j> (~x) and integrate over a rectangle CO,T] X A , where A contains supp cj> , then 5 r r (1-2) A u <j> (x) dx = Ox ( u A<J>(x)dx)ds Jo J A S X f ( u )<j>(x)dW sx sx where the l a s t i n t e g r a l on the r i g h t i s the multiparameter stoc h a s t i c i n t e g r a l i n the sense of Ito discussed i n Walsh (1986) Chapter 1. Equation (1-2) may be derived from (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 of the middle term by par t s . An existence theory f o r (1-1) has been developed i n the case d = 1 , when f : R -> R , i s a L i p s h i t z - continuous function which grows at most l i n e a r l y at i n f i n i t y . Dawson (1972) established existence and uniqueness under these conditions, using a H i l b e r t space approach. Funaki (1983), established the same r e s u l t , with j o i n t continuity of sample paths i n t and x . Walsh (1981) established a modulus of continuity i n t and i n x f o r solutions of an equation s i m i l a r to (1-1) and investigated f i n e r sample path properties, under the same conditions on f . In Chapter Three of t h i s thesis an existence r e s u l t is- established f o r (1-1) assuming only continuity and l i n e a r growth of f for d = 1 . may be regarded as a d e r i v a t i v e of order d + 1 of a continuous function of I cl unbounded v a r i a t i o n (the Brownian Sheet) on R x R . When d >_ 2 there i s no hope of f i n d i n g a continuous function u to s a t i s f y the equation, even i n the weak sense of (1-2) . The most we can hope f or i s to f i n d a continuous process v , such that u may be regarded as a d e r i v a t i v e , i n the sense of The s i t u a t i o n f o r d >_ 2 i s e n t i r e l y d i f f e r e n t . The term W tx i n (1-1) 6 d i s t r i b u t i o n s , of order d - 1 , of v . In t h i s case u w i l l not, i n general, have point values, and i t i s d i f f i c u l t to see what sense can be made of the term f ( u ) occurring i n the stochastic i n t e g r a l on the r.h.s. of sx (1-2). In order f o r our theory of stochastic i n t e g r a t i o n to make sense of (1-2) we would need f ( u ) to be an adapted continuous process. This i s out of the question i f f i s supposed to be a r e a l function of the (non-existent) po i n t values of u . The functions f f o r which (1-2) can be reasonably expected to make sense are the constant functions. Walsh (1984) has shown existence and uniqueness of solutions to (1-2) i n t h i s case. We were not able to extend h i s r e s u l t s (see Appendix B). 7 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 M e a s u r e - V a l u e d B r a n c h i n g P r o c e s s e s (MB P r o c e s s e s ) were f i r s t o b t a i n e d by J i r i n a (1958) as 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 p a r t i c l e s . 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 (1968) and c o - w o r k e r s , 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 have been o b t a i n e d by Dawson and Hochberg (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 w i t h t h e case we s h a l l s t u d y h e r e , hence we r e f e r t o i t o f t e n as t h e "Dawson P r o c e s s " . We w i l l however most o f t e n make use o f a m a r t i n g a l e c h a r a c t e r i z a t i o n 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 t h e case o f a i n i t i a l Lebesgue measure i s as f o l l o w s . L e t p a r t i c l e s be d i s t r i b u t e d i n i t i a l l y on R^ o r a p o r t i o n t h e r e o f , a c c o r d i n g t o a P o i s s o n p o i n t p r o c e s s w i t h i n t e n s i t y A . Suppose t h a t t h e r e a f t e r each p a r t i c l e i n d e p e n d e n t l y e x e c u t e s a B r o w n i a n m o t i o n on R . A l s o suppose t h a t each p a r t i c l e i n d e p e n d e n t l y undergoes c r i t i c a l b r a n c h i n g w i t h r a t e u , i . e . a t f i x e d o r e x p o n e n t i a l l y d i s t r i b u t e d t i m e s , whose number i n a u n i t t i m e i n t e r v a l has e x p e c t a t i o n p , t h e p a r t i c l e d i e s o r s p l i t s i n t o two p a r t i c l e s , each outcome b e i n g e q u a l l y l i k e l y . I f t h e p a r t i c l e s p l i t s , b o t h d a u g h t e r p a r t i c l e s b e g i n i n d e p e n d e n t c a r e e r s from t h e p o i n t o f b i f u r c a t i o n . We now 1 d a s s i g n a mass, of' — t o each p a r t i c l e , and o b t a i n a random measure on R a t A each t i m e . Now suppose t h a t b o t h t h e i n i t i a l d e n s i t y o f p a r t i c l e s A, (which i s t h e r e c i p r o c a l o f t h e w e i g h t a s s i g n e d t o each p a r t i c l e ) and t h e b r a n c h i n g r a t e u, 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 such a way t h a t —• i s c o n s t a n t . Then t h e r e A d 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 t h e space o f p o s i t i v e measures on R Dawson (1972) i n d i c a t e d a c o n n e c t i o n i n d = 1 between t h i s p r o c e s s 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 s y s t e m , and t h e s o l u t i o n t o t h e SPDE 8 d-3) | E = + ^ „ , 9t . 2 t x 9x b u t t h i s c o n n e c t i o n has n o t y e t been made r i g o r o u s , s i n c e t h e f u n c t i o n f : u — > /u i s n o t L i p s h i t z , and hence (1-3) does n o t f a l l under t h e p u r v i e w o f t h e e x i s t e n c e t h e o r y d i s c u s s e d i n 1.2. W i t h t h e e x i s t e n c e theorem i n c h a p t e r t h r e e , we may now c l o s e t h e gap i n C h a p t e r Five,' and e s t a b l i s h t h e i d e n t i t y o f t h e s o l u t i o n o f (1-3) w i t h t h e MB p r o c e s s i n one d i m e n s i o n . As 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 t o make sense o f an e q u a t i o n l i k e (1-3) i n d i m e n s i o n s d >_ 2 . However 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 be c o n s t r u c t e d , and i t s s o l u t i o n may be shown t o c o i n c i d e w i t h t h e Dawson p r o c e s s . T h i s c o n s t r u c t i o n i s used i n C h a p t e r F o u r t o e s t a b l i s h s e v e r a l p r e v i o u s l y unknown r e s u l t s on t h e sample p a t h s o f t h e Dawson P r o c e s s . S p e c i f i c a l l y we show t h a t i ) any g i v e n Lebesgue n u l l s e t i s a.s. n e v e r c h a r g e d , i i ) t h e mass on any g i v e n Lebesgue s e t i s a.s. a c o n t i n u o u s f u n c t i o n o f t i m e , i i i ) . c onvergence o f a sequence o f bounded f u n c t i o n s i n measure, i m p l i e s t h e a.s. convergence o f t h e i n t e g r a l s o f t h o s e f u n c t i o n s w i t h r e s p e c t t o t h e random measure, u n i f o r m l y on f i n i t e t i m e i n t e r v a l s . 9 1.4 The Fleming-Viot Process In Fleming and V i o t (1979) a measure-valued process was introduced as a l i m i t , under suitable scalings of time and space, of the Ohta-Kimura stepwise mutation model. Further r e s u l t s on the structure of the sample paths have been obtained by Dawson and Hochberg (1982). Among these r e s u l t s i s the f a c t that f or f i x e d times the support of the random measure has Hausdorff dimension not greater than 2, almost surely. In Chapter Six we e s t a b l i s h t h i s r e s u l t f or a l l times simultaneously using Non-standard methods to amplify some of the ideas of Dawson and Hochberg (1982). For further information on the Fleming-Viot process, the reader i s r e f e r r e d to section 6.1. CHAPTER TWO Non-Standard Analysis and Probability 2.1 Some Definitions and Notations from Non-Standard Analysis For a r e a l introduction the ideas of non-standard analysis, with a minimum of technical apparatus, we refer the reader to Cutland (1983). An int 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 transfer 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 useful when dealing with hyper-finite c o l l e c t i o n s , which may be treated as f i n i t e sets, though they are generally i n f i n i t e . We w i l l usually 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 for standard objects. The embedding of a standard object into the non-standard universe w i l l be denoted by an asterisk (*) to the l e f t . We say that x_ e *R i s i n f i n i t e s i m a l i f |x| < ^ for every n e N ; we denote t h i s x ~ 0 ; x s y means x_ - y_ « 0 . We say x e *R i s i n f i n i t e , i f |x| > n , for every n e N . I f x e *R i s f i n i t e (not i n f i n i t e ) then there i s a unique x e R such that x_ « *x . We say that x i s near standard and c a l l x the standard part of JC , denoted °x or st(x) . These concepts may be extended, i n the obvious fashion, to any space. In p a r t i c u l a r , i f f e C [ R k ; R n ) and F e *C(. Rk;, R") , then st(F) = f <=> ° F (x) = f(°x) for a l l nearstandard x e *R We s h a l l require the.following axiom of saturation (see Cutland 1983, 1.9). 11 I f {A } n neN 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 s e t s , t h e n n neN A i s n o t empty. Two consequences o f t h i s a r e n a) Denumerable Comprehension. F o r e v e r y i n t e r n a l s e t A , and e v e r y f u n c t i o n f : N -> A , t h e r e i s an i n t e r n a l f u n c t i o n F : *N -»- A w h i c h e x t e n d s f . b) I n f i n i t e s i m a l U n d e r f l o w . L e t S be i n t e r n a l , S £ *R , and suppose f o r some a > 0 , x e S whenever 0 < °x <_ a . Then f o r some e & 0 , x e S whenever e <_ x_ <_ *a . A u s e f u l n o t i o n i s S - c o n t i n u i t y (S f o r s t a n d a r d ) . D e f i n i t i o n 2.1.1. An i n t e r n a l f u n c t i o n F : E c *R n i s c a l l e d S - c o n t i n u o u s i f f x ~ y => F (x) PS F (y_) e . ns (* - R n ) ' .. That t h i s i s t h e a p p r o p r i a t e n o t i o n o f c o n t i n u i t y t o l i n k s t a n d a r d and non- s t a n d a r d , - . i s shown by , Theorem 2.1.2. ( C u t l a n d (1983), Theorem 1.6). L e t F : E <= *R *R be i n t e r n a l . Then °F e x i s t s and i s c o n t i n u o u s i f f F i s S - c o n t i n u o u s on E . N o n s t a n d a r d P r o b a b i l i t y Theory r e a l l y came i n t o i t s own a f t e r - t h e development o f Loeb Measure. Theorem 2.1.3. (see C u t l a n d (1983), Theorem 3.1). E v e r y i n t e r n a l * f i n i t e l y a d d i t i v e measure space (|~, F, y_) g i v e s r i s e : t o a c l a s s i c a l o - a d d i t i v e measure space (|J, L ( F ) , L (y_)) , such t h a t F c L ( F ) , and i f A e F , t h e n L (A) = °y_(A) . I f A e L ( F ) , t h e n L(y_) (A) = inf{°y_(B) | A £ B , A e F} . F u r t h e r , i f L ( y ) (A) < 0 0 , t h e n t h e r e i s B e F , w i t h L (y_) (B A A) = 0 . The (complete) measure space ( f , L ( F ) , L(v_)) i s c a l l e d t h e Loeb e x t e n s i o n o f Cf , F, y_) . I n t h e c a s e o f p r o b a b i l i t y measures, we w i l l denote t h e Loeb e x t e n s i o n o f (ft, F, P) by (Q, F, P) , c o n t r a r y t o o u r u s u a l c o n v e n t i o n . 12 Theorem 2.1.4. (see C u t l a n d (1983) Theorems 3.1, 3.5). I f E i s an i n t e r n a l f i e l d o f s u b s e t s o f [ , and F : [ •+ i s an i n t e r n a l E-measurable f u n c t i o n t h e n t h e p r o j e c t i o n s t ° F : [ + R u {°°} i s L ( E ) m e a s u r a b l e . C o n v e r s e l y i f f : [ •-> i s L(E) m e a s u r a b l e , and [ i s a - f i n i t e w i t h r e s p e c t t o L(y_) (as i s a l w a y s t h e case h e r e ) , t h e n t h e r e i s an i n t e r n a l E-measureable f u n c t i o n F : [~ -> *R^ such t h a t s t ° F ( x ) = f ( x ) L ( y ) - a.e. Such an i n t e r n a l F i s c a l l e d a l i f t i n g o f f . I f f has f i n i t e s u p p o r t , i . e . L ( y ) ({xj f (x) ^ 0}) < 0 0 , t h e n f a d m i t s a l i f t i n g F w i t h t h e same p r o p e r t y . I f T £ *R^ i f f : T R i s c o n t i n u o u s , t h e n we may o b t a i n a l i f t i n g F w h i c h i s S - c o n t i n u o u s and f o r w h i c h s t ° F ( x ) = f(°x) , V 21 e r n s t - 1 ( R d ) (the n e a r - s t a n d a r d p o i n t s ) . Such an F i s c a l l e d a u n i f o r m l i f t i n g o f f . D e f i n i t i o n 2.1.5. An i n t e r n a l f u n c t i o n F on ( f , F, y) i s c a l l e d S - i n t e g r a B l e , i f f F dp i s f i n i t e , and F ( x ) dy + |F(x) I dp & 0 , f o r a l l {x: 1F(X) I < 6} {x: | F ( X ) | > H> i n f i n i t e s i m a l 6 and i n f i n i t e H . F i s s a i d t o be S - I J w i t h r e s p e c t t o p_ , i f | F | ^ i s S - i n t e g r a b l e . Theorem 2.1.6. (see C u t l a n d (1983) Theorem 3.9). I f F i s S - i n t e g r a b l e on ( f , F , p) , t h e n f o r a l l A e F , F dy = F d L ( y ) A A D e f i n i t i o n 2.1.7. An i n t e r n a l measure space (T, F, y_) i s c a l l e d 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 a t o p o l o g i c a l measure space (E, F, y) i f f ( i ) T i s a h y p e r f i n i t e i n t e r n a l s u b s e t o f *E ( i i ) F i s t h e i n t e r n a l power s e t o f [ ( i i i ) A s e t B c E i s y-me a s u r e a b l e , i f f s t "*"(B) n [ i s L (y) - m e a s u r e a b l e . I n t h a t case y(B) = L (y_) ( s t - 1 (8) n D . The c a n o n i c a l example o f 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 i s t h e d i s c r e t e r e p r e s e n t a t i o n o f Lebesgue measure on R , w h i c h we w i l l use f r e q u e n t l y . Theorem 2.1.8. (see C u t l a n d (1983) Theorem 4.1). L e t Ax. be any i n f i n i t e s i m a l , i = l , . . . , d . L e t M. e *N\'N be i n f i n i t e , l l i = 1, 2,...,d such t h a t °M Ax. ^ 0 . Then l e t l I X = {k. Ax.,...,k.Ax.)Ik. e *Z, k.I < M.} . D e f i n e a measure A on X by 1 i j D ' l l ' l — d s e t t i n g t h e v a l u e o f A_ on each p o i n t x_ e X t o be II Ax- • Then i = l (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 Lebesgue measure. 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 out S - c o n t i n u i t y o f f u n c t i o n s f r o m h y p e r f i n i t e g r i d s X t o *R . 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 i n R Theorems about 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 X , can be t r a n s l a t e d i n t o theorems a b o u t t h e weak convergence o f a sequence o f p r o c e s s e s on a sequence o f f i n i t e g r i d s t o a c o n t i n u o u s l i m i t . We w i l l n o t e x p l i c i t l y make such a t r a n s l a t i o n . 