{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Mathematics, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Lo\u0301pez, Miguel Martin","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2009-03-17T18:46:16Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1996","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Dawson-Watanabe superprocesses are stochastic models for populations undergoing spatial migration\r\nand random reproduction. Recently E. Perkins (1993, 1995) introduced an infinite\r\ndimensional stochastic calculus in order to characterize superprocesses in which both the reproduction\r\nmechanism and the spatial motion of each individual are allowed to depend on the\r\nstate of the entire population, i.e. superprocesses with interactions.\r\nThis work consists of three independent chapters. In the first chapter we show that interactive\r\nsuperprocesses arise as diffusion approximations of interacting particle systems. We construct\r\nan approximating system of interacting particles and show that it converges (weakly) to a limit\r\nwhich is exactly the superprocess with interactions. This result depends very intimately upon\r\nthe structure of the particle systems.\r\nIn the second chapter we study some path properties of a class of one-dimensional interactive\r\nsuperprocesses. These are random measures in the real line that evolve in time. We employ\r\nthe aforementioned stochastic calculus to show that they have a density with respect to the\r\nLebesgue measure. We also show that this density, function is jointly continuous in space and\r\ntime and compute its modulus of continuity. Along with the proof we develop a technique that\r\ncan be used to solve some related problems. As an application we investigate path properties\r\nof a one-dimensional super-Brownian motion in a random environment.\r\nIn the third chapter we investigate the local time of a very general class of one-dimensional\r\ninteractive superprocesses. We apply Perkins' stochastic calculus to show that the local time\r\nexists and possesses a jointly continuous version.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/6160?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"Extent":[{"label":"Extent","value":"5340672 bytes","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/extent","classmap":"dpla:SourceResource","property":"dcterms:extent"},"iri":"http:\/\/purl.org\/dc\/terms\/extent","explain":"A Dublin Core Terms Property; The size or duration of the resource."}],"FileFormat":[{"label":"FileFormat","value":"application\/pdf","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/elements\/1.1\/format","classmap":"edm:WebResource","property":"dc:format"},"iri":"http:\/\/purl.org\/dc\/elements\/1.1\/format","explain":"A Dublin Core Elements Property; The file format, physical medium, or dimensions of the resource.; Examples of dimensions include size and duration. Recommended best practice is to use a controlled vocabulary such as the list of Internet Media Types [MIME]."}],"FullText":[{"label":"FullText","value":"P A T H P R O P E R T I E S A N D C O N V E R G E N C E OF I N T E R A C T I N G S U P E R P R O C E S S E S by MIGUEL MARTIN LOPEZ M.Sc. (Mathematics) The University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1996 \u00a9 Miguel M . Lopez, 1996 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence 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 Mathematics The University of British Columbia Vancouver, Canada Date S Abstract Dawson-Watanabe superprocesses are stochastic models for populations undergoing spatial mi-gration and random reproduction. Recently E. Perkins (1993, 1995) introduced an infinite dimensional stochastic calculus in order to characterize superprocesses in which both the re-production mechanism and the spatial motion of each individual are allowed to depend on the state of the entire population, i.e. superprocesses with interactions. This work consists of three independent chapters. In the first chapter we show that interactive superprocesses arise as diffusion approximations of interacting particle systems. We construct an approximating system of interacting particles and show that it converges (weakly) to a limit which is exactly the superprocess with interactions. This result depends very intimately upon the structure of the particle systems. In the second chapter we study some path properties of a class of one-dimensional interactive superprocesses. These are random measures in the real line that evolve in time. We employ the aforementioned