@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Mathematics, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Reimers, Mark Allan"@en ; dcterms:issued "2010-08-18T19:11:22Z"@en, "1986"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description "In this thesis we introduce Non-Standard Methods, in particular the use of hyperfinite difference equations, to the study of space-time random processes. We obtain a new existence theorem in the spirit of Keisler (1984) for the one dimensional heat equation forced non-linearly by white noise. We obtain several new results on the sample path properties of the Critical Branching Measure Diffusion, and show that in one dimension it has a density which satisfies a non-linearly forced heat equation. We also obtain results on the dimension of the support of the Fleming-Viot Process."@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/27515?expand=metadata"@en ; skos:note "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) W^ , 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 W^ , 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 W^ 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 (~x) and integrate over a rectangle CO,T] X A , where A contains supp cj> , then 5 r r (1-2) A u (x) dx = Ox ( u A(x)dx)ds Jo J A S X f ( u )(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 u^ _ 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 £ £ 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) = 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 . 1T ) ^ ' 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 e C ( R) , C3-4) utx<$> (x) dx - u d> (x) dx = Ox u Ad)(y)dyds sy ft 0 f ( u )(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 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 | 2 q ( l + E\\K | 2 q ) ( Q - \" - ) 2 7^) q - - 0 < s < t y e x ^ *-y A * < c C l 0 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 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 . | 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 ~ 1^ + |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 $)...) (x) = (x) +$ (x-Ax) ] / A x 2 = - ^ r - (°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 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\" (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 ) (x) dx -t x R R u c|> (x) dx a = S ' Ox R u (j) (x) + sx 0 R f ( u )\"-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 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 = 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 ) = 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 5^ .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 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 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 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 1 s s -. -p j?. 2q _ 2q E X - X < c E ( ) M At) -' t r 1 - - L s r° 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) 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 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 ( 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 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 X^ _ 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 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 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 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 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 = r 4> 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 ~ 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 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 F^ _) 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 0 L> — Ax < K 2 • t • ( 2 L ) d • ^ A x d (4-23) + K 2 I X ^ ' - L } A t 0 = = ) X A t t t „ L s 1 1 - 0 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 x^ _ 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 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 , 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) - ( 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 X^ _ -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 e C ( R) c t Ao 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 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 A^^} 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/A^ } 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 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 Y^ _ 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 NY^ _ , 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 NY^ _ 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 Y^ _ . 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 = t 2y Y (1-Y ) d t 0 * (6-9) d 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 b^ _ = 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 B^ _ , 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 = 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 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 b^ _ 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 ; = 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 x) ( B s + A \" B ) . 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 < K / ( t - r ) / A t 0 r ( i . e . r < t/2) , t h e n I I < < ^ > - Q | ) 2 0> Y U A x d Lv 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 . An e x a m i n a t i o n o f t h e p r e d i c t a b l e ,* y X A $ ^ s 0