14 2.2 Non S t a n d a r d W h i t e N o i s e A n d e r s o n (1976) i n t r o d u c e d 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 B r o w n i a n M o t i o n , namely an 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 o f s i z e /Kt i n a t i m e s t e p A t . I m p l i c i t i n t h i s c o n s t r u c t i o n was 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 o f w h i t e n o i s e on t h e l i n e , as a sum o f I I D random v a r i a b l e s £ , each o f mean 0 and v a r i a n c e A t , on a h y p e r f i n i t e t i m e - l i n e o f s p a c i n g A t . R e c e n t l y Andreas S t o l l (1985) has 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 t o . a r b i t r a r y a - f i n i t e Radon s p a c e s . F o r o u r p u r p o s e s we o n l y need r e p r e s e n t a t i o n s o f w h i t e n o i s e on r e c t a n g l e s i n R^ , o r on a l l o f R^ . L e t X be a h y p e r f i n i t e l a t t i c e as d e s c r i b e d above, and l e t ft be an i n t e r n a l space•on w h i c h a r e d e f i n e d a f a m i l y {£ } o f I . I . D . S-L * X£X 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 E(£ ) = 0 and var (5 ) = 1 . F o r — 21 x. most o f o u r a p p l i c a t i o n s we w i l l need 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, as w e l l , x The e x i s t e n c e o f s u c h a space ft may be shown by example. L e t ft = {-1, 1} . L e t F be t h e f a m i l y o f i n t e r n a l s u b s e t s o f ft , and d e f i n e A P_CA). = J — L f o r A e F . Then ( f t, F, P) i s an i n t e r n a l p r o b a b i l i t y s p a c e , |ft| and we may d e f i n e £ as t h e c o o r d i n a t e maps. We w i l l most o f t e n be u s i n g a * c o u n t a b l e l a t t i c e X = { (k^Ax, •. - ,k_.Ax) | k l ' " * " ' k d e * z ^ • A n exemplary space ft i s t h e n {-1, 1} . W e may t a k e F t o be t h e * a - f i e l d g e n e r a t e d by t h e * f i n i t e s u b s e t s o f ft ( c l o s e d under * a - u n i o n s and *a - i n t e r s e c t i o n s ) . The i n t e r n a l p r o b a b i l i t y measure P_ may be d e f i n e d as i n t h e s t a n d a r d a n a l o g u e : P_ i s t h e u n i q u e * a - a d d i t i v e measure 15 on F such t h a t , (<4 : x e X} a r e i n d e p e n d e n t and f o r any x e X , P({co = l } ) = P ( { w = - l } ) = i . x x 2 The t r a n s f e r p r i n c i p l e g u a r a n t e e s t h a t the Kolmogorov E x t e n s i o n Theorem c a r r i e s o v e r t o the n o n - s t a n d a r d s e t t i n g , and t h e r e f o r e t h a t s u c h a measure P_ e x i s t s . I n t h i s case a g a i n we w i l l t a k e t h e p r o b a b i l i t y space ( f i , F, P) t o be t h e Loeb e x t e n s i o n o f ( f i , F, P_) . G i v e n s u c h a space fi , and random v a r i a b l e s £ , d e f i n e f o r i n t e r n a l x r A ~ s e t s A c X , W(A) = } E, IT Ax ; t h e map W : A -»- W(A) i s c a l l e d — x . . i xeA — i = l d - d i m e n s i o n a l S-white n o i s e on X . S t o l l (1986) shows Lemma 2.2.2. I f °A_(A) < °° and _A(A A B) = 0 , t h e n °W(A) = °W(B) P-a.s. Thus we may make D e f i n i t i o n 2.2.3. F o r each Loeb measureable s e t A c X , w i t h L (A_) (A) < °° , a s t a n d a r d random v a r i a b l e w(A) i s w e l l d e f i n e d (up t o a n u l l s e t ) by w(A) = °W(A) P - a . s . whenever A i s i n t e r n a l and L(X_) ( A A A ) = 0 . Theorem 2.2.4. ( S t o l l (1986) Theorem 2 .5). The f a m i l y (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 o f (X, X) 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 Stochastic Integrals 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 of spacing At representing a l i n e segment i n R + , and l e t X be a l a t t i c e of spacing Ax i n each d i r e c t i o n representing a rectangle i n R D . Let 9, be an i n t e r n a l space supporting a c o l l e c t i o n E, , (t,x) e T x X , of I.I.D. tx - - 2 S-L random va r i a b l e s , as described above. In t h i s section we w i l l use the notation dW to represent tx F v^At Ax d . We define an i n t e r n a l f i l t r a t i o n F , t e T on ft to be the tx t — algebra of i n t e r n a l sets generated by {W 0 < s < t, x e X} . On the Loeb sx — — space Cft, F, P) we define a f i l t r a t i o n F , t e st(T) , by F = n a(F ) v W , where W i s the c o l l e c t i o n of P-null sets. For t °t>t ± properties of t h i s f i l t r a t i o n see Hoover and Perkins (1983) §3. We say that an i n t e r n a l process u t x ^ w ^ a l i f t i n g of a process U 2 I({|x| > H ; u^Cw) on R + x R d x fl , i f for H e *NXN and n e N , t <_ n}) dX dX « 0 , and LC*. x X x P)C(Ct, x, u) | °U (u) ? u (w)}) = 0 . '-t - X — — — t X o. o - - — _t, X We say that an i n t e r n a l process u t x ^ o n ^ x ^ i s F^-adapted, i f TJ , 0 < s < t , y e X , i s F measureable. sy — t_ We s h a l l c a l l a process û _ on R x R , F^_-adapted, i f , for each t e R + , u (jo). , (s e i-0, t ] , x e R D , a) e ft), i s B([0,t]) x B( R D ) x f sx t measureable. Since our f i l t r a t i o n F i s continuous and we do not wish to integrate discontinuous integrands, we s h a l l not make d i s t i n c t i o n s between adapted, progressively measureable, op t i o n a l , and predictable processes. We s h a l l u s u a l l y be w o r k i n g w i t h F ^ a d a p t e d l i f t i n g s o f F^-adapted p r o c e s s e s . Theorem 2 . 3 . 1 . Suppose u^- x^w^ ;"' s a n L F^_-adapted p r o c e s s on R x R x and t h a t an i n t e r n a l F^ ada p t e d p r o c e s s U (to) l i f t s u , and i s — t x S - L 2 ( T x X x n; L(A' t *'A x . p ) ) . Then f o r any t e R + and t ss " t C2-1) a.s. u dw sx s x y y u dw 0 < s < t x'x S5 sx where the i n t e g r a l on t h e l e f t i s i n the sense o f I t o . P r o o f : We f i r s t e s t a b l i s h an i s o m e t r y p r o p e r t y : I I U dW 0<s<t xeX § - § - J I I I UsxdWsxV+ 2 X I I { I UsxdO 4 I U s ' \o£s.<t (xeX §S s*j 0 < 3 < £ s<s'<t\xeX 2 - IxeX - dW x s ( s i n c e d w s i x i s ( c o n d i t i o n a l l y ) i n d e p e n d e n t o f F g ) = E / y y [ u 2 dw 2 + y u u ,aw dw 1 + 2 x 0 SA- v s x sx sx sx sx s x ' \0<s<t xeX — — x'/x — — X " x:JteX I I E ( U 2 ) A x d A t + 0 ( u s i n g E(dW 2 ) = A x d A t ) 0<s<t xeX - - ^ 1 1 1 2 | u | | L (TxXxfi) Hence t h e mapping U > £ £ U dW a c t s i s o m e t r i c a l l y f r o m t h e ^ ^ v s x s x 0<s<t xeX -- 2 2 space o f F -adapted' S-L (TxXxo,) p r o c e s s e s t o *L (fi) . Hence, i f U and ' 2 U a r e any two S-L F^-adapted l i f t i n g s o f u , y I D dw - y yudw 0<s<t xeX — — 0<s<t xeX — — I I E | U . - U' | 2 A x d A t 0<s<t xeX t x t x 1 Hence °y £ U dW 0<s<t xeX a . s . o I I u' dW 0<s<t xeX § x § x A s i m i l a r argument shows t h a t i f t 1 RJ t , t h e n a.s. y y u dw tv sx sx 0 , so t h a t t h e r . h . s . o f (2-1) i s w e l l d e f i n e d t , < s < t xeX — up t o a n u l l s e t . We must 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 t h e I t o i n t e g r a l . C o n s i d e r a p r o c e s s u o f t h e form u. (o>) = I r j _ " . . ( t ) • I . (x) • I_(u>.) , t x It ,t ) A K where R i s an F measureable s e t i n fi , and A i s Lebesgue measureable 1 i n R . P i c k %2 ~ ^2 ' a n d P^c^- A i n t e r n a l i n X such t h a t L U x ) ( A A s t 1 ( A ) ) .= .0; . By Theorem 3.2 i n Hoover and P e r k i n s ( 1983), we may ' f i n d ' £ t - and an i n t e r n a l s u b s e t R o f fi - , s u c h t h a t R e and P (R A R) = 0 . _ 1 L e t u\ (to) = I t x Then u d w = lv(w) s x sx K dw s x R [ t l ' t A t 2 ) > < A a . s . 1 T ) ^ ' I I d w » (by. d e f i n i t i o n o f t h e w h i t e t <s<tAt 2- n o i s e dw. ) xeA t x I I 0<s<t xeX U dW sx s x Thus t h e l e f t hand s i d e and r h s o f Theorem 2.3.1 c o i n c i d e f o r s i m p l e f u n c t i o n s u 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 c o m b i n a t i o n s t h e r e o f . 2 Such l i n e a r c o m b i n a t i o n s a r e c l e a r l y dense i n t h e H i l b e r t space o f L 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 s t o c h a s t i c i n t e g r a l . Hence t h e two s i d e s must c o i n c i d e f o r a l l such u and U by t h e i s o m e t r y p r o p e r t y o f t h e s t o c h a s t i c i n t e g r a l , and o f t h e h y p e r f i n i t e sum. CHAPTER THREE The Heat Equation with Non-Linear Stochastic Forcing 3.1 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 existence theorem for solutions of equations of the form (3-1) |» f ( u ) * , dt 2 tx dx + * + on R x R . Here W i s "white noise" on R x R . We suppose the function tx f to be continuous, and to s a t i s f y a growth condition: there i s a r e a l K , such that (3-2) f 2 ( u ) £ K ( l + u 2 ) for a l l u . We solve C3-1) subject to an i n i t i a l condition C3-3J. u(0,x) = u (x) , where u^(x) i s a bounded continuous function. With minor changes i n notation f could be made to depend on t and x as well as u , and u^(x) could be taken as a random function. With more s i g n i f i c a n t changes, the same construction can be made to work i f (3-1) i s modified by the introduction of an a d d i t i o n a l f o r c i n g term + g(t, x, u) , or by the introduction of a bounded, non-zero, continuous non-linear function 9 2u of u m u l t i p l y i n g the term — — . With appropriate i n e q u a l i t i e s analogous 3x to those i n Appendix A, a treatment very s i m i l a r to the remainder of t h i s chapter can be done for equation (3-1) on a s t r i p R + x [a, b] with D i r i c h l e t o r Neumann t y p e boundary c o n d i t i o n s s p e c i f i e d a t a and b . A random i n i t i a l c o n d i t i o n u i n d e p e n d e n t o f t h e w h i t e n o i s e W may be 0 t x h a n d l e d by e n l a r g i n g t h e p r o b a b i l i t y space 0, . As d i s c u s s e d i n s e c t i o n 1.2, (3-1) i s o n l y s o l v a b l e i n a 'weak' s e n s e , oo t h a t i s , i f <|> e C ( R) , C3-4) utx<$> (x) dx - u d> (x) dx = Ox u Ad)(y)dyds sy ft 0 f ( u )<j>(y)dw sy s y T h i s i s c a l l e d t h e "weak" form o f ( 3 - 1 ) . The main theorem (3.9.2) o f t h i s c h a p t e r a s s e r t s t h a t t h e r e e x i s t s a space Q , such t h a t , f o r any f , a s t o c h a s t i c p r o c e s s u , j o i n t l y c o n t i n u o u s i n t and x , may be d e f i n e d t x 00 on 9, , f o r w h i c h (3-4) h o l d s f o r any (f> e C (R) . c S e c t i o n 3.2 i n t r o d u c e s t h e p r o b a b i l i t y space ft and d i s c u s s e s t h e c o n s t r u c t i o n o f a w h i t e n o i s e on 0, . S e c t i o n 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) and shows how i t may be s o l v e d i n t e r n a l l y , f o r a s o l u t i o n U t x S e c t i o n 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 w h i c h a r e u s e d s u b s e q u e n t l y . P r o o f s 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 on 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 U t x , w h i c h a r e n e c e s s a r y f o r 3.6 and 3.7. S e c t i o n 3.6, o b t a i n s bounds on t h e moments o f s p a t i a l d i f f e r e n c e s U. - U . I n s e c t i o n 3.7-we o b t a i n bounds on t h e moments o f t e m p o r a l t x t y d i f f e r e n c e s U\ - U . W e use the r e s u l t s o f 3.6 and 3.7 i n 3.8 t o show t x r x t h a t U i s , w i t h p r o b a b i l i t y 1, a l i f t i n g o f a j o i n t l y c o n t i n u o u s p r o c e s s t x 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 u a c t u a l l y s a t i s f i e s ( 3 - 4 ) , hence i s a weak s o l u t i o n o f ( 3 - 1 ) . 23 3.2 W h i t e N o i s e on t h e space ft L e t a p o s i t i v e i n f i n i t e s i m a l Ax be g i v e n . L e t X be the s e t {kAx|k e *Z> . Now p i c k aa such t h a t 0 < °a < - j , and l e t A t = a A x 2 . Now suppose t ^ > 0 i s g i v e n and l e t t be any number o f t h e f o r m { k A t , k e *N} s u c h t h a t t _ ~ tc . L e t T be { k A t l k e *N, k < t , / At} . - f t 1 — - f We have 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 space r -,TxX (such as {-1, 1} ) on w h i c h a r e d e f i n e d a f a m i l y o f I . I . D . random 2 v a r i a b l e s {E, t 6 T, x e X} such t h a t E ( E ) = 0 & E (£ ) = 1 V ( t , x ) , t x - - — t x — t x - - and s u c h t h a t 5^ p o s s e s s e s f i n i t e h i g h e r moments o f a l l o r d e r s , t h e n t h e random ( i n t e r n a l ) s e t f u n c t i o n A > T E /AtAx i n d u c e s a , • \ •» t x ( t , x ) e A — " w h i t e n o i s e " {w^(oj) |A e B(['0, t f ] !x R) } . F u r t h e r , t h i s w h i t e n o i s e i s a d a p t e d t o the f i l t r a t i o n d e r i v e d as p e r s e c t i o n 2.3 from t h e i n t e r n a l f i l t r a t i o n F g e n e r a t e d by t h e v a l u e s {E, s < t } . I t i s w i t h r e s p e c t t o t s x - _ - t h i s w h i t e n o i s e t h a t we s h a l l s o l v e ( 3 - 1 ) . The method we s h a l l use i s s i m i l a r i n s p i r i t t o t h a t employed i n K e i s l e r (1983) f o r s t o c h a s t i c O.D.E.'s,/ 24 3.3 Hyper-Finite Difference Equations Let F(u) be a uniform l i f t i n g of f ( u ) , subject also to the growth condition (3-2). Let be a uniform l i f t i n g of u^ which i s also uniformly bounded on X . Then consider the h y p e r f i n i t e analogue of (3-1) , T T a + U U . - 2 U + U F(U ) £ t+At,x t,x t,x+Ax t,x t,x-Ax tx tx (3-5) At . 2 Ax /At AJ or equivalently, U,.,.., = U + At Ax 2 [ u . - 2 U + U . ] + F(U ) E — t+At,x t,x t,x+Ax t,x t,x-A x tx *tx Ax We may solve (3-5) i n p r i n c i p l e i n d u c t i v e l y . The s p e c i f i c a t i o n of U U / X and E Q ^ for x e X gives us enough information to f i n d . Knowing U.. and £, , we may solve for U . , and so on. At,.- At,- 1 2At,-, Continuing i n t h i s manner, we define an i n t e r n a l process U (u) . We may f i n d a closed form expression f o r t h i s inductive d e f i n i t i o n as follows. From (3-5) i t i s clear that the value F(U ) £ enters into the d e f i n i t i o n tx tx of U , then U . . , U , U . , and then f i v e t+At,x ' t+2At,x-A x t+2At,x t+2At,x+Ax values of U , and so on. The c o e f f i c i e n t Q n A t with which F(U ) E t+jAt, X~Y tx tx enters into the d e f i n i t i o n of U • „ ,may be found as the s o l u t i o n of a .trHiAt,y difference equation: = 1 ; = 0 Vx ^ 0 ; , (n+l)At nAt L „ x^nAt ^ ^nAt (3-6 Q = a Q . + (l-2a)Q + a Q . *x • *x-Ax x x+Ax B u t we may r e c o g n i z e (3-6) as the d i f f e r e n c e e q u a t i o n g o v e r n i n g t h e d e n s i t y o f a Markov P r o c e s s , i n f a c t , a random w a l k . Thus we have t Lemma 3.3.1. The c o e f f i c i e n t s Q- a r e the " 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 on^ t h e l a t t i c e X , s t a r t i n g a t x = 0 , a t time A t , and t a k i n g one s t e p t o the r i g h t ( o r l e f t ) i n ea c h ' t i m e p e r i o d A t w i t h p r o b a b i l i t y 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 u tx y l Q- - F(U ) L L v ^x-y S y 0<s<t yeX - i -I ? 1^ sy •* Ax + I Q t + A t u y,X ° l The s econd term on t h e r . h . s . 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 the h y p e r f i n i t e h e a t e q u a t i o n (3-5), w h i c h we w i l l d e s i g n a t e h e r e a f t e r U t x 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 f i n d t h e f o l l o w i n g i n e q u a l i t i e s h e l p f u l i n t h e n e x t t h r e e s e c t i o n s . P r o o f s a r e d e f e r r e d t o App e n d i x A, s i n c e t h e y do n o t i l l u m i n a t e the p a r t i c u l a r s u b j e c t m a t t e r o f t h i s c h a p t e r . Lemma 3.4.1. There i s a f i n i t e c o n s t a n t K , depen d i n g on a , such t h a t f o r a 2 i t/A?t c *N , I (Q-) < K ^ v x — a J t xeX Lemma 3.4.2. There i s a f i n i t e c o n s t a n t , depe n d i n g on a , such t h a t 2 f o r a l l t , I I CQ-j < K / t / A t . " 0<s<t x€X S - a - Lemma 3.4.3. There i s a c o n s t a n t K , depen d i n g on a , s u c h t h a t V z e X a I I CQ- - Q-A ) 2 < K |z| / Ax . 