stochastic calculus to show that they have a density with respect to the Lebesgue measure. We also show that this density, function is jointly continuous in space and time and compute its modulus of continuity. Along with the proof we develop a technique that can be used to solve some related problems. As an application we investigate path properties of a one-dimensional super-Brownian motion in a random environment. In the third chapter we investigate the local time of a very general class of one-dimensional interactive superprocesses. We apply Perkins' stochastic calculus to show that the local time exists and possesses a jointly continuous version. i i Table of Contents A b s t r a c t i i Tab le o f Con ten t s i i i Acknow ledgemen t \/ i v C h a p t e r 0. In t roduc t ion 1 0.1 Review ..' 2 0.2 Summary of the Main Results ; 8 C h a p t e r 1. W e a k Convergence of In teract ing B r a n c h i n g Pa r t i c l e Systems 11 1.1 Introduction 11 1.2 Interactive Branching Particle Systems '. 14 1.2.1 The Particle Picture 14 1.2.2 An Equation for KN 23 1.3 Tightness of the Normalized Branching Particle Systems 29 1.3.1 Convergence of the Projections 30 1.3.2 Compact Containment Condition 33 1.3.3 Relative Compactness of (KN) : ' 36 1.4 Identification and Uniqueness of the Limit 36 C h a p t e r 2. P a t h P rope r t i es of a One-D imens iona l Superd i f fus ipn w i t h Interac-t ions 41 2.1 Introduction and Statement of Results 41 2.1.1 Main Result 41 2.1.2 Historical Stochastic Calculus 46 2.2 Some Auxiliary Processes 50 2.3 A Generalized Green's Function Representation for X 61 2.4 Proof of the Main Result 63 2.5 Examples: Super Brownian Motions with Singular Interactions ; 75 C h a p t e r 3. L o c a l T i m e s for One-D imens iona l In teract ing Superprocesses 80 3.1 Introduction and Statement of Results 80 3.2 Existence and Regularity of Local Times 84 B i b l i o g r a p h y 100 iii Acknowledgement I would like to thank all the individuals and institutions that made this journey possible. M y most sincere thanks go to my advisor, professor E d Perkins. It has been a privilege to work under his direction. It is also a pleasure to thank the probability group at U B C . They and their numerous visitors were a constant source of inspiration and high quality mathematics. In particular I acknowledge some useful conversations with professor D. Dawson. I should like to express my gratitude to the Mathematics Department at Universidad de los Andes, especially to professor S. Fajardo. I am indebted to them for showing me the beauty of mathematics and for teaching me the concept of proof. During these last years I have received much encouragement from my family and friends. Thanks to mom*and dad for that.and for the rest. I thank also the hospitality of professors R. Adler (Chapel Hil l ) , J-F. Le Gall (Paris), T. Lindstrjsm- (Oslo) and B. Rozovskii (Los Angeles). M y best thanks to Anders Svensson, I^jXpert extraordinary, for showing me the path to com-puter, guruhood. I gratefully acknowledge the financial support of Dr. E . Perkins and The University of British Columbia. Last but not least I thank Laura for her love, support and for putting up with me. iv Chapter 0 Introduction This work is devoted to the study of superprocesses with interactions. Superprocesses (or measure-valued branching diffusions) are measure-valued processes that model populations un-dergoing random branching and spatial motion. By a population we mean a system involving a number of similar particles. We are interested in the approximations that are possible when the number of particles is large. Consider.for example a large population of goats. Individual goats reproduce, die (branching) and wander around (spatial motion). They also interact with each other in many ways. For example they live in clusters or clans. They also have memory and tend not to return to fields they have grazed until some time has passed. Suppose we are asked to implement a computer simulation of the evolution in time of the population. We are interested not only in the total number of goats, but in their geographical locations as well. To this end, we would place a square grid over the region of interest and assign to each square (or pixel) a height (or color) proportional to the number of goats within. Due to the births-deaths (i.e. branching) and spatial motion of each animal the color of each pixel would change as time goes on. We are looking at the evolution in time of the density of goats. Note that the value of such random process at any given instant is not a number but a colored map. Therefore we must give a mathematical interpretation to \"colored map\"-valued processes. One way to do it is to regard each map as a measure. This example allows us to see why measure-valued processes may arise naturally and are not a mere technical device. We wish to know if after appropriate rescaling, the density of goats can be well approximated by some diffusion process. That is, we are after a limit theorem. We quote the great mathematician A . N . Kolmogorov: The epistemological value of the theory, of probability is revealed only by limit the-orems. Moreover, without limit theorems it is impossible to understand the real content of the primary concept of all of our sciences - the concept of probability. Suppose that we have established the fact that as the number of goats goes to infinity a limit process does exist. We call it the goat superprocess. It is then natural to ask, how does it look like? More specifically, do the measures corresponding to the values of the goat superprocess have density functions? If so, are they continuous? Since continuous functions can oscillate wildly, how continuous? Of course, we require a rigorous formulation of all these questions. The subject of superprocesses is a rapidly developing field. It has been stimulated from sev-eral different directions including population genetics, branching processes, interacting particle systems and stochastic partial differential equations. For example the so-called Fleming-Viot superprocess is a generalization of the Wright-Fischer model (for random replication with er-rors) and has been studied in connection with population genetics (Hartl and Clark 1989). Rogers and Williams (1986) wrote in the preface of their book: Here are some guidelines ori what you might move on when your reading of our book is done. (vi) Measure-valued diffusions, random media, etc.. Durrett (1985) and Dawson and Gartner (1987) can be your 'open sesame' to what is sure to be one of the richest of Aladdin's caves. We believe that the monograph of Dawson (1993) proves that Rogers and Williams were correct. In this thesis we wil l give a rigorous description of a general class of models that hopefully includes what our intuition says the goat superprocess should be. We will then answer some specific questions about certain subsets of the class of models. In Section 0.1 we survey some of the theory of superprocesses. In Section 0.2 we explain our results. 0.1 Review In this Section we review the most basic concepts needed to understand both the meaning and relevance of our results. Most of the background information is very recent and known only by a handful of experts. The ideal preparation is furnished by a thorough reading of the papers Perkins (1993, 1995). These in turn can be well understood after reading either of our favorites Walsh (1986) or Dawson (1993). Our notation is consistent with that of Perkins (1995), and a reader familiar with this material can safely skip the rest of this Section. We have made an effort to give an intuitive understanding of the ideas presented below. However many of them are accompanied by heavy technical baggage, and these technicalities are'important for a careful examination of this thesis. They cannot be avoided. Some of these ideas may seem obscure at a first glance, but they contain large amounts of valuable information. This is hardly a surprise since we wil l be studying some fairly complex objects. We encourage the reader to refer to the excellent references when in doubt or need. We assume a basic knowledge of Probability theory on the part of the reader. We expect the reader to be acquainted with 1. Martingales, Brownian motion, weak convergence of probability measures on metric spaces, critical branching processes (the standard Galton-Watson process will suffice), the Poisson process. 2. Martingale inequalities (Doob's, Burkholder's and maximal), stopping times, stochastic integration (Ito's lemma), local times (Tanaka's formula). Most of these topics (certainly those in 1) are covered in a first graduate course in Probability theory. Those not familiar with the topics listed in 2 should still be able to understand the statement and meaning of the most important theorems. They should also be able to read the remainder of this Section. The subjects listed in 1 and 2 are the absolute minimum required to understand most of the proofs. Familiarity with superprocesses and\/or stochastic partial differential equations (abbreviated SPDE) is highly recommended. 