0<s<t -xeX x ~ a " Lemma 3.4.4. There i s a c o n s t a n t K , de p e n d i n g on a , such t h a t , f o r a l l a t e T , and r < t I I CQ-"") - Q- " ) 2 < K / ( t - r ) / A t 0<s<r x eX We s h a l l 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 B u r k h o l d e r (1973, Theorem 2.1.1., s p e c i a l i z e d s l i g h t l y ) . Theorem 3.4.5. L e t M , n e N be a h y p e r m a r t i n g a l e and l e t <M> n e N n 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 s q u a r e f u n c t i o n , and l e t p > 1 be f i n i t e . Then t h e r e i s a f i n i t e c o n s t a n t K d e p e n d i n g o n l y on p , such t h a t P E ( M P ) < K E C < M > P / / 2 ) + K E(max |M - M | P ) . - n - p - n P - Q < k < n k+1 k 27 3.5 Bounds on Moments o f U t x P i c k q > 1 , and l e t R ( t , x ) = E | U | 2 q . L e t H (t) = sup R ( t , x ) q - - - t x q - q - - C o n s i d e r any f i x e d ( t , x ) / and l e t ji be th e measure on [ 0 , t ] x X 2 d e f i n e d by y ( { ( s , y ) } ) = (Q-" §) — . L e t l y l denote y ( [ 0 , t ] x X). - - X ~ Y Ax 1 1 L e t c d e n o t e , i n what f o l l o w s , a f i n i t e c o n s t a n t , d e p e n d i n g o n l y on q and t , w h i c h may change i t s e x a c t v a l u e from l i n e t o l i n e . Now 1/2 R Ct,x). < c E q - - - - I I ECU ) Q±~ S- E ( f ) 0<s<t y £X S Y X " X sx A x 2 < 5 r |2q + c U 1 t x 1 I n t h e summation above, t h e term c o r r e s p o n d i n g t o ( s , y) may be r e g a r d e d as a m a r t i n g a l e i n c r e m e n t w i t h r e s p e c t t o the f i l t r a t i o n F . Hence we may s a p p l y B u r k h o l d e r ' s i n e q u a l i t y (3.4.5) t o o b t a i n R (t , x ) . < c E[ I I F 2 ( U ) y ( s , y ) ] q (3-8) + c E | sup| F(u ) |E / y ( s , y ) | 2 q + c lu | 2 q . — s y sy - - 1 t x 1 0<s<t - 1 — yeX The se c o n d t e r m on 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 2 2 ^ c E[ max F ( U ) E y ( s , y ) ] w h i c h i s bounded by - 0 < s < t ?y - - c E[ I I F 2(u ) E2 y ( s , y ) ] q . Thus " 0<s<t veX a S " R ( t , x ) < c E | I I F 2 ( U ) ( 1 + E2 ) Hil^-|q| y|g + c|u I2q 0<s<t YeX tl 51 | y 28 A p p l y i n g J e n s e n ' s I n e q u a l i t y t o the p r o b a b i l i t y measure y ( - ) / | y | , we f i n d R Ct,x) < C E [ I I |F(U ) | 2 q ( 1 + \ ^ \ ) H i t f l ] | y | q + C|S \ 2 * . q " " ~ ~ 0 < s < t y £ X ?y \V\ ' t x Now we use t h e f a c t s t h a t | y | q / | y | = | y | q 1 , t h a t |y| i s bounded u n i f o r m l y f o r °t <_ t by Lemma 3.4.2, t h a t t h e moments o f £ a r e f i n i t e , 1 sy t h a t 5 i s i n d e p e n d e n t o f U , and t h a t U i s bounded u n i f o r m l y sy s y t x i n Ct,x) , s i n c e U Q i s bounded, t o o b t a i n R ( t , x ) < c U + I I E | F l t r ; > | 2 q ( l + E\K | 2 q ) ( Q - " - ) 2 7^) q - - 0 < s < t y e x ^ *-y A * < c C l 0<s<t I yeX K 2 q ( l + E | U -' sy |2q) (Q^) *x-y Ax ; <_ c C l + I I R (s,y) CQ- - ) 2 T^) 0<s<t yeX q - - x-y Ax < c a + l H (s) c I — f p - At) . 0<§<t q yeX A X Now u s i n g Ax = / A t / a , and lemma 3.4.1 we f i n d Lemma 3.5.1. r -1/2 (3-9) H ( t ) = sup R ( t , y ) < _ c ( l + I H ( s ) ( t - s ) At) , f o r a l l t i n y q °fs<t [ 0 , t ^ l , where c i s a c o n s t a n t d e p e n d i n g on a , q and t f . Now, i t e r a t i n g (3-9) and i n t e g r a t i n g by p a r t s , H (t) < c I (1 + c I (1 + H ( u ) ) ( s - u ) 1 / 2 A t ) ( t - s ) " 1 / 2 A t 0<s<t 0<u<s < c ( ^ t j + I (1 + H ( U ) ) ( I (s-u) 1 / 2 ( t - s ) 1 / 2 A t ) A t 0<u<t '* u<s<t ~ ' (3.10) < . c ( l + I (1 + H (u)) ( I s V 2 ( t f - s ) 1 / 2 At) A t 1 c ( l + I H (u) At) 0<u<t q where c i s a n o t h e r c o n s t a n t d e p e n d i n g o n l y on a, q, and t ^ . We now r e q u i r e a t y p e o f G r o n w a l l ' s lemma. Lemma 3.5.2. There i s a c o n s t a n t c d e p e n d i n g o n l y on q, maxlu | 2 q . a y 0y and t f , s u c h t h a t E_| | <_ c e x p ( c t ) , f o r t <_ t , and f o r any x e P r o o f . We may t a k e c t o be the maximum o f H (0)(= maxlu | ) 2 q and t h e q y 1 Oy 1 * + o f e q u a t i o n ( 3 - 1 0 ) . We p r o c e e d by i n d u c t i o n , on n e Z . F o r n = 0 , t h e lemma h o l d s f o r t = nAt . Suppose now t h a t f o r n e N H CkAt) <_ c Cl + c A t ) k , f o r k = 0, 1, . . . , n-1 . Then HCnAt) <_ c (.1 + c £ (1 + c A t ) k At) (by (3-10)) 0<k<n = c ci + c ^—r-1— A t c A t (summing a g e o m e t r i c s e r i e s ) = c C l + c A t ) n T h i s i s t h e i n d u c t i o n s t e p . By t h e t r a n s f e r p r i n c i p l e , we may c o n c l u d e t h a t t h i s i n t e r n a l argument v e r i f i e s H(nAt) ' <_ c (1+cAt) n , f o r a l l n e *N such t h a t nAt <_ t . Then we n o t i c e t h a t ( 1 + c A t ) n £ e x p ( c n A t ) , ( i n f a c t t h e y a r e i n f i n i t e s i m a l l y c l o s e ) . • 31 3.6 Bounds on Moments o f S p a t i a l D i f f e r e n c e s L e t U = U + V , where V r e p r e s e n t s t h e c o n t r i b u t i o n f r o m t h e 1—X L.X "CX random f o r c i n g . U i s S - c o n t i n u o u s ( i n f a c t S-smooth), and we a r e i n t e r e s t e d i n t h e c o n t i n u i t y o f V . 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 on t h e moments o f t h e d i f f e r e n c e s V - V t x t y Lemma 3.6.1. There i s a c o n s t a n t c , de p e n d i n g o n l y on q > 1 , max|u 2 q| , a , and t ^ , such t h a t , f o r a l l x, y i n X , and f o r 0 <_ t <_ t , i 12q I I q - | v t x - v ± °lx - zl • P r o o f : We may w r i t e V - V as a sum o f m a r t i n g a l e i n c r e m e n t s w i t h t x t y r e s p e c t t o t h e i n t e r n a l f i l t r a t i o n F ; by (3-7), s V - V = y l (Q-~- - Q ^ ) F ( U ) £ ( — ) V 2 £ a. 0 < s < t zex V* V«' H We w i l l e s t i m a t e t h e 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 I n e q u a l i t y CTheorem 3.4.5). We w i l l d e s i g n a t e by yCs,z) t h e measure on [0, t ] x X w h i c h a s s i g n s t - s t - s 2 A t t o e a c h p o i n t ( s , z ) t h e w e i g h t (Q- - - Q- -) — - - x-z y-z Ax Hlv^ " V t y | 2 q < c EC F 2(u ) d y ( s , z ) ) q s z - - [0,t)xX 2 2 + c E( max F (u ) £ y ( s , z ) ) q " 0<s<t ^ -S-Z " - 32 < c E( [ 0 , t ] x X F 2 ( u , dy_ ( s , z ) ) q i i j I q -- y + C E ( [ F 2 ( U ) K2 ^ S ' 5 ) , q | y | q s z s z < C E C F 2 ( U ) ] q [ i + | 5 | 2 q ] T ^ - ( § ' 5 ) - | y | q s z s z I I (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 (s) - ^ { s - ' * ] • | y | q q " y ( u s i n g independence o f U and E , and a l s o f i n i t e moments o f E ) s z s z s z <_ c I x - y | A t ( u s i n g lemmas 3.5.2 and 3.4.3, n o t i n g t h a t — = ctAx ) • 33 3.7 Bounds on Moments o f Temporal D i f f e r e n c e s 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 on t h e moments o f t h e d i f f e r e n c e s V - V t x r x Lemma 3.7.1. There i s a c o n s t a n t c , depe n d i n g o n l y on q > 1 , max |TJ | , a , and t , such t h a t , f o r a l l t , r <_ t , and a l l x e X , uy r — — - r - I I2q | ,q/2 E V - V < c t - r —' t x r x ' — '- P r o o f : We may suppose w.1.o.g. t h a t r < t . We may w r i t e V - V as a sum o f m a r t i n g a l e d i f f e r e n c e s w i t h t x r x r e s p e c t t o t h e i n t e r n a l f i l t r a t i o n F s S £ x 0 < s < r zeX IS ^ 25 ^ ^ u ^. ~V x-z s z s z Ax r<s<t Z £ A - - We w i l l d e s i g n a t e by y ( s , z ) t h e measure on [ 0 , t ) * X w h i c h a s s i g n s t - s r - s ^ A t t o e a c h p o i n t ( s , z ) t h e w e i g h t (Q- - - Q- -) -— , i f s < r , and t h e - - x-z x-z Ax - - 2 w e i g h t ^2 X_|^ Ax ' ^ - - - < *~ ' T ^ e n u s i - n 9 " B u r k h o l d e r ' s I n e q u a l i t y CTheorem 3.4.5) I l 2 c * E V - V < c E ( — t x r x — — J [ 0 , t ) x X F 2 ( U ) d y ( s , z ) ) q s z + c E( max F 2 ( U ) £ 2 y ( s , z ) ) q - 0 < s < t ~ ~ < C E ( F 2 ( U ) ) q ( i + \K |2 q) - ^ ' 5 > . | y | q y ( u s i n g J e n s e n ' s I n e q u a l i t y ) < c E K 2 Q H (s) ̂  • | y l q (using independence of U , and E, , and the sz sz f initeness of E I E, I 2q) — sz1 < c(t-r) q/2 (using lemma 3.5.2, and lemmas 3.4.2 and 3.4.4, with 34 35 3.8 S - C o n t i n u i t y and t h e S t a n d a r d P a r t The 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 , the s o l u t i o n t o t h e 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 (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 n o n - s t a n d a r d v e r s i o n o f Kolmogorov's C o n t i n u i t y C r i t e r i o n : * Theorem 3.8.1. L e t : fi x r -»- R be an i n t e r n a l p r o c e s s on a h y p e r f i n i t e l a t t i c e F w h i c h r e p r e s e n t s a f i n i t e r e c t a n g l e i n R . I f t h e r e e x i s t p o s i t i v e r e a l numbers 3 , . . . , 3 , y , . . . , y , K such t h a t f o r k = l , . . . , d h- d 1 d k I i k U u x _ U I <_ K I x - y | , whenever x, y e T a r e such t h a t x - y l i e s a l o n g t h e k*^1 c o o r d i n a t e a x i s , t h e n i f 6, < f, /$, , k = l , . . . , d t h e r e k k k o i s a s e t fi' c fi o f Loeb 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 on fi' , , 6 k and a c o n s t a n t c , su c h t h a t f o r k = l , . . . , d U - U < c x - y ' x y' — ' - - 1 whenever x, y e T, | x - y | < 6(w) and x - y l i e s a l o n g t h e k ^ 1 c o o r d i n a t e a x i s . I n p a r t i c u l a r U i s a.s. S - c o n t i n u o u s on T . P r o o f : See S t o l l (1984)t Lemma 3.2. The r e s u l t he s t a t e s i s n o t as d e t a i l e d as 3.8.1, b u t h i s p r o o f i s s u f f i c i e n t . Theorem 3.8.2. The h y p e r f i n i t e p r o c e s s u - ( - x ^ a ) ^ c o n s t r u c t e d by t h e s o l u t i o n o f C3-5), i s a.s. S - c o n t i n u o u s on ne a r s t a n d a r d p o i n t s i n T x X . Moreover, i f < 4" a n d ^2 < \ a n d A c X i s a r e c t a n g l e whose s i d e s have f i n i t 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 1, a p o s i t i v e r e a l f u n c t i o n 6(a)). on fi' , and a c o n s t a n t c , depen d i n g on 3 , 3 max U , a , and 1 2 yeX °l V t , s u c h t h a t w e f i ' , x, y e A , t , r < t 36 1 y l U t x ' U r y ' - c ( ' t ~ 1̂ + |x - y| ) , i f | t - r | + |x-y| < 6 . P r o o f : P i c k q e R + such t h a t 8 < - — — and 8 < q 2 . By lemmas 1 2q 2 2q 3.6.1 and 3.7.1, t h e r e a r e c o n s t a n t s c such t h a t I |2q I 12+(q/2 - 2) E V - V < c t - r ^ ; — 1 t x r x 1 — '- -' I I 2 q I i 2+(q-2) — 1 t x t y — 1 - - 1 Hence by Theorem 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 V i n p l a c e o f U on any s e t T x A , where A i s an i n t e r n a l f i n i t e r e c t a n g l e i n X ,- b u t t h e n e a r s t a n d a r d p a r t o f X i s a a - u n i o n o f such A . Now U = U + V and U 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 i s - a - s o l u t i o n - t o t h e h e a t e q u a t i o n . 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 f orm (3-7) f o r U y i e l d s q u i c k l y t h a t U. - U < c ( t - r + x - y ) t x r y — — — — — Hence t h e theorem i s t r u e f o r U = U + V . • I n g e n e r a l t h e exponents j and ^ a r e b e s t p o s s i b l e (see Walsh (1986), C o r o l l a r y 3 . 4 ) . We may a l l o w s l i g h t l y more g e n e r a l i n i t i a l c o n d i t i o n s , i f we a r e p r e p a r e d t o r e l a x t h e c o n c l u s i o n s l i g h t l y . The arguments i n lemmas 3.6.1 and 3.7.1 depend o n l y on t h e boundedness o f max |u I . Thus t h e boundedness o f U y £ X ' °T 0 i s enough t o e n s u r e t h e S - c o n t i n u i t y o f U - U . However, i n t h i s c a s e U i t s e l f w i l l n o t be S - c o n t i n u o u s i n t h e monad o f z e r o . I f TJ i s bounded b u t 0 d i s c o n t i n u o u s t h e c o n c l u s i o n o f t h e theorem w i l l have t o be r e s t r i c t e d t o 3 7 °t, °r > 0 . R e t u r n i n g t o t h e case when i s c o n t i n u o u s , we f i n d t h a t a n o t h e r way 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 U i s n e a r s t a n d a r d a l m o s t s u r e l y i n * C ( [ 0 , t f ] x R : R) . Hence we may d e f i n e a p r o c e s s U(OJ) o as t h e s t a n d a r d p a r t o f U(OJ) , o r e q u i v a l e n t l y u = U f o r a l l o o t x t , x t , x , a.s. I t i s c l e a r t h a t t h a t s e t ft1 = {co :\/n e N, 3k e N , V x, y £ X , t , r < t , | x - y | < — , It - r l < — =>|u - U | < —} i s i n t h e - - — - f 1 - - ' k 1 - - — k t x £ Z n a - a l g e b r a g e n e r a t e d by t h e i n t e r n a l s e t s , hence i s L o e b - m e a s u r e a b l e . Thus t h e p r o c e s s u t x(o» = i °Utx^oO" f o r any ( t , x ) K ( t , x ) , i f co e ft' 0 , i f co i Q1 has sample p a t h s i n C ( [ 0 , t ] x R : R) . We a r e o f c o u r s e r e a l l y i n t e r e s t e d i n s o l u t i o n s f o r a l l t . Thus t a k e t h e h y p e r f i n i t e t i m e l i n e T up t o some i n f i n i t e number L , and t a k e ft 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 T x X . C o n s t r u c t t h e s o l u t i o n as b e f o r e . A l l theorems 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 any f i n i t e t i m e t . Thus we have C o r o l l a r y 3 . 8 . 3 . The s o l u t i o n on T x X , where T now i s an i n f i n i t e s i m a l g r i d r e p r e s e n t i n g R + , c o n s t r u c t e d from t h e d i f f e r e n c e e q u a t i o n s C3-5I i s S - c o n t i n u o u s i n (°|x| < «>} n {°'t < °°} , a.s. o We n o t e a l s o t h a t u i s 0CF ) m e a s u r e a b l e f o r any s w i t h s > t t - s - - hence by d e f i n i t i o n Cin s e c t i o n 2 . 3 ) u i s F^_-adapted. 38 3.9 S o l u t i o n o f t h e SPDE We now show t h a t t h e p r o c e s s u o f s e c t i o n 3.8 i s i n f a c t a s o l u t i o n t x o f (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 0, i n s e c t i o n s 2.2 and 3.2. We must check c o n d i t i o n ( 3 - 4 ) . We need f i r s t a new d e f i n i t i o n . D e f i n i t i o n 3.9.1. An i n t e r n a l f u n c t i o n $ on an i n f i n i t e s i m a l l a t t i c e X £ R d , i s c a l l e d a l i f t i n g t o o r d e r k o f a f u n c t i o n cj> : R d -> R , i f °6 (. . . (6 $)...) (x) = <J) (° x) f o r a l l x e X . Here 6 x. x. 3x, . . . 3 X . - x. *1 \ \ \ i s t h e f i n i t e d i f f e r e n c e o p e r a t o r i n t h e d i r e c t i o n x.: (6 ( $ ) ( x ) = i x. l [§(x,,...,x.+Ax,...,x ,) - $ ( x , , . . . , x ) ] / A x . A consequence o f t h i s d e f i n i t i o n -± ~ i -d - l -d i s t h a t f o r n e a r s t a n d a r d x, °[$(x+Ax) - 23>(x) +$ (x-Ax) ] / A x 2 = - ^ r - <j>(°x) 8x i f $ i s a l i f t i n g t o o r d e r 2 o f (j) . We n o t e t h a t e v e r y C f u n c t i o n cj> has a c a n o n i c a l l i f t i n g t o o r d e r k , * namely <f> r e s t r i c t e d t o t h e l a t t i c e . oo Now f i x any e C ( R) , and l e t $ be a l i f t i n g t o o r d e r 2 w h i c h i s o e x a c t l y 0 on v a l u e s o f x whose s t a n d a r d p a r t s x l i e o u t s i d e t h e c l o s e d s u p p o r t * (j) , ( t o a v o i d ( u n n e c e s s a r y ) c o n c e r n o v e r t h e convergence o f - c o u n t a b l e sums) . Then U • $ (x) i s a ( u n i f o r m ) l i f t i n g o f u d> (x) a.s. f o r any t ss t . Thus t x - t x | u t x < H x ) d x - I u 0 x«Mx)dx a = S * °l ( U t x - U 0 x ) «(;, Ax 39 xeX 0<s<t u 4 - u s+At,x s,x A t •At • $ ( x ) - A x ' I 1 0<s<t xeX U . 2 U + u SyX+Ax s x s,x-Ax Ax F(U ) sx $(x) + — T I Z I Z - ? $ ( X ) /At Ax -- Ax A t (3-11) I 0<s<t xeX 0(x+Ax) - 2$(x) +'$(x-Ax) I U _ - Ax A t sx Ax + ° I I F ( U g x ) $(x) /AtAx E 0<s<t xeX sx Now U C$ (x+Ax) - 2$(x) + $ ( x - A x ) ] / A x i s a u n i f o r m l i f t i n g a . s. o f u <f>" (x) s x sx Hence t h e f i r s t t e r m on t h e r . h . s . o f (3-11) i s a.s. u c(>"(x)dx • R sx Now F ( U ) $ (x) i s a (u n i f o r m ) F - a d a p t e d l i f t i n g o f f ( u )4>(x) sx — s sx Hence by Theorem 2.3.2, t h e second term on t h e r . h . s . o f (3-11) i s a.s. e q u a l 't t o 0 R f (u )<j)(x)dw . Thus we have, d) e C ( R) t e R sx s x c (3-12) u <i> (x) dx -t x R R u c|> (x) dx a = S ' Ox R u (j) (x) + sx 0 R f ( u )<Mx)dw S , X SX Theorem 3.9.2. There i s a Loeb space , on w h i c h any e q u a t i o n o f t h e form (3-1) has a s o l u t i o n j o i n t l y c o n t i n u o u s i n t and x w i t h r e s p e c t t o t h e c a n o n i c a l w h i t e n o i s e on 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 i n t e r n a l t r a n s f o r m a t i o n p r i n c i p l e t o t h i s Loeb space fi (see K e i s l e r (1984)),. I n t h i s c a s e , t h e e q u a t i o n (3-1) has a s o l u t i o n w i t h r e s p e c t t o any w h i t e n o i s e on R s u p p o r t e d by 0, . Walsh (1986) has 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 case when f i L i p s h i t z . P r e s u m a b l y t h i s i s f a l s e i n g e n e r a l b u t we do n o t know a c o n v e n i e n t c o u n t e r e x a m p l e . 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 I n t r o d u c t i o n The 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 s u c c e s s f u l i n C h a p t e r 3 f o r one s p a t i a l d i m e n s i o n may be a t t e m p t e d f o r h i g h e r d i m e n s i o n s as w e l l . T h i s a p p r o a c h does n o t s u c c e e d t o the same e x t e n t as i t does i n one d i m e n s i o n , as i s s p e l l e d o u t i n A p p e n d i x B. However i t does s u c c e e d w i t h t h e 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 , as w i l l be e x p l a i n e d h e r e . I t 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 n o n - s t a n d a r d l y , by p l a c i n g , on a h y p e r f i n i t e g r i d X , r e p r e s e n t i n g a p o r t i o n o f R , d->"-l , an i n f i n i t e number o f p a r t i c l e s , each e x e c u t i n g an i n f i n i t e s i m a l random w a l k , and u n d e r g o i n g b r a n c h i n g . I f we l e t U ^ s t a n d f o r the d e n s i t y _of„particles a t any g r i d p o i n t x e X , a t t i m e t , t h e n , i f the i n i t i a l , d e n s i t y ;. i s - t a k e n l a r g e enough, i t i s p o s s i b l e t o show t h a t U s a t i s f i e s a 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 o f t h e f o r m °* t u = A U + Z - t x where Z i s an i n t e r n a l n o i s e w i t h E(Z. ) = 0 , E (Z u t x ' ' t x | U t x ' _ d * — — AtAx Hence Z may be w r i t t e n as /u w where W i s l i k e an S-white n o i s e i n many r e s p e c t s . T h i s n o i s e W i s however, a l i t t l e awkward t o work w i t h , and t h e r e f o r e we 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 s e t o u t i n 4.2. I n 4.4 we examine t h e t o t a l mass o f t h e p r o c e s s c o n s t r u c t e d i n 4.2, and use t h i s i n 4.5 t o e s t a b l i s h 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 e a s i l y t h a t t h e s t a n d a r d p a r t i s w e l l - d e f i n e d . I n 4.6 we v e r i f y t h a t t h i s s t a n d a r d p a r t 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 s e v e r a l new r e s u l t s a b o ut 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 , u s i n g n o n s t a n d a r d c o n s t r u c t i o n . 43 4.2 A 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 L e t Ax be any i n f i n i t e s i m a l , and l e t X be { x i x = (k Ax,...,k Ax) , k. e Z , d > 1} so t h a t X r e p r e s e n t s R . - - 1 d I W(: w i l l 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 the Dawson P r o c e s s o n l y on t h e whole o f R . A v e r y s i m i l a r t r e a t m e n t i s p o s s i b l e i f r e f l e c t i n g boundary c o n d i t i o n s 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 R d , o r a l o n g t h e edges o f a r e c t a n g l e i n R , 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 i n d e e d depend on t h o s e f o r t h e unbounded domain. L e t A t be an i n f i n i t e s i m a l , s u c h t h a t A t / A x d £ 0 . T h i s makes some 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 . I n cas e d = 1 we r e q u i r e 1 2 A t < j & , as i n C h a p t e r 3. L e t T be a h y p e r f i n i t e t i m e l i n e o f s p a c i n g A t : T = { t : t = k A t , k e * Z + , k £ M} . We w i l l suppose 0 < ° (MAt) < °° . L e t t = MAt . TxX I t i s e a s i e r , and 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} and t o l e t E (co) be t h e c o o r d i n a t e map, a s - . o u t l i n e d i n s e c t i o n 2.2. Thus t x - •. . PCE = 11 = P{E = -1} = i . - t x t x 2 The a n a l o g u e o f e q u a t i o n (3-5) i n h i g h e r d i m e n s i o n s i s : C4-11 C6.U) = CAU) + A — - I E , t t x - t x \ ./ d 2At I t x o r e q u i v a l e n t l y , U , = U + AtCAu) +1 / A 1 E , U t + A t , x u t x t x % A 2 / t x (r:•) where 6 I s a f i n i t e d i f f e r e n c e a n a l o g u e o f — : t 9 t ( 6 t U ) t x = ^ u t + ^ t x - u t x ^ / A t , and A i s t h e f i n i t e d i f f e r e n c e a n a l o g u e o f t h e L a p l a c i a n i n R : C4-2) CAU) 2 / n ( U t ( x ,...,x.+Ax,...,x ) x j_i=l - -1 - l -d + U 4 - / A \ 2d U . t ( x ,...,x.-Ax,...,x ) t ( x , , .. .,x ) - -1 - i -d - -1 -d The t e r m /u A t u t x i s s u b s t i t u t e d f o r s i m p l y / U A t t x i n ( 4 - 1 ) , Ax - - * A x - i n o r d e r t o e n s u r e , t h a t , a n o n - n e g a t i v e v a l u e a t ( 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 t i m e s t e p . The v a l u e s o f U f o r w h i c h t h e l i n e a r t e r m i s t a k e n a r e 4 A t 0 < U < Ax d ' F o r an i n i t i a l c o n d i t i o n f o r (4-1) we may use any n o n - n e g a t i v e i n t e r n a l def r . d f u n c t i o n on X , r e q u i r i n g o n l y t h a t I u Ax , t h e t o t a l xeX Ox i n i t i a l mass, be f i n i t e and t h a t t h e mass on p o i n t s o f X w h i c h a r e n o t n e a r -^standard be i n f i n i t e s i m a l i n sum. We may r e p r e s e n t any f i n i t e p o s i t i v e d d B o r e l measure on R (we w i l l denote t h e space o f a l l s u c h measures ( R )) by such, an i n t e r n a l f u n c t i o n U Q (see C u t l a n d (1983) Theorem 4.7 and p r e c e e d i n g r e m a r k s ) . 45 4.3 The C o e f f i c i e n t s Q "u At U SY sy From (4-1) and (4-2) we o b s e r v e t h a t each v a l u e o f / A — — £ . d 2 sy Ax -- e n t e r s i n t o t h e d e f i n i t i o n o f su b s e q u e n t U 's . We denote t h e c o e f f i c i e n t t x t - s o f t h e f o r m e r t e r m 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 0~ ~ , o b s e r v i n g t h a t x-y t h e s e c o e f f i c i e n t s a r e homogeneous i n space and t i m e . We may t h e n w r i t e t h e an a l o g u e o f a Green's f u n c t i o n f o r m u l a : U A t U sy £ t x K~y \ * A D 2 I sy y x-y Oy 0<s<t Y £ / ^ - - v Ax / -- yeX - - t+ A t Lemma 4.3.1. The c o e f f i c i e n t s Q- 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 i n t e r n a l i n f i n i t e s i m a l random w a l k B on X . The s t a n d a r d p a r t o f B t t i s a.s. d - d i m e n s i o n a l B r o w n i a n m o t i o n o f r a t e 2. P r o o f : As 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 f o r t h e c o e f f i c i e n t s Q- from (4-1) and ( 4 - 2 ) . We o b s e r v e t h a t t h e s e d i f f e r e n c e x e q u a t i o n s c o r r e s p o n d t o a Markov p r o c e s s , B , on X , w i t h p a r a m e t e r t e T where B̂ _ s t a r t s a t 0 e X , and a t each t i m e s t e p B t a k e s a s t e p t o one o f t h e 2d n e a r e s t n e i g h b o u r s o f i t s c u r r e n t p o s i t i o n , w i t h i n f i n i t e s i m a l 2 p r o b a b i l i t y a = At/Ax f o r each o f t h e 2d p o s s i b i l i t i e s . The p r o c e s s s t a y s p u t w i t h p r o b a b i l i t y 1 - 2da ~ 1 . Q- +^ t = P(B = x) . X t ~ i t h L e t B denote t h e i c o o r d i n a t e o f B . S i n c e s t e p s i n o p p o s i t e t t d i r e c t i o n s have e q u a l p r o b a b i l i t y , B 1 i s an i n t e r n a l m a r t i n g a l e . A l s o E ( :(Bt+A-t " B^). 2 | B s ; 0 <_ s <_ t ) = 2 a A x 2 = 2At . Hence t h e ( i n t e r n a l ) 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 < B l > t = 2 t • U s i n g B u r k h o l d e r ' s I n e q u a l i t y i ± 2 on t h e h i g h e r moments o f , we may c o n c l u d e t h a t (B^_) i s S - i n t e g r a b l e t . Now a s t e p i n one d i r e c t i o n e x c l u d e s a s t e p i n any o t h e r d i r e c t i o n d u r i n g t h e same ti m e s t e p . Hence E[ (B-1 . - B 3) ( B 1 „ - B 1) ] = 0 i f i ^ j . t+At t t+At t S u c c e s s i v e s t e p s a r e i n d e p e n d e n t , so t h a t t h e i n t e r n a l p r o c e s s <B 1, B^> = 2 6 . . t . Now we i n v o k e Hoover and P e r k i n s (1983) Theorem 8.5 t o a s s e r t t h a t B has a.s. a s t a n d a r d p a r t b , and we o b s e r v e t h a t b s a t i s f i e s i ) b ^ i s a m a r t i n g a l e i = l , . . . , d i i ) <b 1, b 3> = 26 t i , j = l , . . . , d t i : ( a g a i n w i t h r e f e r e n c e t o Hoover and P e r k i n s ( 1 9 8 3 ) ) . Now i ) and i i ) above c h a r a c t e r i z e d - d i m e n s i o n a l B r o w n i a n M o t i o n . • We need a l i t t l e more i n f o r m a t i o n a b out t h e Q's . o Q - 2 x" II || Lemma 4.3.2: I f °t > 0 , — = P (°x) , where P (x) = 1 exp{ } : — ^ — ' ~ i d t — t r— 4 t Ax - » 4TT t Q-x i s 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. Hence i s S - c o n t i n u o u s i n A x d t f o r °t > 0 . P r o o f : 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 f o r B i s t h e same as t h a t f o r bQ^_ , w h i c h has d e n s i t y P0^_ , f o r any °t > 0 . The s t a t e m e n t o f t h e lemma w i l l f o l l o w t h e n , i f Q-/Ax d i s s - c o n t i n u o u s f o r °t > 0 . x 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 h o l d s : C4-4I ( 6 t Q x ) = (A Q ^ ) x Now i t i s c l e a r from t h e d e f i n i t i o n o f t h e random w a l k B t h a t t h e c o e f f i c i e n t s Q - a r e s y m m e t r i c w i t h r e s p e c t t o change o f s i g n on any o f th e d i n d i c e s x C l a i m : F o r each t , x , Q - > Q - . _ f o r k = l , . . . , d , where e - - x — x+Axe, k - . k r e p r e s e n t s a u n i t v e c t o r i n t h e x 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 .. We p r o v e t h i s by * - f i n i t e i n d u c t i o n on t . I t i s c l e a r l y t r u e f o r t = A t At x Q = P(B = x) = 6 - . Suppose t h e c l a i m h o l d s f o r some t e T . Then by x 0 - 0 - * (4-4) and ( 4 - 2 ) , a C4-5) Q t + t = ( l - 2 d a ) Q - + a ( £ Q- . ) , and x x . . x.+ Axe j = l d t+At t , r t (4-6) Q _ . = ( l - 2 d a ) Q - _ . + a ) n- _ Now i f x 5̂ .0 t h e n a l l terms a p p e a r i n g i n (4-5) and (4-6) above l i e -k i n t h e same h a l f - p l a n e ; each t e r m i n (4-6) i s s h i f t e d by e Ax r e l a t i v e t o , k and hence, by th 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 t e r m i n ( 4 - 5 ) . Hence Q- > Q- , - . x - x+Axe, - k Now suppose ( w . l . o . g . k = 1) x = 0 . We p i c k one o f t h e two p o s s i b i l i t i e s f o r $ and s t i c k w i t h i t . L e t y be a (k-1) t u p l e . By o u r a s s u m p t i o n s on A t , i n any d i m e n s i o n d , (2d+l ) a <_ 1 . Hence (4-7) C l - ( 2 d + l ) a ) Q § x > ( l - ( 2 d + l ) a ) Q ^ x > ( 1 - ( 2 d + 2 ) a ) Q | x ^ + a p * ^ Now a d d i n g a O- + a 07 + a Q~ t o b o t h s i d e s o f (4-7) and u s i n g Ax,y -Ax,y t t , Q- = Q- , we o b t a i n ' -Ax,y Ax,y (4-8) (l-2dcO + a(Qj + Q - ) >_ (l-2da)Cj£ + a (Q^ + Q | ) Oy Ax,y -Ax,y Ax,y Oy 2Ax,y Now by t h e i n d u c t i o n a s s u m p t i o n d t d t C4-9) a 1 Q 5 y + A x e . > « I 2Zx,y +Axe ' j = 2 u y - j j=2 j A d d i n g (4-8) and (4-9) we o b t a i n t + A t t+At 2 5y ^Zx.y * T h i s e s t a b l i s h e s t h e c l a i m f o r { Q ^ + ^ t | x e X} . 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 f o r a l l t e l . Now by Lemma 4.3.1, f o r any i n t e r n a l r e c t a n g l e A X d d xeA Ax (4-10) I —=- Ax c P (x)dx , f o r °t > 0 I n l i g h t o f t h e m o n o t o n i c i t y c l a i m j u s t e s t a b l i s h e d , t h i s means °(Q-/Ax d) = P Q (°x) i f °t > 0 , and x e ns(X) . T h e r e f o r e a l s o i s S - c o n t i n u o u s i n t f o r t > 0 , s i n c e P (x . d - - t Ax i s c o n t i n u o u s 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 bounds i n s e c t i o n 4.5. I t has 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 t h e n o t a t i o n A : Y - xeA V Y Lemma 4.3.3. There i s a c o n s t a n t K < °° and a p o s i t i v e i n f i n i t e s i m a l A t ' 49 such t h a t f o r a l l i n t e r n a l s e t s A , and a l l y e X , whenever t , r , s e T and r - s > t - r > A t ' , t - s r - s Q- - - Q- -A:y A:y ( t - r ) < K-- ( r - s ) P r o o f : Suppose f i r s t t h a t °(t-r) > 0 Now c l e a r l y f o r f i x e d t , r and s , max IQ- - - Q- -1 = Q- - - Q- - A,y A : y - A ^ Vt,r,s:y \'t,r,s:y where A = {x e X|Q- § < Q- -} y , t , r , s - x-y x-y The s t a n d a r d p a r t o f t h i s s e t i s easy t o i d e n t i f y , u s i n g Lemma 4.3.2. I t i s t h e b a l l A ^ = {x e R d| Mx-y|| 2 < 2d ^ - s ) ( r - s ) c ^ r y , t , r , s 1 1 1 J n 2 ~ ( t - r ) r - s ' where y = °y , t = °t , r = °r and s = °s . The r a d i u s 6 o f t h i s - - - - t , r , s b a l l i s computed by s o l v i n g P^ ^(x) = P (x) - i . e . [ 4 T r ( r - s ) ] d/2 e x p [ - 4 ( r - s ) -] = [ 4 7 T ( t - s ) ] d/2 exp[- 4 ( t - s ) -] d/2 r - s 4 ( r - s ) ( t - s ) .2 o J ( t - s ) r - s . r t - s , <=> 6 = 2d logC ] t - s r - s Now (P (x-y) - P ( x - y ) ) d x r - s J t - s y , t , r , s 50 a - l ' • t , r ' S 1 , 2,«< „ d - l exp(-p / 4 ( r - s ) ) p dp , (4tr) d/2 J ( r - s ) d/2 d - 1 1 (4TT) .d/2 TT) } t . r . s 1 , 2,., d - 1 ' ' exp(-p / 4 ( t - s ) ) p dp ( t - s ) d/2 where S i s t h e a r e a o f t h e s u r f a c e , S, , , o f t h e u n i t b a l l i n R d - l d-1 Now l e t p1 = p^/tTs" i n t h e f i r s t i n t e g r a l , and p' = p/2/r-s i n t h e s e c o n d , t o o b t a i n (4TT) d/2 j [ t , r , s / 2 / r - s ^ _ p2 P e t , r ,s/2/t -s d - 1 -p S i n c e t h e f u n c t i o n f(p) = p e i s bounded, we may f i n d K t o bound t h i s i n t e g r a l by K 6 ( ~ i t , r , s / r— v r - 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 bound t h i s f u r t h e r , by KpEE ( ( !2 } " ( r ' s ) _ ) r S / t - r ( / t ^ s ~ + / r - s ) < K t - r r - s Thus, i f °(t-r) > 0 (4-11) r - s > t - r = > sup \c£ - ( t - r ) Q£-s. I < K - 7 ^ T - , Vy e X AcX A:y A:y — ( r - s ) 51 S i n c e t h e i n t e r n a l s t a t e m e n t (4-11) i s t r u e f o r t - r > e f o r a l l r e a l e > 0 , t h e n by t h e p r i n c i p l e o f i n f i n i t e s i m a l o v e r f l o w i t must h o l d f o r t - r > A t ' ss o . • We b e l i e v e t h a t t h i s Lemma i s t r u e f o r t - r down t o A t b u t an i n t e r n a l p r o o f o f t h i s i s n o t e a s y . 