2 B y a measure-valued process we mean a random process whose state space is Mp(E), the space of finite measures on some complete, separable, metrizable topological space (E, 13(E)). As usual, B(E) denotes the Borel a-field and Mp(E) is endowed,with the topology of weak convergence. Superprocesses are related to branching processes, population genetics models, stochastic partial differential equations and interactive particle systems. The canonical example is d-dimensional super Brownian motion, which we now describe. F i x a positive integer N, a positive real number 7 and a finite measure m on JRd. At time t=0, N particles are located in H d with law m(- ) \/m(R d ) . They move independently according to d-dimensional Brownian motions. If a particle is located at position a; at time \u00a3, then let the probability that it dies before time t 4- dt be ~fNdt + o(dt). If it dies, a fair coin is tossed and the particle is replaced by 0 or 2 identical particles situated at the position of death of their parent. The new particles then start undergoing independent Brownian motions and the process continues in the same fashion, with particles moving, dying and branching ad infinitum. In this model we want to keep track of the number of particles as well as their locations. Let I(N, t) be the total number of particles alive at time t, and let Z\\, i = 1,... ,I(N,t), label their locations. Consider the rescaled measure-valued process \u2022\u2022 ' HN,t) *N(t) = jj \u00a3 6ZV (O-1) 1=1 where Se denotes a unit mass at e. When the number of particles is large, i.e. as N' \u2014>\u2022 60, the particle system XN is approximated by a measure- valueddiffusion X which we call super Brow-nian motion with branching rate 7. In fact, the sequence of probability measures (P(XN \u20ac \u2022))N converges weakly to a law P m on C([0,00), Mf(JRd)) (Dawson 1993). Following the usual con-vention, we endow C([0,00), E), the set of continuous paths t Xt \u20ac E, with the compact-open topology and D([0,00), E), the space of cadlag paths t \u2022->\u2022 xt G E, with Skorohod's Ji-topology. Super Brownian motion X can be characterized through a martingale problem (Dawson 1993). A typical example of a martingale problem is Levy's characterization of Brownian motion. It says that if an IRd-valued random process B is a continuous martingale with square function {Bl,B^)t'= tdij (here 6ij denotes Kronecker's delta), then B is a d-dimensional Brownian mo-tion. The following is a martingale problem that uniquely characterizes the law P m : If is a measure we write \/j,(
) = X0( d(i and Mp{E) denotes the set of finite Borel measures on a metric space E and it is endowed with the weak topology. T h e o r e m 1.1 ( S u p e r - B r o w n i a n M o t i o n w i t h B r a n c h i n g R a t e 7). Let SQ denote a unit mass at 0. There is a continuous M f ( I R d ) valued, adapted process Xt defined, on a filtered probability space ( f i ,^\" , ( ^ ) , P ) such that j. TP(X0 = 60) = 1 2.1f(t>\u00a3 C 6 2 (M d ) , then XM = XoW + f\\s^)ds:+Zt{4>), ( M F ) A , 7 11 where Zt( ? - (ZN(4>))t) : N \u20ac N}, ' {\/\u00ab,... , O ( Z f ( 0 ) 2 - (ZN( )2 \u2014 (Z(4>))t)t>o is a ^\"^-martingale. This concludes the proof of the Proposition. \u2022 37 P r o p o s i t i o n 1.29. \u2022{Z( 0 a.s. (1:40) Jo . . n-\u00bboo For every integer t there is a j = j(t) such that t A Tj \u2014 t. Moreover, the integral in.(1.40) is increasing as a function of t. Therefore \\K?fr{s, Kn, \u2022)<\/>(-)%- Ks(7(s,Kr) e C%(1R.) jt{4>)' = f Pt (t,yt) = ip(P,Y0) + J i-\u00a3(s,ys) + -^(s,ya)j d s + J .-^(s,ys)dy(s) = rb(0,Y0) + fi ?jt(s,ys)dy(s) for -Kj-a.e. y Vt G [0,1] a.s., (since \u2014(s,x) = --^(s,x)). Moreover, by the same lemma \u00a3t(l,u,y) = 1 - \u00a3s(l,u),y)b(s,uj,y)dy(s) + b(s,uj,y)2\u00a3s(l,u,y)ds 54 for Kt-a..e.-y Vt \u20ac [0,1] a.s. Integration by parts, justified once more by Ito's lemma 2.17 yields \u00a3T{\\,LO,y)^{t,yt) = ^(0,y 0 ) + J (\u00a3s(l)^(s,ys) - ^{s,ys)\u00a3s(\\)b(s))dy(s) + ( 6 ( 5 )2 ^ ( 1 )^ (5 ,^ ) - 6(5)^(1)1^(5,^))^ for Kt-&.e. y Vt \u20ac [0,1] a.s. Now apply historical Ito's lemma 2.18 to obtain J \u00a3t(l)y>(t,yt)Kt(dy) = j 1>(0,x)p(x)dx + J j5,(1,L0,y)^{s,ys)ZK{ds,dy) Vte'[0,1] a.s. Recalling the definition of ip and jn, we see that this last equation is exactly (2.12). \u2022 R e m a r k 2.33. If b = 0 and 7 = 1 then Proposition 2.32 provides a very simple proof for the usual Green's function representation of super Brownian motion (compare with that of Konno and Shiga 1988, p 212). P r o p o s i t i o n 2.34. (a) For each 0 > 0 let \u00a3t(6,LO,y) = exp (-6IR(b,t,u,y) + (o - y ) j f 0(5,LO,y) 2 ds) . Then for any 6 > 0, \u00a3sA.