52 4.4 The T o t a l Mass P r o c e s s M Now l e t M = ) U Ax be t h e t o t a l mass o f t h e i n t e r n a l measure t L v t x xeX — whose d e n s i t y U i s o b t a i n e d by s o l v i n g (4-1) i n d u c t i v e l y . We have by d e f i n i t i o n , /U A t U 0<s<t yeX x eX i ~ l y Ax r r r t-s / s x d I n what f o l l o w s c d e n o t e s a f l e x i b l e c o n s t a n t whose v a l u e s depend o n l y on q and t , and w h i c h v a l u e s may change f r o m l i n e t o l i n e . Lemma 4.4.1. There i s a c o n s t a n t c d e p e n d i n g on q > 1 , M , and t ^ , such t h a t \jt <_ t , E ( M 2 q ) < c e C t P r o o f : U s i n g Theorem 3.4.5 ( B u r k h o l d e r ' s I n e q u a l i t y ) , t h e f a c t t h a t v t - s ) Q- - = 1 and t h e f a c t t h a t f o r any a, b, a A b < a v x-y — xeX - i E ( M 2 q ) < c E ( | I I 1 • U A t A x d | ) - t — - 1 ^ L sy 1 0<s<t yeX -£ d ,2 • * (4-12) + c E ( max U AtAx £ ) " 0 < s < t ^ 5* yeX + c M 2 q 0 Now t h e s e c o n d t e r m i n (4-12) above may be bounded by c E ( max | M A t | q ) <_ c E| \ M A t | q . Hence (4-12) becomes 0<s<t - 0<s<t - 0<s<t r 0 E ( M 2 q ) < c EI T M A t - t — - „ _ s < c E| I M A t l ^ + c(M 2 q+D ~ ' 0<s<t 5 0 (4-13) < c y E ( M 2 q ) A t + c ( M 2 q + l ) — « . s 0 0<s<t f o r a l l t <_ t f . We complete t h e p r o o f w i t h an a p p e a l t o t h e a p p r o p r i a t e v e r s i o n o f G r o n w a l l ' 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 M, - M as f o l l o w s , i n o r d e r t o t r 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 . L e t q > 2 , and w . l . o . g . t a k e 0 <_ r < t <_ t • Then ( 4 - 1 4 ) E | M - M I 2 5 < c E I J 1 U A t A x d | q t r ' — - L i_ v sy 1 r<s<t yeX -- ( u s i n g B u r k h o l d e r ' s I n e q u a l i t y and a p p l y i n g t h e same r e a s o n i n g as i n g o i n g f r o m ( 4 - 1 2 ) t o ( 4 - 1 3 ) ) q < c E | I M " ( t - r ) q r<s<t ^ ( t - r ) 1 - - < c E ( I M q - ^ ) ( t - r ) q s sx.  s t - r r<s<t - - - ( a p p l y i n g J e n s e n ' 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 — — ) on t h e i n t e r v a l r < s < t ) t - r - - - 54 < c I Ellfl) ( t - r ) q ~ r<s<t " S ( t - r ) - - (4-15) <_ c e x p ( c t f ) ( t - r ) H , by Lemma 4.4.1. Thus we have Lemma 4.4.2. The t o t a l mass p r o c e s s M / d e f i n e d above, i s a. s . S - c o n t i n u o u s on 0 < t < t . P r o o f : A p p l y t h e Kolmogorov C o n t i n u i t y C r i t e r i o n (3.8.1) t o ( 4 - 1 5 ) . • I n f a c t M i s S-Holder c o n t i n u o u s o f o r d e r — - e f o r any E > 0 . 55 4.5 S - C o r i t i n u i t y o f t h e P r o c e s s F o r i n t e r n a l s e t s A , and i n t e r n a l f u n c t i o n s F l e t t h e n o t a t i o n s X^ and X*" s t a n d f o r T U A x d and J F ( x ) U A x d . We i n t r o d u c e t h e t u t x L v - t x x£A — xeX c l a s s o f ( s t a n d a r d ) f u n c t i o n s C 2 = { f e C 2 ( R d) | 3K < °° such t h a t | f ( x ) | + I A f (x) | <_ K \/x e R d} . R e c a l l from s e c t i o n 3.9 t h a t i f an i n t e r n a l f u n c t i o n F l i f t s t o o r d e r 2 a f u n c t i o n f e C, „ , t h e n b, 2 °(AF)(x)= Af(°x) , Vx e n s ( X ) . Lemma 4.5.1. I f F i s an i n t e r n a l f u n c t i o n s u c h t h a t | F ( X ) | + | A F ( X ) | <_ K , Vx e X , t h e n t h e p r o c e s s X F i s S - c o n t i n u o u s on T a.s. P r o o f : F o r r < t e T , 7* - X F = Y F ( x ) (U - U ) A x d t r y - t x r x x e X r s+At,x s x - i l IF(x) L ————J AtAx r<s<t x r r/Usx Usx i = y i F ( X ) (Au ) - A ; , ^ ^ L . - S - X V A A d 2At sx r<s<t x — — V A+Ax A t A x d (4-16) = I I ( A F ) ( x ) U A x d A t , S X r<s<t x u sx . d + I [ F ( x ) ( ^ A _ p g / A t A x d ^ ^ L - sx 2 A t sx r<s<t x 56 The f i r s t t e r m i n (4-16) above we r e c o g n i z e as I X A t r<s<t - AF Now X- < K M . Hence i f q > 1 s s - . -p j?. 2q _ 2q E X - X < c E ( ) M At) -' t r 1 - - L s r<s<t - + c E( I I F 2 ( x ) u A x d A t ) q r < s < t xeX ( u s i n g B u r k h o l d e r ' s I n e q u a l i t y f o r t h e second t e r m on t h e r i g h t and i n c o r p o r a t i n g t h e t e r m i n v o l v i n g "max" as we d i d i n (4-12)) < c I E ( M 2 q ) ( t - r ) r<s<t 2q s t - r - -' + c I E ( I F 2 ( x ) U A x d ) q A t — \/ — S X r<s<t xeX r | F 2 , ^ F 2 2 (we now r e c o g n i z e t h e s e c o n d terms as l E | X | A t and X <_ K M ) r<s<t - - - (4-17) < c ( t - r ) 2 q + c ( t - r ) q , i f t t , u s i n g J e n s e n ' s I n e q u a l i t y a s e c o n d t i m e . Now we a p p l y t h e N.S. v e r s i o n o f t h e Kolmogorov C o n t i n u i t y C r i t e r i o n (Theorem 3.8.1). • 57 A Lemma 4.5.2. U m sup °X n = ° a - S * where n-*>° t e [ 0 , t f ] " A = {x e X| |x| >_ n} ; t h a t i s , X i s n e a r s t a n d a r d i n M ( R d) f o r a l l n t F t , a.s. P r o o f : L e t H be i n f i n i t e . L e t F be an i n t e r n a l f u n c t i o n s u c h t h a t 0 < _ F < _ 1 , F = 1 on A , F = 0 on A^ and AF i s bounded. Then H H —1 — E(<> < E C X / " 1 ) - 1 ( 1 Q- t + A t) U A x d . Now by Lemma 4.3.1, f o r each y e ns (X ) r - t - s r „t+At ) Q s 1 , so t h a t e = ) Q- z 0 . By a s s u m p t i o n U Q i s n e a r s t a n d a r d l y c o n c e n t r a t e d , so t h a t l i m °( I U A x d ) = 0 . n - X » y£A - n Hence, (4-18) Etxf) < I e U ~ 0 => xl s 0 a.s. t " y e X X °Z t F Now by Lemma 4.5.1 X i s a.s. S - c o n t i n u o u s . There 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 w h i c h (4-18) h o l d s . Hence A °X F = O U t e T , a . s . => sup °X H = 0 a.s. ~ " t e C 0 , t f ] £ A ° n P T h e r e f o r e sup X — > 0 as n -> 0 0 , t e C O,t f] £ A A n n + l , n b u t s i n c e X > X we must have convergence a .s. U 58 Theorem 4.5.3. X : T M ( R ) i s a.s. S - c o n t i n u o u s . Thus X i s F n e a r s t a n d a r d i n * C ( R ; ( R d ) ) , and ( ° X ) q = L ( X ) 0 s t - 1 f o r a l l t t e [ 0 , t ] a.s. P r o o f : R e c a l l t h e weak t o p o l o g y on Mp( R d) : + x <=> f o r a l l bounded c o n t i n u o u s f , x ( f ) -»- x ( f ) . L e t {(j) } be a c o u n t a b l e c o l l e c t i o n o f n T k 2 cl CT _ ( R ) f u n c t i o n s w h i c h c o n s t i t u t e a convergence d e t e r m i n i n g c l a s s f o r b, 2 M ( R ) . L e t {$ } be l i f t i n g s t o o r d e r 2 o f t h e {cj> } . Then by Theorem •t k k $ 4.5.1 e a c h X^_ i s a . s . S - c o n t i n u o u s . Now by t h e Loeb c o n s t r u c t i o n (see s e c t i o n 2.2) (4-19) <h d L ( X ) o s t = °X k T h e r e f o r e t h e l . h . s . o f (4-19) i s c o n t i n u o u s a .s. f o r each <j) i n t h e convergence d e t e r m i n i n g c l a s s . Hence, a l m o s t s u r e l y L ( X ) ° s t - 1 = L ( X ) 0 s t - 1 t s f o r a l l t s s e T . * d From Lemmas 4.4.1 and 4.5.2, X i s i n f a c t n e a r s t a n d a r d i n ( R ) f o r a l l t , a . s . , so t h a t ° (X^) e x i s t s and e q u a l s L ( X f c ) ° st~J- \^ t a.s. • Now we show a s t r o n g e r form o f c o n t i n u i t y . * d Theorem 4.5.4. L e t X be n e a r s t a n d a r d i n M ( R ) (as d i s c u s s e d a t t h e ' ' 0 F end o f s e c t i o n 4 .2). L e t A be i n t e r n a l . Then X i s S - c o n t i n u o u s f o r t °t > 0 on a c o a r s e r g r i d T' c T o f i n f i n i t e s i m a l s p a c i n g A t ' i n d e p e n d e n t o f A P r o o f : We t a k e A t ' from Lemma 4.3.3. L e t Y 1 = T Q - + A t U A x d and l e t t L v A:y Oy TEA - - 59 U Y 2 = t I I Q j . - ( Jv AtAx' 0<s<t yeX A : Y sy • ^ A x d ) £ 2 sy Then X A = vJ; + Y 2 t t t From Lemma 4.3.3, t h e r e i s a K s u c h t h a t , i f r < t < 2 r i n 7" , |Q- - Q- I < K ( t - r ) / r . Hence i f t ~ r and °r > 0 - - - A:y A:y — - - - - I |0j - O f | U A x d y e X A : x A : y °y < — ( t - r ) M — r - - 0 ~ 0 . Thus Y^ i s S - c o n t i n u o u s i n T' n s t - 1 ( t > 0 ) . Now suppose £ < t e T ' , t - r < l , 0 < y < l f and q >_ 2 . Then u s i n g B u r k h o l d e r ' s I n e q u a l i t y and a b s o r b i n g t h e terms i n v o l v i n g a maximum i n t o t h e summations, as we d i d i n s e c t i o n 4.4, ((4-12) and ( 4 - 1 3 ) ) , i 2 2 i 2 c2 E Y I - Y < c E — t r 1 — + C E + c E < C E 0<s<r - ( t - r ) Y ye Y (Q- - - Q - -) U AtA: u ^ A:y A:y sy d q I v yeX X Y sy r - ( t - r ) ' < s < r i I I (Q | . 5 ) 2 U AtA < ^. A:y sy r<s<t y£A u -± I 2-2Y Ct-r) M A t 0<s< r - ( t - r ) (we i n v o k e Lemma 4.3.3 and n o t e ( r - s ) > ( t - r ) .) 60 + C E q 1 1 • M A t r - ( t - r ) Y < s < r - ( s i n c e |pjf - - -| < 1) 1 A:y A:y' — + c E T M A t 1 r<s<t - q (4-20) < c ( ( t - r ) ( 2 - 2 Y ) q + ( t - r ) Y q + ( t - r ) q ) ( f o l l o w i n g t h e u s u a l J e n s e n ' s i n e q u a l i t y t r a i n o f development, by now so f a m i l i a r ) . Our f a i t h f u l s e r v a n t t h e c o n s t a n t c has now a c q u i r e d dependence on Mo ' 0 1 ' - f ' a n d q ' k u t n o t o n t a n d - • 2 The b e s t c h o i c e o f y f o r (4-20) seems t o be — . U s i n g t h e Kolmogorov c r i t e r i o n (Theorem 3.8.1) we g e t 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 i f p u t s a l l i t s mass on a n u l l s e t A , we g e t f a i l u r e o f S - c o n t i n u i t y i n t h e monad { t « o} . C o r o l l a r y 4.5.5. L e t F be any bounded i n t e r n a l f u n c t i o n on X . Then F i — 1 t h e p r o c e s s X̂ _ i s S - c o n t i n u o u s on T n s t (t>0) , a.s. P r o o f : L e t K be a bound f o r | F | . L e t X F = y y F ( x ) o~ + A t U A x d , and t v v ~ x-y Oy yeX xeX - *• - U l e t zf = I I • ( I F ( x ) QI~JT) ( /u_AtAx d A A x d ) ^ yeJ t w v - x - y sy 2 sy 0<s<t yeX xeX _ £ -£ z °£ F " F ~ F . „. • -r< Then X* = X + X f c . Now i f t > r e T 61 |x F - x F | < K | X { f > 0 } - x { F > 0 } | + K | X { f < 0 } - x { F < 0 } 1 t r 1 — 1 t r ' 1 t r F o l l o w i n g t h e f i r s t p a r t o f Lemma 4.5.4, X F i s S - c o n t i n u o u s i n T ' n {°t > 0} . Now i f q > 1 , E | X F - X F | 2 q < c E| I 1 ( 1 F ( x ) ( Q ^ - Q x : | ) ) 2 U A t A x d | 2 q 0<s<r yeX x e X - £ - - -£ + c E | I I K 2 u A t A x d | 2 q " r<s<t yeX Now C £ F ( x ) CQ- - - Q|~|) ) 2 < 2 K 2 sup (Q| - - <g~-)2 < c — by Lemma 4.3.3. Hence f o l l o w i n g t h e s e c o n d p a r t o f t h e argument i n 4.5.4 we o b t a i n t h e s i m i l a r r e s u l t I~F ~ F I 2 q , ,2q/3 E X - X ^ < c ( t - r ) - t r — - - and t h u s ~X i s S - c o n t i n u o u s on T a.s. • Remark: We b e l i e v e t h a t X i s H o l d e r S - c o n t i n u o u s o f i n d e x 2 ~ z ^ o r any z > 0 . Remark: I f we t o o k t f t o be h y p e r f i n i t e , °t = °° , t h e n 4.5.3, 4.5.4 and 4.5.5 w i l l h o l d a .s. u n t i l a l l f i n i t e t i m e s , and hence a.s. on n s ( T ) . We w i l l use 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. 62 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 P r o b l e m We show now t h a t t h e m e a s u r e - v a l u e d p r o c e s s x , w h i c h we c o n s t r u c t e d i n t h e l a s t s e c t i o n as L ( X t ) o s t _ 1 i s i n f a c t t h e measure d i f f u s i o n s t u d i e d by Dawson and o t h e r s . We w i l l use one o f t h e m a r t i n g a l e c h a r a c t e r i z a t i o n s 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 ) , w h i c h a r e r e f i n e m e n t s o f t h e 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 S t r o o c k (1978), Dawson and K u r t z (1982) and o t h e r s . The 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 b r a n c h i n g measure d i f f u s i o n x^ g i v e n i n R o e l l y - C o p p o l e t t a (1986; Theorem 1.3, c o n d i t i o n i i i ) ) i s as f o l l o w s : Theorem 4.6.1. x i s t h e (unique) M ( R d) v a l u e d p r o c e s s w i t h w e a k l y t F c o n t i n u o u s p a t h s , and t h e g i v e n i n i t i a l c o n d i t i o n s u c h t h a t f o r a l l $ e D(A) (the domain o f t h e L a p l a c i a n A <j> d e f „ a ) mt = Xt " xo x A$ ds i s a c o n t i n u o u s m a r t i n g a l e . 0 s <mr> t t ,2 x • ds . 0 s We know a l r e a d y (Lemma 4.5.3) t h a t o u r p r o c e s s i s w e a k l y c o n t i n u o u s i n M C R ^ ) • We w i l l v e r i f y a) and b) above f o r f u n c t i o n s <|> e ( R ^ ) o n l y . F b / 2 The e x t e n s i o n t o D(A) (which i s t h e c l o s u r e under A o f C, „( R ) i n b, 2 C (R ) ) may be done by a l i m i t i n g argument. I t i s i n f a c t u n n e c e s s a r y t o b 2 d Ms. R o e l l y - C o p p o l e t t a ' s argument i n h e r Theorem 1.3 t o go beyond C ( R ) . b, 2 F o r s u c h a <{> l e t $ be a l i f t i n g t o o r d e r 2, and suppose |$| <_ K . Then x<f> = r <M°x)d L ( X t ) , t e T' , a.s. °t h s (X) ( r e c a l l f rom 4.5.2 t h a t t h e mass on X \ ns (X ) i s a l w a y s i n f i n i t e s i m a l a.s.) ° r d ) $(x) U Ax , by Loeb L i f t i n g Theorem (Theorem 2.1.6) X — t x o $ = X t Hence, <j> 4> <j) m t = x t " x o t A * x ds 0 5 i s i n d i s t i n g u i s h a b l e from t h e s t a n d a r d p a r t o f $ d e f $ $ r A* M = x - x - J x f A t - 0<s<t - = I $ ( x ) ( U - U ) A x d - I I u (A$)(x) A x d A t xeX - Ox sx - - (U , A i ~ U ) s+Atx sx = I I *(£) rr A t A x d 0<s<t xeX A t d Y l $(x) (A U ) AtAx 0<s<t xeX - - ( u s i n g t h e i n f i n i t e s i m a l a n a l o g u e o f i n t e g r a t i o n - b y - p a r t s ) (4-21) y y *(x)[7u A t A x d A HsX A x d ] ? « . L Y - s x 2 — sy 0<s<t xeX 1 $ Now i s a m a r t i n g a l e w i t h r e s p e c t t o t h e i n t e r n a l f i l t r a t i o n F̂ _ . Hence irr = M V t e T , a.s. i m p l i e s t h a t irr i s an F - m a r t i n g a l e , by °t - " fc t Theorem 5.2 o f Hoover and P e r k i n s (1983). T h i s v e r i f i e s a) o f Theorem 4.6.1. F u r t h e r , s i n c e m^ i s a.s. c o n t i n u o u s i n t , t h e p r e d i c t a b l e s q uare f u n c t i o n <m̂ > c o i n c i d e s a.s. w i t h t h e s q u a r e f u n c t i o n [m^] . Now by ^ t a n o t h e r r e s u l t o f Hoover and P e r k i n s (1983) (Theorem 6.7) [ s t ( M t ) ] = s t [ M ] , t h e s t a n d a r d p a r t o f t h e i n t e r n a l s q u a r e f u n c t i o n , and by Theorem 8.5 o f t h e same p a p e r , 64 CM ] T ~ <M > v t , a . s . , $ 2 $ p r o v i d e d (M^) i s S - i n t e g r a b l e , where <M > i s t h e i n t e r n a l p r e d i c t a b l e ( w i t h r e s p e c t t o F̂ _) square p r o c e s s . Now by 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 E ( sup ( M $ ) P ) < c E " t < t £ ~ P " T l $ 2 ( x ) U A x d A t u •• - s x 0<s<t xeX P/2 < c K E - p \ M A t 0<s<t p/2 (where M i s t h e t o t a l mass p r o c e s s ) s w h i c h i s f i n i t e by Lemma 4.4.1. $ 2 Now a p p l y Chebychev's I n e q u a l i t y t o c o n c l u d e t h a t (M ) i s S - i n t e g r a b l e as r e q u i r e d . Now <M $ > 0<s<t xeX (4-22) \ I $ (x) U AtA? 0<s<t xeX I I $ 2 ( x ) I tr sx d ( — — - U ) A t A x d v - r 4 A t , 4 A t sx 0<s<t xeX {u < — - i ~ sx A d Ax Now t h e second term above may be bounded by r, - j. ~v sx , 4Atl 0<s<t xeX — {u < TJ sx A d r I I i - A X { X < L} AtAx M l 2 I I u i / u A x d A t 0<s<t xeX — {u < — - r } { x > L> — Ax < K 2 • t • ( 2 L ) d • ^ A x d (4-23) + K 2 I X ^ ' - L } A t 0<s<t - By 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) goes t o 0 a.s. as L -»• 0 0 . The f i r s t t e r m i s i n f i n i t e s i m a l f o r a l l L f i n i t e . Hence, f o r each t 2 ,» . \ <k a.s. 0 ,$ a.s. ° v $ A a.s (4-24) <nr> = <M > = ) X A t t t „ L s 1 1 - 0<s<t -  f - S 2 x ds , 0 s as r e q u i r e d by c o n d i t i o n b) o f Theorem 4.6.1. Hence Theorem 4.6.2. The s t a n d a r d p a r t x f c o f t h e h y p e r f i n i t e p r o c e s s X c o n s t r u c t e d 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 (4-1) f o r an i n t e r n a l d e n s i t y U , i s t h e 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 c o n s i d e r e d by Dawson and o t h e r s . 4.7 New R e s u l t s on t h e Dawson Measure V a l u e d D i f f u s i o n T h i s s e c t i o n r e p r e s e n t s j o i n t work w i t h Ed P e r k i n s , my t h e s i s s u p e r v i s o r . Now t h a t we know t h a t t h e measure v a l u e d p r o c e s s x̂ _ w h i c h we c o n s t r u c t e d v i a s o l u t i o n o f t h e 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 (4-1) f o r an i n t e r n a l " d e n s i t y " TJ , i s i n f a c t a c o n s t r u c t i o n o f the Dawson P r o c e s s , we may use t-X t h i s c o n s t r u c t i o n 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 t h e p r o c e s s . F i r s t we need Lemma 4.7.1. L e t A c X be a Loeb-measureable s e t o f measure z e r o . Then on a s e t fi o f p r o b a b i l i t y 1 L ( X )(A) = 0 ,V t e T* n s t _ 1 ( t > 0 ) . P r o o f . L e t {A } „ be a n e s t e d sequence o f i n t e r n a l s e t s s u c h t h a t A c A n neN - n f o r e ach n e N , and ° (y (A ) ) ->• 0 . E x t e n d {A } t o *N i n such a way x n n t h a t t h e A 's a r e s t i l l n e s t e d ( t h a t t h i s may be done, f o l l o w s from n w ^ - s a t u r a t i o n ) . L e t H be i n f i n i t e i n *N . Then u (A ) ss o . Suppose t e T' , x H — w i t h °t > 0 . Now by Lemma 4.3.2, i f K i s a bound f o r p (x) , Q- t + A t °- H i *€A„ Ax — rl ECX^ H) = EC I A x d ) = I U A x d < K y x ( A H ) M 0 = 0 . x e A H — yeX H — - t A H Thus f o r each t t h e r e i s a s e t fi- o f p r o b a b i l i t y 1 on w h i c h X ~ 0 L e t { t }, be a c o u n t a b l e S-dense s e t i n T' . L e t fi' be t h e s e t o f -k keN , 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 - c o n t i n u o u s f o r t e T . L e t fiTT = n fi n n . Then on fi , (X ) = l i m (x. ) = 0 f o r some k = l %c H t ° t V t ^ -k. - l H \J sequence { t } . Thus X x 0 V t e T' , hence by i n f i n i t e s i m a l u n d e r f l o w — . t ~~ 1 - f o r each to e ft , t h e r e i s an i n f i n i t e s i m a l e(co) such t h a t rl A F o r n e *N , l e t Y = sup X n A 1 . Then Y i s S-L f o r each n , and n teT' t the sequence {Y } i s i n t e r n a l . Now E(Y ) x o , f o r any i n f i n i t e n ne N H J H e *N . Hence ° (E (Y )) 4- 0 as n =° t h r o u g h N , and th u s °Y 4-0 on - n n A a s e t ft o f p r o b a b i l i t y one. Now L ( X ) ( A ) <^L(X )(A )= °X n < • °Y V/t e T' , V/n e N • Hence L ( X )(A) = 0 , \ / t e T' on ft . • t - A Hence, we may draw C o r o l l a r y 4.7.2. I f A c R d i s a Lebesgue n u l l s e t (X(A) = 0) , t h e n x (A) = 0 Vt > 0 , a . s . P r o o f : st_1CA) i s a Loeb n u l l s e t . • We i n t r o d u c e t h e n o t a t i o n s x^ = t f dx. and x = x (A) d f Theorem 4.7.3. I f f : R •+ R i s a bounded m e a s u r e a b l e f u n c t i o n , t h e n x i s a . s. c o n t i n u o u s i n t > 0 . I n p a r t i c u l a r x̂ _ i s a.s. c o n t i n u o u s i n t > 0 f o r any Lebesgue s e t A . P r o o f : L e t F : X *R be a l i f t i n g w i t h r e s p e c t t o L ( y ) o f f . Then x l e t A be {x : x e X °F(x) ^ f(°x)} w h i c h i s a Loeb n u l l s e t . By Lemma 4.7.1, L ( X ) CA) = 0 \/t e T' a.s. Now F i s S - L 1 o f (X,Xfc) V t a.s. ( s i n c e M i s f i n i t e Vt a.s.) . Hence x f = f °F(x)dL(X ) = °X F Vt e T' t v - o f c j - t t - F a.s. By Lemma 4.5.5 X i s S - c o n t i n u o u s a .s. t 68 Theorem 4.7.4. L e t <j> be a bounded measureable f u n c t i o n on R , and suppose a sequence {A } o f u n i f o r m l y bounded m e a s u r e a b l e f u n c t i o n s k keM c o n v e r g e s t o 6 i n Lebesgue measure. Then f o r any e > 0 sup Ix - x | — > 0 , H e , t f ] t t P r o o f : L e t $ and $ k e N , be l i f t i n g s o f <)> and <j> , k e N JC JC r e s p e c t i v e l y . E x t e n d {$, }, t o an i n t e r n a l sequence {$ } k e *N . k keN k Then i f °t > 0 ' E ( X k ) = ° J u" ( * ( X ) - $ ( x ) ) A x d t v t x k — xeX u (<)>(x) - <j> ( x ) ) d x where u = s t ( U ) . -i ° x R d t x -> 0 as k -»• 0 0 , s i n c e u i s bounded and L 1 ( R) f o r t x t > 0 . * Hence i f H e N i s i n f i n i t e , (4-25) ° E ( X H ) = 0 => X H ~ 0 a.s. Vt e T . $ $ H i -1 Now X^ and X̂ _ -are b o t h S - c o n t i n u o u s on T n s t (t>0) a.s. by Lemma 5.5.2. T h e r e f o r e s i n c e (4-25) h o l d s f o r a c o u n t a b l e S-dense s u b s e t $ " $ H o f T , hence X ~ Q Vt e T' a . s . , and de f „ H Y„ = max X ^ 0 a.s. H t e T ' £ T h e r e f o r e Y^ — > 0 as k ->- 0 0 t h r o u g h N . 69 Now we have s e e n (Lemma 4.7.3) t h a t f o r k £ N ( r e s p . x^_) and k $ s t ( X t ) ( r e s p . s t ( X t ) ) a r e i n d i s t i n g u i s h a b l e p r o c e s s e s . Hence I k 4> I a.s. i ki ° P „ n sup | x^ - x Y = o max X = Y — > 0 • t € [ e , t f j t t teT' * k 70 CHAPTER 5 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 D i m e n s i o n 5.1 I n t r o d u c t i o n I n C h a p t e r Three we o b t a i n e d an e x i s t e n c e theorem f o r SPDEs o f t h e fo r m 3u — = Au + f ( u ) w t 3 t t x where f 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 o f i m p o s i n g a L i p s h i t z c o n d i t i o n on f . As d i s c u s s e d i n C h a p t e r One, t h e one d i m e n s i o n a l 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 has been b e l i e v e d t o s a t i s f y t h e SPDE ,,. l i 3 2 u A /- • (.5-1) — = — - + Zu w d t „ 2 t x d X However t h e t h e o r y o f such an e q u a t i o n has n o t been w e l l - k n o w n s i n c e t h e f u n c t i o n u -> i s n o t L i p s h i t z . I n t h i s s e c t i o n we use t h e r e s u l t s o f C h a p t e r Three t o show t h a t t h e 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 o f Dawson does i n d e e d s a t i s f y (5-1) i n one s p a t i a l d i m e n s i o n , and t h u s i t has a.s. a j o i n t l y c o n t i n u o u s d e n s i t y . 5.2 The SPDE and t h e Measure D i f f u s i o n TxX L e t d = 1 , and l e t fi be t h e space {-1, 1} as d e s c r i b e d i n s e c t i o n 4.2. L e t U 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 d i f f e r e n c e t x e q u a t i o n s ( 4 - 1 ) : U £ t x . t x (5-2) 6 U. = A U + ( / r T - A — - -*-) " , t t x - t x t x 2 A t v ^ x - and l e t X be t h e m e a s u r e - v a l u e d 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 U . 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 x^ = °(X^) was i n f a c t t h e 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 t h e cas e d = 1 however, we have t h e a d d i t i o n a l i n f o r m a t i o n t h a t i f U i s S - c o n t i n u o u s and S-L''" , t h e n U 0 t x i s S - c o n t i n u o u s on T x X a . s . by C o r o l l a r y 3.8.3. F u r t h e r m o r e u = °U s a t i s f i e s ( 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 o f u as the s o l u t i o n t o ( 5 - 1 ) , we need o n l y n o t e t h a t , s i n c e x 0 , t h e n f o r u e ns T R) On t h e o t h e r hand, s t a r t i n g from t h e SPDE (5-1) we note t h a t t h e 2 m a r t i n g a l e p r o b l e m i s e a s i l y s a t i s f i e d s i n c e f o r <f> e C ( R) c t Ao <Kx)dw 0 s x s x R 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 2 u A ( x ) d s d x sx R I f we a l s o assume t h a t u Q i s L^ , t h e n rt o u dsdx < 0 0 a.s. sx and a d o m i n a t e d convergence argument e n s u r e s t h a t t h e m a r t i n g a l e p r o b l e m i s 2 s a t i s f i e d f o r <J> e C ( R) (which c o i n c i d e s w i t h D(A) i n t h i s case) . b, 2. Thus we have Theorem 5.2.1: L e t u_ (x) be c o n t i n u o u s and L ( R) . L e t w^ be the — 0 t x TxX w h i t e n o i s e c o n s t r u c t e d i n s e c t i o n 2.2 f r o m t h e Loeb space o f {-1, 1} o n t o R + x R . There i s a j o i n t l y c o n t i n u o u s n o n - n e g a t i v e p r o c e s s u s u c h t h a t U 0 x = U 0 ( X ) ' and f o r a l l 6 e „( R) , and a l l t e R + b, z C5-3) u t x<t)(x)dx = U 0 x ^ ̂  d x + R 32d) u — - (x) dxds s x „ 2 R dx 0 R /u * (x) dw sx s x Moreover any s o l u t i o n t o (.5-3) i s t h e j o i n t l y c o n t i n u o u s d e n s i t y o f t h e un i q u e m e a s u r e - v a l u e d s o l u t i o n t o the m a r t i n g a l e p r o b l e m 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 c o n t i n u o u s d e n s i t y h o l d s f o r a l l r e a l i z a t i o n s o f t h e o n e - d i m e n s i o n a l Dawson b r a n c h i n g d i f f u s i o n , n o t o n l y t h e one c o n s t r u c t e d h e r e . T h i s f o l l o w s from the f a c t t h a t t h e e x i s t e n c e o f a j o i n t l y c o n t i n u o u s 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 t h i s p r o c e s s Csee C u t l e r ( 1 9 8 5 ) ) . CHAPTER SIX The S u p p o r t o f t h e F l e m i n g - V i o t P r o c e s s 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 As m e n t i o n e d i n C h a p t e r One, the F l e m i n g - V i o t p r o c e s s i s t h e l i m i t i n g c a s e o f a model u s e d i n t h e o r e t i c a l g e n e t i c s : t h e Ohta-Kimura model f o r d q u a n t i t a t i v e c h a r a c t e r s . B r i e f l y , i n t h i s model, the t o t a l number o f i n d i v i d u a l s i s c o n s e r v e d , and the dynamics i n v o l v e two p r o c e s s e s , g e n e t i c " d r i f t " , and m u t a t i o n . M u t a t i o n i s m o d e l l e d by a random w a l k on Z " D r i f t " i s m o d e l l e d by r e p l a c i n g i n d i v i d u a l s a t random by new i n d i v i d u a l s whose g e n e t i c t y p e matches a n o t h e r i n d i v i d u a l c h osen a t random from t h e r e s t o f t h e p o p u l a t i o n . I f we denote t h e t y p e s by p o i n t s k e Z d , t h e n we may denote by 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 k a l i v e a t t i m e t e R + , d i v i d e d by N , t h e t o t a l number o f i n d i v i d u a l s ( c o n s e r v e d ) , t h e n p C t , - ) forms a c o n t i n u o u s - t i m e c o u n t a b l e s t a t e space Markov jump p r o c e s s w i t h g e n e r a t o r : C6-1) L f f p ) . = I l ^ j e Z [y p ( i ) p ( j ) + D p ( i ) 9 i . ] ( f ( p 1 D ) - f (p)) where p 1" 1 (k) p ( k ) + i - , i f k = j { p ( k ) - , i f k = i P ( k ) o t h e r w i s e , and f 1, i f = 1 e -2d, i f = 3 0, o t h e r w i s e , and 74 y and D a r e p o s i t i v e c o n s t a n t s d e s c r i b i n g t h e r a t e s o f " d r i f t " and m u t a t i o n r e s p e c t i v e l y . E a c h o f the N i n d i v i d u a l s t a k e s one s t e p ("mutates") i n one o f t h e 2d p o s s i b l e d i r e c t i o n s a t P o i s s o n t i m e s whose r a t e i s D/N . S i m i l a r i l y each i n d i v i d u a l d i e s and i s r e p l a c e d by a n o t h e r whose t y p e c o i n c i d e s w i t h t h a t o f a g i v e n o t h e r o f t h e N-1 i n d i v i d u a l s ( i . e . an " o f f s p r i n g " o f t h a t 9 i n d i v i d u a l ) 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 w i t h r a t e Y / N To 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 r e - s c a l e t i m e and space by 1/N 2 and 1 / N 1 / 2 r e s p e c t i v e l y : f o r A e B( R d) l e t (6-2) X A = T p ( N 2 t , k) — eA t a k e s v a l u e s i n M^( R d) , t h e space o f p r o b a b i l i t y measures on R d . The r e s u l t o f F l e m i n g and V i o t (1979) i s t h a t i n t h e l i m i t as N •+ 0 0 , t h e m e a s u r e - v a l u e d p r o c e s s d e f i n e d by (6-2) c o n v e r g e s i n t h e space o f (R ) - v a l u e d p r o c e s s e s p r o v i d e d t h e i n i t i a l measure c o n v e r g e s i n M ( R d) . We r e f e r t o F l e m i n g and V i o t (1979) and Dawson and Hochberg (1982) f o r more d e t a i l s on t h i s . However, a n o n - s t a n d a r d c o n s t r u c t i o n o f t h e F l e m i n g - V i o t p r o c e s s i s immediate from ( 6 - 2 ) . We s i m p l y t a k e N i n f i n i t e and l e t t h e s t a t e space be *Z . S t a t e d i n n o n - s t a n d a r d language t h e r e s u l t o f F l e m i n g and V i o t i s * + Theorem 6.1.1. F o r N i n f i n i t e , and t e n . s . ( R ) (6-2) d e f i n e s a.s. a p r o c e s s w i t h S - c o n t i n u o u s p a t h s i n n s ( ( R )) , i f i n i t i a l l y a l l b u t an * d r- i n f i n i t e s i m a l f r a c t i o n o f t h e 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 . Note 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 space i n (6-2) i s * + -1 + t e c h n i c a l l y R , t h a t X ~ X"_ i n t h e weak t o p o l o g y whenever t ~ r e s t (R ) , a.s. Hence X has a s t a n d a r d p a r t , x , w h i c h i s a m e a s u r e - v a l u e d S t r o n g Markov p r o c e s s c a l l e d t h e F l e m i n g - V i o t P r o c e s s . 7 5 6.2 The D i m e n s i o n o f a P u t a t i v e S u p p o r t S e t We use h e r e t h e t e r m i n o l o g y o f Dawson and Hochberg (1982) and some o f t h e i r r e s u l t s . What we a i m t o show i n t h i s c h a p t e r i s t h a t a n o n - s t a n d a r d c o n s t r u c t i o n makes much o f t h e i r work more n a t u r a l (and a good d e a l eas i e r ' . ) as w e l l as e x t e n d i n g t h e i r r e s u l t s . The main r e s u l t (Theorem 6.4.4) o f t h i s c h a p t e r a s s e r t s t h a t t h e d i m e n s i o n o f t h e s u p p o r t o f t h e F l e m i n g - V i o t p r o c e s s i s a t most 2 f o r a l l t i m e s s i m u l t a n e o u s l y , a . s . (Dawson and Hochberg were a b l e t o show t h i s o n l y f o r f i x e d t i m e s ) . I n t h i s s e c t i o n we d e r i v e t h e 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 w i l l be shown t o be t h e s u p p o r t i n g s e t . C o n s i d e r a p a r t i c l e ( o r i n d i v i d u a l ) a l i v e a t some t i m e t . As t h e p r o c e s s e v o l v e s and t h e p a r t i c l e wanders, a t some t i m e r > t t h i s p a r t i c l e may d i s a p p e a r ( 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 somewhere e l s e ) , o r e l s e i t may s e r v e as t h e 'type-model' f o r t h e r e p l a c e m e n t o f some o t h e r p a r t i c l e w h i c h d i s a p p e a r s a t t i m e r . I n t h i s l a t t e r c a s e we say t h a t b o t h p a r t i c l e s a t time r a r e ' d e s c e n d a n t s ' o f t h e o r i g i n a l p a r t i c l e a t t i m e t . F o r ease o f t e r m i n o l o g y we w i l l say t h a t t h e p a r t i c l e a t any t i m e s > t , up u n t i l t h e t i m e o f d i s a p p e a r a n c e o f t h a t p a r t i c l e , i s t h e d e s c e n d a n t o f t h a t p a r t i c l e a t t i m e t . 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 r e l a t i o n . F u r t h e r m o r e e v e r y p a r t i c l e a t t i m e t has a u n i q u e a n c e s t o r a t any t i m e s < t ; i f we f o l l o w t h e p a t h s o f p a r t i c l e s b a ckward i n t i m e , t h e y may c o n v e r g e , b u t t h e y w i l l n e v e r s p l i t . Two p a r t i c l e s a t a t i m e r > t a r e s a i d t o have a common a n c e s t o r a t time t , i f t h e y a r e b o t h d e s c e n d a n t s o f a g i v e n p a r t i c l e a t t i m e t . We w i l l 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 N p a r t i c l e s a t any t i m e , by l o o k i n g f o r a s m a l l ( f i n i t e ) s e t o f a n c e s t o r s , a t an e a r l i e r t i m e whose d e s c e n d a n t s c o m p r i s e a l l o f t h e N p a r t i c l e s a t t h e l a t e r t i m e . 