(6) is a P s -martingale starting at 1. (b) There is a function K : [l,oo) x [0,00) \u2014> R.+ , non-decreasing in each variable such that Ss(l)p <*(p,s)\u00a3s(p) \u2022 (c) The second term on the r.h.s. of (2.12) is a square integrable martingale null at zero. P r o o f , (a) Recall that by Proposition 2.14 nt = yt \u2014 yQ \u2014 flb(s,LO,y) is a P^ -Brownian motion. Since b is bounded, \u00a3sA.{0) is an exponential martingale. (b) We estimate \u00a3s(l)p = \u00a3s{p)e{^-p)I\u00b0ku)2du <\u00a3s{p)es^-p)cBu = K(p,s)\u00a3s(p). (c) Note that HPt-s^lloo < 00. Moreover P p Ks(\u00a3s(l)2)ds^ < pP*[\u00a3s(2M2,s)}ds -0 P - a.s, (2.25) Using (2.24), (2.25) and Ito's lemma.for historical integrals 2.18 we obtain (2.23) \u2022. 2.4 Proof of the Main Result In this section we put together the results from Sections 2.2, and 2.3 to prove Theorem 2.9. We begin with some technical lemmas. -L e m m a 2.43 ( K o l m o g o r o v ) . (a) Let [B(x) : x G P d ) be a family of random variables in-dexed by x \u20ac P d . Suppose thai there exists a real p > 0 and two constants Co, $ > 0 swc\/i that \u2022-'\u2022r . V x , x \u20ac P d , E[jB(x) - \u00a3(x)|p] < C0|jx - x\\\\d+3.. ' Then the process (B(x) : .x \u20ac P d ) has,a continuous version which is globally Holder with exponent a, for any a < (3\/p. \u2022 (b) Let I C P . 3 be the product of 3 intervals (either closed, open or semi-open). Let [B(x) : x \u00a3 I) be a 3-dimensional random field. Suppose that forany k > 13 there is a, constant Co > 0 . such that \" ' ' E[|B(xi ,x 2 ,x 3 ) - B(xi,x3,xi)\\k] < Co(|x i - x i | | ^ V | x 2 - x 2 | ^ + |x3 - x 3 | V ) . for any. ( x i , x 2 , x 3 ) , (x i ,X2 , x 3 ) G I such that 0 < |xi \u2014 xi|,|x2 \u2014x2|,|x3 \u2014 x3| < 1. Then the process (B(x) : x \u20ac I) has a continuous version, which is Holder, with exponent a, for any a < 1\/4. Moreover, for any X2,x3 fixed, the map x\\ i-4 i? (x i ,x 2 , x 3 ) is Holder-a for any a< 1\/4 and the map x 2 >-\u00bb l ? (x i ,x 2 ,x 3 ) is Holder-a for any a < 1\/2. If we know that the process (-B(x) \u2022 x \u20ac P d ) is continuous to begin with then there is no need to take a version. 63 P r o o f . Although Kolmogorov's theorem is not generally stated in this form, the standard proof (Revuz and Yor 1991) works equally well. \u2022 L e m m a 2.44. (a) Denote p'(t,x) = Dxp(t,x). For any 0 < e < 1, t \u20ac [0,1] rt roo I \\p'(t \u2014 s,x + e) \u2014 p'(t \u2014 s,x)\\dxds < C2AA.iy\/s. JO J-oo '(b) Let 0 < s < t < 1. Then rs roo I \\p'(t~ r i x ) \u2014p'(s ~ r,x)\\dxdr < c2.44.2Vt \u2014 s. Jo J-oo (c) Let T{n) := mf{t > 0 : Ht(l) > n} A 1. There is a function 0 : N x [0,oo) \u2014\u2022 1R+, non-decreasing in each variable such that X;(l) < e{n,s) on {T(n)>s}. Notice also that K* (1) =X*(1). P r o o f , (a) Estimates (a) and (b) should be well known. We prove them since we don't know a reference. We need the following elementary estimate (e.g. Ladyzenskaja 1968 p. 274) ' \\D?D?p{x,t)\\.^Cn^-^-^exp^-Cn^y) . (2.26) Let 8 > 0, 8 < t. We estimate rt roo ft-S roo \/ \/ \\p'(t - s,x + e) - p'{t - s,x)\\dxds = \/ \/ \\p'(t-s,x + e)-p'{t-s,x)\\dxds Jo J-00 ' Jo J-00 rt roo + \\p'[t - s,x + e) \u2014 p'(t - s,x)\\dxds \u2022 Jt-S J-00 \u2022 ' rt\u2014S roo < \/ \\p'(t - s,x +.e) -p'(t - s,x)\\dxds -. Jo J-00 rt roo + 2 \/ \\p'(t- s,x)\\dxds Jt-S J-00 = Il+h. Use estimate (2.26) to check that rt. ^ -\u00ab$ y\/t - S < Ci\\T8. J 2 < C ft ~T^=ds 64 To estimate I\\ we use the fundamental theorem of calculus followed by a linear change of variables, rt\u20146 roo . rl h = \/ \/ \u00a3 \/ D2p(t - s; x + ze)dz JO J-oo' Jo dxds ft\u20146 rl roo . <\u00a3 \/ \/ \\D2p(t-s,x + JO JO J-oo ze)\\dxdzds (by Fubini) ft-8 rl < e I \/ dzds Jo Jo t-s (using estimate (2.26)) ^ e l o g Q ) . {a) j CatAT[n)HtAT[n){dy)da = J J cp(a)C1AT{n)daHtAT{n)(dy) (by Fubini) = J-J 4>{Ys)o2{Xs,Ys)dsHtAT{n){dy) (by the density of occupation formula). We can apply Ito's lemma for historical integrals to this last H-integral to obtain: ftAT(n) n ptAT(n) f ps (3.42) r rt l( ) ptAT(n) f JJ (Yr)o2(Xr,Yr)drZH(dsdy) ptAT(n) r + Jo J (a)Xs(da)ds = f 4>{a) [ o--2{Xs,a)Lads{X)da Jo J Jo = J cp(a)Lat(X)da. (by (3.18)) -P r o o f of T h e o r e m 3.4. Propositions 3.16 and 3.19 show that the family of random variables (Lf(X)) satisfies the conditions (i) and (ii) of Definition 3.1. \u2022 99 Bibliography [1] Adler, R . J . (1992). Superprocess local and intersection local times and their corresponding particle pictures, in Seminar on Stochastic Processes 1992, Birkhauser, Boston. [2] Barlow, M . T . , Yor, M . (1981). Semimartingale inequalities and local times, Z. Wahrschein-lichkeitstheorie verw. Gebiete 55, 237-254. [3] Dawson, D . A . (1993). Measure-valued Markov processes, Ecole d'ete de probabilites de Saint Flour, 1991, Lect. Notes in Math. 1541, Springer, Berlin. [4] Dawson, D . A . , Gartner, J . (1987). Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics #0,247-308. [5] Dawson, D.A\\, Perkins, E . A . (1991). Historical processes. Mem. Amer. Math. Soc. 454. [6] Dawson, D . A . , Perkins, E . A . (1996). Measure-valued processes and stochastic partial dif-ferential equations. Preprint. [7] Dellacherie, C . , Meyer, P .A . (1982). Probabilities and Potential B, North Holland Mathe-matical Studies No. 72, North Holland, Amsterdam. [8] Durret, R. (1985) Particle systems, random media, large deviations, Contemporary Maths. 41, Amer. Math. Soc, Providence, R. I. [9] Ethier, S .N. , Kurtz , T . G . (1986). Markov processes: characterization and convergence, Wiley, New York. [10] Hart l , D . L . , Clark, A . G . (1989). Principles of population genetics, second edition, Sinauer Associates, Inc., Sunderland, Masachussetts. [11] Jacod, J . , Shiryaev, A . N . (1987). Limit theorems for stochastic processes, Springer-Verlag, New York. [12] Konno, N . and Shiga, T . (1988). Stochastic differential equations for some measure valued diffusions, Probab. T h . Rel. Fields 79, 201-225. . [13] Ladyzenskaja, O . A . , Solonnikov, V . A . and Ural'ceva, N . N . (1968). Linear and quasilinear parabolic equations of parabolic type, Transl. Math. Monographs Vol 23, Amer. Math. Soc. [14] Le Gal l , J . F . Perkins, E . A . , Taylor, S. J . (1995) The packing measure of the support of super-Brownian motion, to appear in Stoch. Process. Applications. [15] Meleard, S. and Roelly, S. (1990). Interacting measure branching processes and the associ-ated partial differential equations., Stochastics and Stochastic Reports. [16] Perkins, E . A . (1988) A space-time property of a class of measure-valued branching diffu-sions, Trans. Amer. Math. Soc. 305, 743-795. [17] Perkins, E . A . (1993) Measure-valued branching diffusions with spatial interactions, Probab. T h . Rel. Fields 94, 189-245. 100 [18] Perkins, E . A . (1995). On the martingale problem for interactive Measure-Valued Branching Diffusions, Memoirs of the Amer. Math. Soc. No. 549, 1-89. [19] Reimers, M . (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion, Probab. T h . Rel. Fields 81, 319-340. [20] Revuz, R . J . and Yor, M . (1991). Continuous martingales and Brownian motion, Springer-Verlag, New York. [21] Rogers, L . C . G . and Williams, D . (1986). Diffusions, Markov processes and martingales, Vol 2., Wiley, New York, [22] Shiga, T . (1994). Two contrasting properties of solutions for one dimensional stochastic partial differential equations, Canadian Journal of Math. Vol. 46 No. 2, 415-437. [23] Sugitani, S. (1988) Some properties for the measure-valued branching diffusion process, J . Math. Soc. Japan 41. 437-462. [24] Sznitman, A-S . (1991). Topics in the Propagation of Chaos, Ecole d'ete de Probabilites de Saint Flour, L . N . M . 1464. [25] Walsh, J .B . (1986). An Introduction to Stochastic Partial Differential Equations, Ecole d'ete de Probabilites de Saint Flour, L . N . M . 1180. [26] Yor, M . (1978). Sur la continuity des temps locaux associes a certaines semi-martingales, in Temps Locaux, Asterisque 52-53. [27] Zvonkin, A . K (1974). A transformation of the phase space of a diffusion process that removes the drift, Math. USSR Sbornik, Vol. 22 No. 1, 129-149. 101 ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. There is no restriction on the nature of this information, e.g., it could be plain text, hypertext, or an image; it could be a definition, information about the scope of a concept, editorial information, or any other type of information."}],"Genre":[{"label":"Genre","value":"Thesis\/Dissertation","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","classmap":"dpla:SourceResource","property":"edm:hasType"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","explain":"A Europeana Data Model Property; This property relates a resource with the concepts it belongs to in a suitable type system such as MIME or any thesaurus that captures categories of objects in a given field. It does NOT capture aboutness"}],"GraduationDate":[{"label":"GraduationDate","value":"1996-11","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","classmap":"vivo:DateTimeValue","property":"vivo:dateIssued"},"iri":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","explain":"VIVO-ISF Ontology V1.6 Property; Date Optional Time Value, DateTime+Timezone Preferred "}],"IsShownAt":[{"label":"IsShownAt","value":"10.14288\/1.0079976","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","classmap":"edm:WebResource","property":"edm:isShownAt"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","explain":"A Europeana Data Model Property; An unambiguous URL reference to the digital object on the provider\u2019s website in its full information context."