76 C o n s i d e r any f i n i t e t i m e i n t e r v a l [ 0 , T] . L e t e > 0 , and l e t {a^} and Â̂} be s t r i c t l y d e c r e a s i n g sequences o f p o s i t i v e numbers s u c h a2/A t h a t A e n n -> oo and n a2+E/A 0 as n -* 0 0 . (A c o n v e n i e n t c h o i c e w o u l d be a = A (4+e)/ (4+2e) ̂  ̂ n n n n We suppose w . l . o . g . t h a t {T/Â } a r e a l l i n t e g e r s so t h a t f o r each n , t h e ( f i n i t e ) s e t s { t | t = k A , 0 < k < T/A } f o r m a p a r t i t i o n o f n t JC n / Jc n — — n CO, T] . We l e t N be t h e number o f a n c e s t o r s a t t i m e t o f t h e n,k n , k - l N p a r t i c l e s a l i v e a t t i m e t , . L e t A . be t h e u n i o n o f b a l l s o f n,k n,k r a d i u s a , c e n t e r e d a t each o f t h e N , a n c e s t o r s a t t i m e t , , o f n n,k n , k - l t h e s y s t e m o f N p a r t i c l e s a t t i m e t . L e t A be t h e u n i o n o f n,k n,k a. s m a l l e r b a l l s o f r a d i u s —— , c e n t e r e d a t t h e same p o i n t s . ( T e c h n i c a l l y a l l o f t h e above a r e n o n - s t a n d a r d ( i n t e r n a l ) o b j e c t s , b u t s i n c e t h e y a r e a l l n e a r - s t a n d a r d I w i l l make th e d i s t i n c t i o n between s t a n d a r d and i n t e r n a l o n l y when n e c e s s a r y . ) L e t k (t) = Ct/A ] i d e n t i f y i n w h i c h i n t e r v a l Ct , , t , , ) , t n n n,k n,k+l l i e s . Then l e t (6-3) A. = u n s t ( A ) t i n,k ( t ) m=l n=m n w h i c h i s a ( s t a n d a r d ) B o r e l s e t f o r each s t a n d a r d t i n Co, T] . T h i s w i l l t u r n o u t t o be o u r s u p p o r t i n g s e t . The d i m e n s i o n o f t h i s s e t depends on t h e number o f a n c e s t o r s a t t i m e t , , o f t h e N p a r t i c l e s a t t h e t i m e s t , (and n,k * n,k hence o f a l l t h e p a r t i c l e s a l i v e between t i m e s t , and t , ) . n,k n,k Dawson and Hochberg t r e a t e d t h i s p r o b l e m f o r f i x e d t i m e s by c o n s t r u c t i n g an i n f i n i t e p a r t i c l e s y s t e m t o d e s c r i b e t h e F l e m i n g - V i o t p r o c e s s a t t h a t p a r t i c u l a r t i m e . U s i n g o u r h y p e r f i n i t e model, we c a n use t h e same s y s t e m o f N p a r t i c l e s a t a l l t i m e s . Dawson and Hochberg (1982; (6.23)) showed t h a t N t h e d i s t r i b u t i o n o f th e t i m e T t a k e n ( i n r e v e r s e t i m e ) t o re d u c e t h e m number o f a n c e s t o r s o f N p a r t i c l e s t o m o n l y , had L a p l a c e t r a n s f o r m - S T N , (6-4) E ( e = H? . (1 + -f-T,) • k=m+l y k ( k - l ) ( T h e i r argument a p p l i e s verbatim t o o u r h y p e r f i n i t e scheme.) We may use (6-4) t o e s t i m a t e P (N > _ £ . ) = P ( ° T N / a > °A ) f o r i n t e g r a l -2- . From (6-3) n,k A c/A n A n n n (6-5) E ( T N / A ) = I } =J^- 4 ~ ^ , a n d - c/A L y k ( k + l ) Y n Y c n k= + 1 A A n n N N H A C / A _ ~ r . . , . „ . , n 2 k= A n n , c + 1 [ Y k ( k - l ) ] V 1 r 1 l - i 2 3 3' k- - £ + 1 3 Y ( k"D k A 1 n 2 , c 3 ~ 3 J ~ 2 3 Y ) N 3y c A 3/2 oA °A Now by Chebychev's I n e q u a l i t y , P ( T ^ ^ > + h — — j — ) < — , n Y f3yc h hence, t a k i n g h = °/5c (yc-1)/SE~ , n (6-6) P ( N , > T5-) < A n ( f o r c > Y " 1 ) . n,k A - n 3 C ( Y C - 1 ) 78 2 + E L e t 6 > 0 . L e t y £ ( A ) denote t h e H a u s d o r f f x measure o f a s e t Then PC max N , > 6 / a 2 + £ ] k<T/A n ' k — n < P(N > 6 / a 2 + e ) A n , l ' n ' n A < f- 2 _ _ _ ( F R O M ( 6 _ 6 ) ) n n , n . > 3 ~2+7 ( Y ~2+F " } a a n n (6-7) = A A 2 3 6 ( Y 6 ~2+F ~ 1 } a a n n 0 as n -> 0 0 s i n c e A n 2+e a n 2+e Hence P( max a N , , , > > < 5 ) " > " 0 » a n c j by t a k i n g a subsequence, i f t£[0,T] n n ' k n ( t ) n e c e s s a r y , we can e n s u r e 0+c (6-8) max a N , #J_, -> 0 a.s. t £ [ 0 , T ] n n ' k n ( t ) oo Now f o r each m , A , ... i s a c o v e r i n g o f n A , . by N , fJ_. m,k (t) n,k ( t ) 1 m,k ( t ) m n=m n m b a l l s o f r a d i u s a . Hence (6-8) e n s u r e s t h a t m y ( n A , ,,,) = 0 u n i f o r m l y i n t , a.s. z m n,k ( t ) 00 00 and hence y ( u n A , ,,..)= 0, f o r a l l t e [ 0 , T] a.s. T h i s i s t r u e e m=l m n - k n ( t ) 79 f o r any E > 0 . Lemma 6.2.1: The random s e t - v a l u e d f r u n c t i o n A d e f i n e d by (6-3) has H a u s d o r f f d i m e n s i o n a t most 2 f o r a l l t i m e , a . s . 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 E q u a t i o n We 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 group o f p a r t i c l e s . A t any t i m e , we s t o p t h e F l e m i n g - V i o t p r o c e s s , d e s i g n a t e n p a r t i c l e s o u t o f t h e N , and r e s t a r t t h i s (Markov) p r o c e s s . F o r t h i s s e c t i o n w . l . o . g . suppose t h e p r o c e s s i s r e - s t a r t e d a t t i m e 0 . L e t Y denote t h e mass a s s o c i a t e d w i t h t h e d e s i g n a t e d n p a r t i c l e s a t t i m e 0 and w i t h t h e i r d e s c e n d a n t s a t t i m e s t > 0 . Y = — (# o f p a r t i c l e s a l i v e a t t i m e t t N w h i c h a r e descended f r o m the o r i g i n a l n) . R e c a l l t h a t we a r e i n c l u d i n g p a r t i c l e s i n t h e o r i g i n a l n t h a t have r e m a i n e d unchanged u n t i l t i m e t , i n t h i s c o u n t . Now Ŷ _ changes o n l y when i ) one p a r t i c l e o u t o f t h e NY f c d i s a p p e a r s , and i s r e p l a c e d by a p a r t i c l e o f a type-model n o t i n c l u d e d i n t h e NŶ _ , o r i i ) one p a r t i c l e o u t o f t h e N-NY^_ e x c l u d e d p a r t i c l e s d i s a p p e a r s 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 o f t h e NŶ _ d e s c e n d a n t p a r t i c l e s . D i s a p p e a r a n c e s and r e p l a c e m e n t s h a p p e n i n g e n t i r e l y w i t h i n t h e c o n t e x t o f t h e NY^ d e s i g n a t e d p a r t i c l e s o r amongst t h e N-NY e x c l u d e d p a r t i c l e s , make no change t o Ŷ _ . Now t h e r e 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 a c c o r d i n g t o t h e t y p e model o f any o t h e r g i v e n p a r t i c l e 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 w i t h r a t e y (see ( 6 - 1 ) ) . There a r e NY • (N-NY^_) p o s s i b l e ways f o r an e v e n t o f t y p e i ) t o o c c u r each h a p p e n i n g 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 w i t h r a t e y , and each c a u s i n g a d e c r e a s e o f s i z e — i n Y. . There a r e (N-NY ) • NY p o s s i b l e ways f o r N t t t an e v e n t o f t y p e i i ) t o o c c u r , each w i t h t h e e f f e c t o f i n c r e a s i n g Y by and o c c u r r i n g a t the same r a t e y as an e v e n t o f t y p e i ) . Thus Y i s an ( i n t e r n a l ) m a r t i n g a l e . The a s s o c i a t e d p r e d i c t a b l e i n c r e a s i n g p r o c e s s i s e a s y t o compute, s i n c e t h e change t o Y a t any t i m e i s t h e sum o f 2NY t(N-NY^) i n d e p e n d e n t P o i s s o n p r o c e s s e s , each o f r a t e y , whose 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 — j . Thus N <Y> = t 2y Y (1-Y ) d t 0 * (6-9) d<Y> t = 2y Y t ( l - Y f c ) d t Hence by Hoover and P e r k i n s (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 * o Y a r e a.s. n e a r s t a n d a r d i n t h e space C((0,°°); R) . L e t y = Y . Then , t f r o m (6-9) d < Y > t = 2 Y Y t ( l - y ) d t and hence b̂ _ = dy i s o / 2 Y y t d - y t ) t a s t a n d a r d B r o w n i a n m o t i o n (up t i l l t h e t i m e o f e x t i n c t i o n o f y) . Thus by e n l a r g i n g o u r p r o b a b i l i t y space we may f i n d a B r o w n i a n m o t i o n b such t h a t (6-10) d y t = /2y y t ( l - y t > d b t F u r t h e r m o r e we have t h e f o l l o w i n g lemma. Lemma 6.3.1: Suppose y = e where 0 < e < — . Then t h e r e a r e f i n i t e 0 2 1 * Not e . Hoover and P e r k i n s (1983) Theorem 8.5 r e f e r s t o - d i s c r e t e t i m e p r o c e s s e s X , s u c h t h a t i f Ax i s t h e change i n X o v e r an i n f i n i t e s i m a l t i m e s t e p A t , t h e n sup °|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 2 , we may e n s u r e sup °~|AY| = 0 a.s. and b r i n g Y i n t o t h e framework o f t h e theorem q u o t e d . 82 c o n s t a n t s K^, K 2 i n d e p e n d e n t o f e , s u c h t h a t P ( 3s e [ 0 , t ] such t h a t y < ^ e) < K. exp C - r ^ - ] . s — 4 — 1 K ^ t P r o o f : W r i t e y f c as t h e t i m e change o f a B r o w n i a n m o t i o n . As l o n g as 3 5 Y f c s t a y s i n t h e range [- e, - E ] t h e d e r i v a t i v e o f t h e t i m e change must be a t l e a s t — ^ — ^ — • Now use t h e e s t i m a t e -2yeU-4e) P ( 3 s e [ 0 , t ] s u c h t h a t l b I > c) < 2 P ( | b I > c) s — ' t 1 — 2 c = 4 P ( b >_ c) < K e 2 t • 83 6.4 V e r i f i c a t i o n o f S u p p o r t We now check t h a t t h e random s e t A f c o f s e c t i o n 6.2 does i n d e e d s u p p o r t t h e measure x f c o f Theorem 6.1.1 (the F l e m i n g - V i o t P r o c e s s ) . Lemma 6.4.1: C o n s i d e r a t t i m e t any p a r t i c l e p . L e t s < t , and t r a c e t h e p a t h o f t h e (unique) a n c e s t o r o f p a t each t i m e r , f o r s <_ r < t . Then t h i s p a t h i s an i n f i n i t e s i m a l random w a l k whose s t a n d a r d p a r t i s a d - d i m e n s i o n a l B r o w n i a n m o t i o n o f r a t e 2D . P r o o f : Between t h e appearance and u l t i m a t e d i s a p p e a r a n c e o f any g i v e n p a r t i c l e , i t t a k e s a s t e p o f s i z e — f r o m i t s c u r r e n t p o s i t i o n t o any one / N o f t h e 2d n e i g h b o u r i n g p o s i t i o n s 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 w i t h r a t e DN . Replacement (which i s b i f u r c a t i o n o f an a n c e s t o r p a r t i c l e ) o c c u r s 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 w h i c h i s i n d e p e n d e n t o f t h e m o t i o n o f t h e p a r t i c l e . I f we i m a g i n e s u c h a m o t i o n c o n t i n u i n g i n d e f i n i t e l y and c a l l t h i s p r o c e s s B̂ _ , and i t s c o o r d i n a t e s B^ , t h e n c l e a r l y each B^ i s an i n t e r n a l m a r t i n g a l e , s i n c e s t e p s t o t h e r i g h t o c c u r a t t h e same r a t e as s t e p s t o t h e 1 2 l e f t . Now <B 1> = 2DNt( — ) = 2Dt , s i n c e B 1 i s t h e sum o f 2 i n d e p e n d e n t 1 / N P o i s s o n p r o c e s s e s , o f r a t e 2DN , and o f a m p l i t u d e —— . The m o t i o n s B 1 and / N ~B? happen a c c o r d i n g t o i n d e p e n d e n t P o i s s o n p r o c e s s e s , hence <B 1, B " 5 > t = 0 i f i ^ j . Thus by Hoover and P e r k i n s (1983) (once a g a i n l ) B^ has a s t a n d a r d p a r t b̂ _ a . s . , and t h i s b ^ s a t i s f i e s t h e c h a r a c t e r i z a t i o n o f d - d i m e n s i o n a l B r o w n i a n m o t i o n : E ( b b 0 < r < s ) = b ; <b> = 2 D I t . • t ' r — — s t 2 A g a i n we must l o o k a t t h e p r o c e s s B t a t d i s c r e t e i n t e r v a l s ( o f s i z e A t = 1/N) t o p u t i t i n t h e framework o f Theorem 8.5 o f t h i s p a p e r . 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 A , and A , . n,k n,k c K l ~ a n / K 2 A Lemma 6.4.2: P ( x ((°A , ) ) > £ ) < — °e n f o r f i n i t e c o n s t a n t s ° n,k — e n,k V K 2 • P r o o f : C o n s i d e r t h e f a m i l y o f d e s c e n d a n t s o f any one o f t h e N p a r t i c l e s n / JC a t t i m e t . . us e d i n d e f i n i n g A I f we t r a c e back f r o m t i m e t , n , k - l 3 n,k n,k t h e movements o f any one o f t h e p a r t i c l e s i n t h i s 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 one a n c e s t r a l p a r t i c l e a t t i m e t , , , n , k - l we f i n d a m o t i o n o f t h e k i n d d e s c r i b e d i n Lemma 6.4.1, whose s t a n d a r d p a r t i s a B r o w n i a n M o t i o n . 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 a t t i m e t n,k from i t s ' a n c e s t o r a t t i m e t ' , i s d i s t r i b u t e d N(0, 2DA i ) . F o r anv n , k - l n J p a r t i c l e p a t t i m e t , , n,k a„ -a 2/K A E ( I A C (P)) £ P ( B D A > - ) < K e A . n n,k Hence N -a 2/K A E ( x U°A f)\< ECX^ C A C ) ) = \ I E I (p.) < °K e n 2 n °, n,k — t . n,k N . - ;o I — 1 t . n,k 1=1 A , n,k n,k and c K-, „ - a 2 / K n A PCx (C°A V ) C ) > e) < - i °e n 2 n . o n,k e fcn,k c K o - V V n Lemma 6.4.3: P ( x ( ( A , ) ) > £ ) < — e o n,k e n,k+l P r o o f : As above, w i t h l a r g e r K 2 , s i n c e d i s p l a c e m e n t s a r e d i s t r i b u t e d N ( 0 , 4D A I ) . n L e t E > 0 . From Lemmas 6.4.2 and 6.4.3 we may deduce P ( max {x (°A C , ) v (°A° )} > T fcn,k n ' k Si.k+l n ' k 2 n K T -a 2/K A < — e n z n - v O as n <*> . - EA n Cas p e r u s u a l , o u r t r u s t y s e r v a n t s , t h e c o n s t a n t s and K 2 a r e c h a n g i n g v a l u e s when n e c e s s a r y ) . Hence a fortiori PC max ( x (°A C ) v x +. (°A C )} > f ) -> 0 . n Now c o n s i d e r t h e p o s s i b i l i t y t h a t t h e s e t s A , c o n t a i n 1 - — o f , n,k 2 t h e mass o f t h e p r o c e s s x a t t i m e s t , and t , , , b u t t h a t more t h a n t n,k n,k+l O E o f mass l i e s o u t s i d e t h e s e t s A , a t some t i m e i n between t , and n,k n,k t , , - L e t s , be t h e f i r s t s u c h t i m e , s i s a s t o p p i n g t i m e f o r n,k+l n,k n,k t h e Markov p r o c e s s x^ and we may c o n s i d e r i t r e s t a r t e d a t t i m e s , f r o m t n,k i t s c o n f i g u r a t i o n a t s , . One o f t h e f o l l o w i n g must o c c u r d u r i n g t h e n,k i n t e r v a l Cs t . ,.) • n,k n,k+l a) a t l e a s t one f o u r t h o f t h e mass e t h a t l i e s o u t s i d e A , i n i t i a l l y n ,k a t r a v e l s a d i s t a n c e — Cto r e - e n t e r A , ) 2 n,k b) t h e mass e t h a t l i e s o u t s i d e A , d e c r e a s e s by a t l e a s t one f o u r t h n,k e 3 Cthat i s t h e — N p a r t i c l e s have a t most e N d e s c e n d a n t s a t t i m e t n,k+l) C o n s i d e r c a s e a ) . F o r e a c h p a r t i c l e p a t t i m e t , , , t h e d i s p l a c e m e n t from i t s n,k+l a n c e s t o r a t ti m e s , i s d i s t r i b u t e d N ( 0 , D ( t , , - s ) I ) . R e c a l l n,k n,k+l n,k fro 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 any g i v e n s u b s e t o f p a r t i c l e s , forms 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 t i m e t • , o f d e s c e n d a n t s o f t h e e N p a r t i c l e s a t t i m e s , , w h i c h a r e n,k + l n n,k 3. o u t s i d e b a l l s o f r a d i u s — 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 i n i t i a l mass e , t i m e s t h e 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 i s o u t s i d e s u c h - a 2 / K 2 A a b a l l , w h i c h p r o d u c t i s bounded by e K e n (as i n Lemma 6.4.2). Hence u s i n g Chebychev's I n e q u a l i t y , " a n / K 2 A n P ( f o r a f i x e d n and k, case a) o c c u r s ) < K^e T K 1 T " a n / K 2 A n Hence P ( f o r some k < -— , s , < t , and case a) o c c u r s ) < — : — e A n,k n,k+l A n n w h i c h goes t o z e r o as n -*• 0 0 . Now by Lemma 6.3.1 T P ( f o r some k < — , s , < t , and case b) o c c u r s ) A n,k n,k+l n < — P ( 3 s < t , - s < A s . t . y = - E I v = e ) — A - n,k+l n , k n ys 4 1 Y 0 b ' n (where y i s t h e p r o c e s s m e n t i o n e d i n . 