}],"Language":[{"label":"Language","value":"eng","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/language","classmap":"dpla:SourceResource","property":"dcterms:language"},"iri":"http:\/\/purl.org\/dc\/terms\/language","explain":"A Dublin Core Terms Property; A language of the resource.; Recommended best practice is to use a controlled vocabulary such as RFC 4646 [RFC4646]."}],"Program":[{"label":"Program","value":"Mathematics","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","classmap":"oc:ThesisDescription","property":"oc:degreeDiscipline"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the program for which the degree was granted."}],"Provider":[{"label":"Provider","value":"Vancouver : University of British Columbia Library","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","classmap":"ore:Aggregation","property":"edm:provider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","explain":"A Europeana Data Model Property; The name or identifier of the organization who delivers data directly to an aggregation service (e.g. Europeana)"}],"Publisher":[{"label":"Publisher","value":"University of British Columbia","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/publisher","classmap":"dpla:SourceResource","property":"dcterms:publisher"},"iri":"http:\/\/purl.org\/dc\/terms\/publisher","explain":"A Dublin Core Terms Property; An entity responsible for making the resource available.; Examples of a Publisher include a person, an organization, or a service."}],"Rights":[{"label":"Rights","value":"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/rights","classmap":"edm:WebResource","property":"dcterms:rights"},"iri":"http:\/\/purl.org\/dc\/terms\/rights","explain":"A Dublin Core Terms Property; Information about rights held in and over the resource.; Typically, rights information includes a statement about various property rights associated with the resource, including intellectual property rights."}],"ScholarlyLevel":[{"label":"ScholarlyLevel","value":"Graduate","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","classmap":"oc:PublicationDescription","property":"oc:scholarLevel"},"iri":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the scholarly level of the author(s)\/creator(s)."}],"Title":[{"label":"Title","value":"Path properties and convergence of interacting superprocesses","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/title","classmap":"dpla:SourceResource","property":"dcterms:title"},"iri":"http:\/\/purl.org\/dc\/terms\/title","explain":"A Dublin Core Terms Property; The name given to the resource."}],"Type":[{"label":"Type","value":"Text","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/type","classmap":"dpla:SourceResource","property":"dcterms:type"},"iri":"http:\/\/purl.org\/dc\/terms\/type","explain":"A Dublin Core Terms Property; The nature or genre of the resource.; Recommended best practice is to use a controlled vocabulary such as the DCMI Type Vocabulary [DCMITYPE]. To describe the file format, physical medium, or dimensions of the resource, use the Format element."}],"URI":[{"label":"URI","value":"http:\/\/hdl.handle.net\/2429\/6160","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#identifierURI","classmap":"oc:PublicationDescription","property":"oc:identifierURI"},"iri":"https:\/\/open.library.ubc.ca\/terms#identifierURI","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the handle for item record."}],"SortDate":[{"label":"Sort Date","value":"1996-12-31 AD","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/date","classmap":"oc:InternalResource","property":"dcterms:date"},"iri":"http:\/\/purl.org\/dc\/terms\/date","explain":"A Dublin Core Elements Property; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF].; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF]."}]} 0). and (QH,nt) = (nHxcd,ntxcd). T h e o r e m 1.31. Any limit point K of the sequence (KN) satisfies the martingale problem: \\\/4>eD0 Zt(0). fi = ( f i , .F , (Tt),TP) wil l denote a filtered probability space satisfying the usual hypotheses. PifFt) denotes the cr-field of (.