6 . 3 . 1 ) -e/K.A T 2 n - A ~ K 1 6 n 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 x^ (°A °, ) < -̂ and ( A C, ) < A t n,k 2 t n,k 2 n n,k n,k+l and 3 s e ( t . , t ) such t h a t x (°A C, ) > e) ->- 0 . n,k n,k+l s n,k 87 » C P Thus sup x ( A , ) — > 0 as n + t e [ 0 , T ] t n ' k n ( t ) By t a k i n g a subsequence, i f n e c e s s a r y , we may e n s u r e ° c a • s sup x ( A . .) —'—$ 0 . A g a i n by t a k i n g a subsequence we may a s s e r t t e [ 0 , T ] r n , K n m V sup x (°A C ) < n i l tcCO.T] t n ' k n ( t ) 0 0 a.s. Then sup x ( A ^ ) t e [ 0 , T ] t o c <_ sup x ( u A . , . ) f o r any m e N , t e [ 0 , T ] Z n=m n ' K n ^ > <_ sup I x (°A C ) te [ 0 , T ] n=m t n ' k n ( t ) < y sup x (°A C ~TL t £ [ 0 , T ] * n ' k n ( t ) -> 0 as m -> 0 0 Thus we have Theorem 6 . 4 . 4 : The random measures x f c , w h i c h a r e t h e r e a l i z a t i o n s o f t h e F l e m i n g - V i o t p r o c e s s , a r e s u p p o r t e d f o r a l l t i m e s t on a s e t A o f d i m e n s i o n a t most 2 , a.s. APPENDIX A Some I n e q u a l i t i e s Used i n C h a p t e r 3 A . l P u r p o s e 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 i n v o l v i n g t h e c o e f f i c i e n t s g Q- w h i c h were 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 some i d e n t i t i e s w h i c h p r o v i d e a n e a t r o u t e t o t h e i n e q u a l i t i e s i n A.3. A.2 Some I d e n t i t i e s s y y As we saw i n Lemma 3.3.1, Q- = p ( B - , = x where B- i s an i n f i n i t e -x-y - s - A t s i m a l random w a l k s t a r t i n g a t y a t t i m e 0 , and t h e r e a f t e r t a k i n g s t e p s o f s i z e Ax t o t h e r i g h t o r l e f t w i t h p r o b a b i l i t y a , a t each t i m e i n t e r v a l 2 1 1 1 A t , where a = At/Ax <_ — . The r e a s o n f o r n o t c o n s i d e r i n g — < a <_ — w i l l become c l e a r l a t e r . As we saw i n Lemma 4.3.1 t h e above remarks a r e t r u e a l s o i n d - d i m e n s i o n s , 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 d i r e c t i o n s , e ach w i t h 2 p r o b a b i l i t y a = At/A^ . The i d e n t i t i e s i n A.2 a r e t r u e i n d-d i m e n s i o n s a l t h o u g h the d - d i m e n s i o n a l v e r s i o n s w i l l n o t be used The f i r s t i d e n t i t y i s t r i v i a l . s Lemma A.2.1. 7 Q- = l , i f — - £ N L x A t x Ax e *Z 2 Lemma A.2.2. £ (Q|) = Q^-~At > i f s/At £ *N x * ~ £ Z Ax 8 9 P r o o f : I (Q|)2 = ( I Q|)2 - I I Qx 9? x - x x y^x - - x - y^x (by A . l ) = 1 - ^ ( B s - A t = ^ ' ^ s - A t ^ x 1 - I ^ _ A t = x) - ^ B ^ ^ - x ) X (by symmetry; t h i s s t e p f a i l s i f we c o n s i d e r a r e f l e c t i n g random w a l k . ) 1 - I £ < B ° _ A t = x) P ( B ° s . 2 A t ^ 0 | B ° _ A t = x ) x 1 - P(B° . f< 0) - 2 s - 2 A t 2 8 - A t y 0 (by d e f i n i t i o n ) . • * * Lemma A.2.3. I f z/Ax e Z , and s / A t e N x/Axe Z - - - P r o o f : 7 (Q- - Q- ) , . L * X x+z x/Axe Z - - - 2 2 = I (Q-) - 2 y Q- Q- + I (Q§ ) ^ x ^ x x+z L x+z x - x - - - x - - 2 - 2 l P ( B ° . 4 T - , , P ( B ° . T T . ; • (by Lemma A.2.2) X (by symmetry) = 2 Q 2 S ~ A t - 2 P(B° O A = z) *0 - 2 s - 2 A t = 2 ( 2 ^ - 2 ^ u z r s Lemma A. 2 .4. I f — , — e N , J ( Q£ +S - Q f ) 2 = Q 2 £ + 2 r A t + 2 s - A t _ f + 2 s - A t x/Axe*Z x x 0 0 0 P r o o f : Y (Q-+- — Q-)2 v x x Ax x - x - - - x - 2 r + 2 , - A t _ 2 l p ( B 0 = 0 = 0 | B 0 0 x - r + s - A t - - r+2s-2At £ + 2 - ^ (by symmetry and A.2.2) 2r+2s-At ,„0 „. „2s-At 2 " " - 2 ^ C B r + 2 s - 2 A t = 0 ) + V A.3 Some I n e q u a l i t i e s I n t h i s s e c t i o n I p r o v e the f o u r 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 ) . There i s a c o n s t a n t K such t h a t t I (QX) < K At/t , i f ^ xeX * e N P r o o f : C l e a r l y xeX x x/Axe Z x 0 By d e f i n i t i o n , o f k + 1 * A t = P (IS, I < zr) , where S i s 0 k — 2 k t h e sum o f k 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 - 1 , 0, +1 w i t h p r o b a b i l i t i e s a , l - 2 a , a r e s p e c t i v e l y . V a r ( S ) = 2ka . k S i n c e °a > 0 , t h e n by C o r o l l a r y 2.2.3 o f B h a t t a c h a r y a and Rao (1 9 7 6 ) , ( k + l ) A t -0 (2 ' /2kct 1 /2TT e dz "2 /2ka < — as k -> Now 2 ' /2ka 1 -z /2, e dz i s a s y m p t o t i c a l l y 1 x , 1, + o ( — ) as k •> 0 0 2/2irka /k •/2ka Thus t h e r e i s a f i n i t e K s u c h t h a t ( k + l ) A t < J C 0 " A ft By t h e t r a n s f e r p r i n c i p l e t h i s must h o l d f o r a l l o f N . Thus ( c h e c k i n g t = A t s e p a r a t e l y ) 2 t - A t < _ J ± _ < A t y 0 - 2 t - 2 A t - t * Lemma A.3.2 ( 3 . 4 . 2 ) . There i s a c o n s t a n t K s u c h t h a t f o r t / A t e N I I ( Q - ) 2 1 K / t / A t . 0<s<t xeX - n 1 P r o o f : F o l l o w s f r o m A.3.1 and t h e f a c t t h a t £ — f _ C Sn f o r a l l n e N , k = l /k * hence f o r a l l n e N . • 5 * Lemma A.3.3 ( 3 . 4 . 3 ) . There i s a f i n i t e c o n s t a n t K such t h a t i f - — e Z , I I Ax t * r t - s s ^ I ? I and - e N , t h e n ) ) (Q- - Q- ) < K. A t 0<s<t X £ A - - - P r o o f : Suppose w . l . o . g . t h a t z _̂ 0 . L e t B g be t h e i n f i n i t e s i m a l random wa l k whose d e n s i t y i s Q- + A^ • x L e t J ( t , x ) = I I ( B s > x) ( B s + A " B ) . 0<s<t - L e t L ( t , x ) = I I (B = x ) A x . 0<s<t - L e t L ( t , x ) = I K B = x ; B = x + Ax)Ax - - „ . S - S+At -0<s<t Then L i s t h e t r u e o c c u p a t i o n d e n s i t y ( " l o c a l t i m e " ) , f o r o u r random w a l k w i t h 'pauses', w h i l e L i s t h e d i s c r e t e a n a l o g u e o f Bro w n i a n L o c a l Time. Now E ( L ( t , x ) ) = 7 P(B = x ; B . - B = Ax)Ax - - - ^ " - s - s+At s 0<s<t - - - 93 Y P(B = x ) a • Ax u s -0<s<t = a E ( L ( t , x ) ) . I n P e r k i n s (1982) Lemma 3.1, Tanaka's f o r m u l a i s e s t a b l i s h e d by i n t e r n a l i n d u c t i o n on t : (B f c - x ) + = J ( t , x ) + L ( t , x ) Thus L ( t , 0 ) - L ( t , x ) = - (B - x ) + - J ( t , 0 ) + J ( t , x ) Now J ( t , 0 ) and J ( t , x ) a r e b o t h i n t e r n a l m a r t i n g a l e s , hence E ( L ( t , 0 ) - L ( t , x ) ) = E [ B p - E [ ( B t - x ) + ] (A.3.1) Now s i n c e 0 < °a < | , Q§ = V ( B s _ A t = 0) >_ P ( B s _ A t = x) = Q| , as shown i n Lemma 4.3.2. Now by A.2.3 I 1 (Q- - Q~ ) 2 < 2 I Q n 22" A t - Q 2 s " A t _ . v x x+z — „ u . 0 z 0<s<t xeX - - - 0<s<t < * i <er4t - c4t» At<s<2t U - = 2 I P(B = 0) - P(B = z) 0<s<2t-2At - - 1 2 E ( L ( 2 t - A t , 0) - L ( 2 t - A t , z ) ) |- E ( L ( 2 t - A t , 0 ) - L ( 2 t - A t , z ) < - , by (A.3.1) ~~ a Ax Lemma A.3.4 ( 3 . 4 . 4 ) . There i s a f i n i t e c o n s t a n t K s u c h t h a t i f r / A t < t / A t a r e i n *N , 2 I I (Q!~- - Qr~~> < K / ( t - r ) / A t 0<s<r xeX P r o o f : The l . h . s . above i s e q u a l t o V r . s + ( t - r ) s . 2 L L (Q~ ~ ~ ~ Q~' i w h i c h by A.2.4 i s bounded by 0<s<r xeX x - (A-2) I ( Q 2 s + 2 ( t - r ) - A t + Q 2 s - A t _ 2 Q ( t - r ) + 2 s - A t ) 0<s<r 0 0 F i r s t , suppose t - r < r and ( t - r ) / A t i s even. Then many o f th e terms i n (A-2) c a n c e l l e a v i n g t - r t - r 0 < s < — — <s<t-r - - 2 2 - I Q2-"AT + I Q 2 S _ A T t - r ° t - r 0 r < s < r + — — r+ — <s<t - - - 2 - 2 ° < 1 s S i n c e a — - , t h e c o e f f i c i e n t s Q- a r e monotone d e c r e a s i n g as 3 0 s i n c r e a s e s . Hence the sum o f the s e c o n d , t h i r d , and f o u r t h terms i n (A-3) above i s bounded above b y 0 . The f i r s t t e r m i n (A-3) i s bounded by KV ( t - r ) / A t by A.3.2, as r e q u i r e d . t - r Now i f t - r < r and ~~7~7~ i s odd t h e sum i n (A-2) i s bounded by Y r n 2 s + 2 ( t - r ) - A t n 2 s - A t - ( t - r ) + 2 s 1 , . . I LQ + Q - - 2 Q - J .. A p p l y i n g t h e same 0<s<r v 2 s ~ A t i c a n c e l l a t i o n argument, t h i s i s bounded by \ Q - < K / t - r / i t t - r 0 < s < - — +At — 2 Now i f t - r > r ( i . e . r < t/2) , t h e n I I < < ^ > - Q | ) 2 0<s<r x eX x x 0<s<r xeX y 2 s + 2 ( t - r ) - A t + Q 2 s - A t ~ 0 0~ 0<s<r (by A.2.2) 2 s - A t < 2 I Q — L o 0<s<r — K AT ( B Y A , 3- 2 ) t - r < K — — . - A t 1 Remark: I f a = — t h e n Lemmas A.3.3 and A.3.4 a r e f a l s e . The p a t t e r n o f n o n - z e r o c o e f f i c i e n t s i n t h e a r r a y {Q-} , s / A t e *N , x/Ax e *Z , i s a c h e c k e r b o a r d p a t t e r n . Thus U i s i n d e p e n d e n t o f U, , and o f any U £ x t - A t , x J s; t - s x-y f o r w h i c h —-— + ~ i s an odd h y p e r - i n t e g e r . Thus t h e moment i n e q u a l i t i e s A t Ax 96 on s p a t i a l and t e m p o r a l d i f f e r e n c e s w i l l f a i l . T h i s i s t h e r e a s o n f o r n o t u s i n g t h e s i m p l e s t f i n i t e d i f f e r e n c e 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 D i m e n s i o n s The s u c c e s s o f t h e 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 a p p r o a c h i n C h a p t e r s 3 and 4 t o the 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 one d i m e n s i o n , and t o t h e Dawson measure d i f f u s i o n i n h i g h e r d i m e n s i o n s , l e a d s one t o wonder i f 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 e q u a t i o n s m i g h t l e a d t o a g e n e r a l 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 . So f a r a t l e a s t , t h i s hope has n o t b o r n e f r u i t . The k i n d o f e q u a t i o n s t h a t we w o u l d be l e d t o c o n s i d e r a f t e r t h e a n a l o g y o f t h o s e i n C h a p t e r s 3 and 4, wou l d be o f t h e form ( B - l ) ! ^ = Au + f (u)dW f c x , t e R + , x e R d and t h e c o r r e s p o n d i n g i n t e r n a l e q u a t i o n t'B-2) (6 U ) = (A U. ) + F ( U ) 5 / A t A x d , f o r t -x t t - x t x t x t e T and x e X , h y p e r f i n i t e g r i d s r e p r e s e n t i n g R and R r e s p e c t i v e l y . Now i t i s easy t o see t h a t an i n t e r n a l s o l u t i o n t o (B-2) e x i s t s , by t h e u s u a l i n d u c t i v e c o n s t r u c t i o n . What i s n o t so c l e a r i s whether o r n o t t h i s i n t e r n a l s o l u t i o n U has a n o n - t r i v i a l s t a n d a r d p a r t u , p r e s u m e a b l y i n some space o f d i s t r i b u t i o n s . I t i s a l s o n o t c l e a r what i t woul d mean f o r such a d i s t r i b u t i o n - v a l u e d p r o c e s s 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 u n d e f i n e d o r d i s c o n t i n u o u s a t b e s t . I t seems p o s s i b l e t o make sense o f ( B - l ) i f f ( u ) i s an o p e r a t o r v a l u e d f u n c t i o n o f u w i t h v a l u e s i n a c l a s s o f 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 p u r s u e t h i s p o s s i b i l i t y w o u l d 9 8 t a k e us t o o f a r a f i e l d f r o m t h e i d e a s o f t h i s t h e s i s . I f we r e s t r i c t o u r s e l v e s t o t h e c a s e where f i s a r e a l v a l u e d f u n c t i o n o f a r e a l v a r i a b l e , and t h a t F i s some n a t u r a l l i f t i n g o f f , s u c h as f , t h e n we may s t i l l a s k w h e t h e r o r n o t t h e i n t e r n a l s o l u t i o n t o (B-2) has an i n t e r e s t i n g s t a n d a r d p a r t . S a d l y , i n s e v e r a l c a s e s t h e answer seems t o be 'no'. The f i r s t c a s e we c o n s i d e r e d i s when f i s a c o n t i n u o u s f u n c t i o n o f compact s u p p o r t , (and F i s an S - c o n t i n u o u s l i f t i n g o f compact s u p p o r t ) . Then s i n c e A t << Ax , changes t o t h e i n t e r n a l s o l u t i o n U t o (B-2) a r e i n f i n i t e s i m a l a t each s t e p and an e a s y i n d u c t i o n argument shows t h a t t h e i n t e r n a l s o l u t i o n U i s bounded by a f i n i t e c o n s t a n t . I f we seek t o t x e s t i m a t e t h e v a r i a n c e o f £ U Ax*3 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 — ( o r o t h e r s e t ) A , t h e n we a r e n a t u r a l l y l e d t o examine E ( F (U )) . We may sy s e t up a d i f f e r e n c e e q u a t i o n f o r t h i s q u a n t i t y , and show t h a t i t i s everywhere a.s. i n f i n i t e s i m a l . Thus 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 f i n i t e r e g i o n o f U , i s i n f i n i t e s i m a l . Thus though t h e v a l u e s t a k e n by U l i e a l m o s t a l w a y s a t e i t h e r end o f t h e ( c o n n e c t e d ) s u p p o r t o f F , t h e s e v a l u e s b a l a n c e v e r y p r e c i s e l y on each monad. The b a l a n c e p o i n t i s i n f i n i t e s i m a l l y c l o s e t o t h e v a l u e o f 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 t h e h e a t e q u a t i o n w i t h t h e g i v e n i n i t i a l c o n d i t i o n , a t t h e s t a n d a r d p o i n t c o r r e s p o n d i n g t o t h e monad i n q u e s t i o n . The o t h e r c a s e s w h i c h we examined were when F(U) = , when e i t h e r 0 < p < i , o r ^ < P ^ _ 1 - W e m a Y t r y t o f o l l o w t h e development o f s e c t i o n 4.4 t o f i n d t h e t o t a l mass. I n t h e c a s e 0 < p < - , we f i n d t h a t 2 E(M - M n ) 2 q i s f i n i t e , a t l e a s t i n t h e c a s e o f r e f l e c t i n g boundary c o n d i t i o n s - t u on a f i n i t e r e c t a n g l e , s i m p l y by b o u n d i n g U by 1 + U . However, a c l o s e r e x a m i n a t i o n o f t h e d i f f e r e n c e e q u a t i o n s ( B - 2 ) , w i l l show t h a t , . t h e v a l u e s U a r e a l m o s t a l w a y s i n f i n i t e o r i n f i n i t e s i m a l . Thus t h e q u a n t i t y £ U 2 p i s a c t u a l l y i n f i n i t e s i m a l l y _ _ s m a l l e r t h a n £ , s i n c e a l m o s t a l l t h e mass comes from p o i n t s x xeX where U i s i n f i n i t e . Thus we end up w i t h , i n f a c t E|M t - M 0 | 2 q << E ( 1 M^) , and 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 examine t h e p r e d i c t a b l e i n c r e a s i n g p r o c e s s a s s o c i a t e d w i t h £ (U - U ) $ ( x ) - £ x A $ A t , as i n s e c t i o n 4.5 we f i n d X "cx ux — ^ . s 0<s<t = i t t o be £ J U 2 p $ 2 ( x ) A x d , w h i c h i s l i k e w i s e i n f i n i t e s i m a l l y c l o s e 0<s<t xeX - - t o 0 . 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 (B-2) f o r F(U) = iP , 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 . I f F(U) = U P , i < p <_ 1 , t h e n , a g a i n U must a l m o s t a l w a y s be i n f i n i t e o r i n f i n i t e s i m a l . I f we o b t a i n an 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 t o t a l mass M , we f i n d E(M - M ) 2 = Y Y U 2 p A t A x d . 0<s<t xeX — Whenever U i s i n f i n i t e U << U 2 p , and hence Y U 2 p A x d >> Y U A x d L v sx L v sx x e A — xeA — The moment bounds on c a n n o t be o b t a i n e d . 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