^-predictable sets in [0, co) x fi and (fi , :F, (Tt)) = (fi x C,T x C, (Tt x F ix 0 < h < t2 < ... < tk and # \u20ac ^ ( M * ) . If V e C, let * ( y ) = * - ( * i , * 2 , ' . . . , * f c ) ( y ) = . * ( y ( t i ) ; . . . , y ( * * ) ) -and denote the first and second order partials of fc-i V * ( t j y ) = E l ( i < \u00ab t + i ) * . i + r ( y ( t A \u00ab i ) , . . . , y ( t A t f c ) ) ; . . i=0 ^ fc - l fc-i . - # ( t , y ) = E 53l(\u00ab< W i A ' t j + i ) * m + u + i ( y ( t A t i ) , ; . . , y ( < A \u00ab f c ) ) . m=0 i=0 Let oo L>0 = ( J { * ( t l , * 2 , - , - * m ) : 0 < t i < t2 < - < . < m > * G C 0\u00b0\u00b0(IRm)} U {1}. ro=l D e f i n i t i o n 2.1 (One-dimens ional H i s t o r i c a l B r o w n i a n M o t i o n ) . Let m be a finite Borel measure on JR. A predictable process (Ht : t > 0) on fi with sample paths on fi# is a one-dimensional historical Brownian motion starting at m iff (Ht) satisfies the following martingale problem: (MP)TAAo is a continuous square integrable ^i-martingale such that (Z?$))t = fi j' $(y)2Hs(dy)ds Vi > 0 a.s. There exists a process H satisfying the conditions of Definition 2.1 and it is unique in law. (Dawson & Perkins 1991). We picture H as an infinitesimal tree of branching one-dimensional Brownian motions. 42 D e f i n i t i o n 2 .2 . Let (Kt : t > 0) be a process o n f i with paths on fi#. A set A C [0, oo) x fi is (K,JP) evanescent (or K evanescent) iff A C A i where A i is (JFt* )-predictable and sup IAJ {U,UI, y) = 0 Kt \u2014 a.e. y Vt > 0 P \u2014 a.s. 0 0) bea process on fi with paths on fi\/\/. A map b : [0,oo)xfi \u2014\u00bb\u2022 JR is if-integrable (respectively K-locally integrable) iff it is (Pf )-predictable and \/ 0 ' Ks(\\b(s)\\)ds < oo (respectively, f\u00a3 \\b(s)\\ds < oo Kt \u2014 a.a. y) Vt > O P \u2014 a.s. a For technical reasons that wil l become apparent later we shall not look directly at (2.1) but rather at a historical version of it. We shall call such version (HSE)b^ and it is defined below. D e f i n i t i o n 2 .4 . Suppose that H is a one-dimensional historical Brownian motion starting at m \u20ac MF(JR). Let b : [0,oo) x MF(JR) x IR\u2014> JR, 7 : [0,oo) x MF(SR) x C\u2014> (0, oc). ' Define the projection Ut : fi\/f -> MF{Wi) by Ui(K)(A) = Kt({y : y(t) \u20ac A}). We say that the pair (K, Y) solves the historical stochastic equation (a) Yt = yt+ f b{s,fls(K),Ys)ds, t > 0, ;\u00b0 (HSE)bn . ( b ) . ' KM) = j 0(y*)7(t,n.(A'),r)frt(dj\/)i 0. and sup | Z f ( ^ ) - Z f ( 0 ) | - ^ O o\u2022 00 Vfc G N . 46 See Perkins (1993, 1995) for details. ( Z \/ ^ ) ) ^ is a continuous local (.T^-martingale with square function (ZK{ib))t= f KsM2)ds. , (2.7) Jo. Since DQ is ftp-dense in bC, can be extended to an orthogonal martingale measure {Z*(<\/>) : t > 0, 0 a.s. P r o o f . See Theorem 2.8 and Proposition 2.13 of Perkins (1995) for the proof.' \u2022 P r o p o s i t i o n 2 .20 ( L e v y ' s m o d u l u s o f c o n t i n u i t y ) . Let h(t) = \\\/t\\og+(l\/t). Define L(6, c) = {y EC : \\y(t) - y(s)| < ch(t - s) Vs, t > 0 satisfying 0o-an + 3 c 2 P rsAT(n) \/ 0-(Xr,Yr)'' Jo dr + 3 P H sAT(n) sAT(n) b(Xr,Yr)dr (by Burkholder) < 3 p f A T ( n ) [ | y 0 - a | 2 ] + c 2 P rsAT(n) \u2022 \/ T ( X r ( l ) ) : Jo dr 85 - H I rsAT(n) 1 \\ + P S A T ( n ) Jo ?(Xr(l))dr2 j (by (3.3)) .< 3 hsxnn)[\\Yo - oi 2 ]+c 2p;;r (\u201e) ^ r \u201e . | 2 l ~ H rsAT{n) \/ T ( G ( n , r ) ) 2 d r Jo + P H sAT{n) rsAT(n) \/ T(6(n,r ) )dr JO (by Lemma 3.7) . < 3 (VsAT(n)[\\Yo -a| 2 ] + c 2 S T ( 0 ( n , S ) ) 2 + s 2 T ( Q ( n , s)) 2 ) C3.12.1 (*,\")\u2022 Similarly, P if, sAT(n) r-sAT(n) \/ s g n ( y r - a ) a ( X r , y r ) d y ( r ) Jo < P \/ rsAT(n) ; HSAT(n) l y a(X r ,y r ) dr = C3.i 2 . 2(5,n). The same reasoning also shows that ftAT(n) P i7 sAT(n) rtAl(n) \/ s g n ( y s - a ) 6 ( X \u201e y s ) d 6 Jo < c 3 . i 2 . 3 (s ,n) . As a consequence of (3.9), (3.13), (3.14) and (3.15) we obtain sup P 0 0 and restrict ourselves to the time interval [0,N]. As usual, some localization procedure is needed. We shall stop the processes H, X, Y and C?. at T(n). Note that \u00a3 r ^ A . 1S ^ o c a ^ t * m e \u00b0* Y?(n) (in Campbell space.) Also, Z^T^ is the orthogonal martingale measure associated with HT(n\\ We shall need the following estimates. < L e m m a 3 . 1 2 . For any p > 1 there are constants C3. i2 . i (n ,N,p ) , 0 3 . 1 2 . 2 (n, TV, p), 03.12.3(^1,TV,p) , such that for any (Ft)-stopping time T < T{n) A TV and any x,z \u20ac IR, s, t 6 [0, TV] and sup P T [ ( \u00a3 \u00a3 ) * ]