doctoral thesis · the world is strange, the whole universe is very strange, but see when you look...

Instituto Nacional de Matematica Pura e Aplicada

Doctoral Thesis

From Reaction-Diffusion Models

to the study ofStochastic Differential Equations

Conrado Freitas Paulo da Costa

Rio de Janeiro

2016

Instituto Nacional de Matematica Pura e Aplicada

Conrado Freitas Paulo da Costa

FROM REACTION DIFFUSION MODELS TO THE STUDYOF STOCHASTIC DIFFERENTIAL EQUATIONS

Thesis presented to the Post-graduate Program inMathematics at Instituto de Matematica Pura e Apli-cada as partial fulfillment of the requirements for thedegree of Doctor in Philosophy in Mathematics.

Advisor: Milton Jara

Rio de JaneiroMarch 16, 2016

Aos que vierem depois de nos, que sejam livrespara a amizade e para o pensamento.

Agradecimentos

Gostaria de agradecer a todos que me ajudaram nessa trajetoria. Em es-pecial, ao Tertuliano Franco, Daniel Valesin, Ivaldo Nunes pelas conversasdurante os cursos de verao.

Ao Francisco Ganacim e Marco Mendes, meus amigos e mestres. Vocesme ensinaram muito sobre a vida e a Matematica.

Ao meu orientador Milton Jara que em um momento crucial me motivoua continuar. Obrigado Milton, eu jamais faria o doutorado em Matematicase nao tivessemos conversado.

Aos professores do grupo de Probabilidade do IMPA, Claudio Landim,Roberto (Imbuzeiro) de Oliveira, Augusto Teixeira pelos incentivos e apoiospara as mais diversas duvidas.

Aos professores Elon Lages Lima, e Emanuel Carneiro e Benar Svaiterpelas licoes de matematica.

Aos funcionarios do IMPA em especial a divisao de Ensino pelos cuidadose orientacoes.

Aos meus amigos de faculdade, economistas que sonham e animam osonho de um Brasil melhor.

Em especial, agradeco ao meu irmao, Bernardo da Costa, que me aju-dou mais do que seria possıvel imaginar, desde a monografia ate a tese dedoutorado, seja por telefone, email ou pessoalmente.

Agradeco aos esforcos cooperativos que construiram a wikipedia.com omath.stackexchange.com, o http://gen.lib.rus.ec e o sci-hub.io.

Agradeco ao Johel Beltran, pela atencao dedicada ao problema da unici-dade de solucoes para a equacao diferencial da tese e por sugerir a leitura doartigo do Yamada e Watanabe de 1971.

Agradeco aos membros da minha banca de tese, Tertuliano Franco, LucaAvena, Bernardo de Lima, Augusto Teixeira, Roberto Oliveira e Milton Jara,pelo comentarios e correcoes indicadas para a versao final da tese.

Gostaria tambem de agradecer a universidade de Leiden e a hospitalidadedo instituto de matematica. Em especial aos professores Luca Avena e Frankden Hollander.

v

vi

E por fim, ao meu Pai e minha Mae, que me ajudaram desde os primeirospassos. Nao sei o que seria sem voces, obrigado.

vii

The World is strange, the whole universe is very strange, but see when you look atthe details and you find out that the rules are very simple of the game, the mechanicalrules by which you can figure out exactly what is going to happen when the situation issimple, is again this chess game... If you were in just a corner where only a few pieces areinvolved you can work out exactly what will happen. And you can always do that whenthere are only a few pieces so you know that you are understanding, and yet in the realgame is so many pieces that you can’t figure out what is going to happen. So there was akind of hierarchy of different complexities. It’s hard to believe it’s incredible in fact mostpeople don’t believe, that the behaviour of, say me, one jack jack and you, nodding andall, this stuff is the result of a lot and lot of atoms all obeying very simple rules... Comeout, that it evolves into such a creature with a billion years of life with its experiencesthat has produced the things with prawns that stick out like this and so on... the real...there is such a lot in the world there is so much distance between the fundamental rulesand the final phenomena that is almost unbelievable that the final variety of phenomenacan come from such a steady operation of such simple rules.

But you have to build from this complex scaffolding to find out the simple rules butit is not complicated, it’s just a lot of it, and if you start at the beginning which nobodywant to do... I mean, you come in to me now in interview and you are asking me about thelatest discoveries that have been made, nobody asks about a simple ordinary phenomenain the street: oh like What about those collors or something like that... I would have anice interview explaining all about the colors, Butterfly wings, whole big deal you don’tcare about that, you want the big final result. That is going to be complicated because Iam at the end of 400 years of very effective method of finding things out about the world.

It has to do with curiosity, it has to do with people wondering what makes somethingdo something and then to discover that if you try to get’s answers they are related to eachother that the things that make the wind make the waves, and the motion of water islike the motion of air and is like the motion of sand. The fact that things have commonfeatures turns out more and more universal. What we are looking for is how every thingworks and what makes everything work, but is curiosity, is the way we are, what we are.It is very much more exciting to discover that we are on a ball, heaven sticked upside downspinning arround in space with as a misterious force which hold us on going arround abig globe of gas that is burning by a fuel by a fire that is completely different that anyfire we can make, but know we can make that fire nuclear fire, you know.... That is muchmore exciting story to many people than the tales which other people used to make up,who were worried about the universe that we were living on back of a turtle or somethinglike that they were wonderful stories, but the truth is so much more remarkable and so,what is the pleasure of physics to me is as is revealed that the truth is so remarkableso amazing and I can’t.... I have this disease and many other people that have studiedfar enough to begin to understand a little how things work are fascinated by it and thisfascination drives them on to such an extent that they’ve able to convince governmentsand so on to keeps supporting them and this investigation that the race is making into it’sown environment.

THE FEYNMAN SERIES - Curiosityhttps://www.youtube.com/watch?v=lmTmGLzPVyM

ix

Abstract

Fix a finite graph (V,E). In this article we construct a family of Reaction-Diffusion models that converge after scaling to a solution to the followingStochastic Differential Equations (SDE’s):dζt(x) =

[∆V ζt(x)− β (ζt(x))k

]dt+

√α (ζt(x))l dBx

t ∀x ∈ Vζ0(x) = ρ0(x)

where α, β > 0 and k, l are positive integers such that k > l.We show that the limiting points of the tight familly of processes are

solutions to a well posed martingale problem and that the limiting measurecorresponds to the law induced by a solution to the SDE that we began with.

Keywords: Reaction diffusion models, Scaling limit of particle systems,Stochastic Differential Equations (SDE’s), Martingale Problems

x

Resumo

Fixe um grafo (V,E). Nesse trabalho iremos construir uma famılia de mod-elos dereacao e difusao que apos rescalados convergem para a solucao dasequinte Equacao Diferencial Estocastica (EDE):dζt(x) =

[∆V ζt(x)− β (ζt(x))k

]dt+

√α (ζt(x))l dBx

t ∀x ∈ Vζ0(x) = ρ0(x)

(1)

onde α, β > 0 e k, l sao numeros naturais tais que k > l, existe uma famılia demodelos de reacao e difusao que converge depois de escalados para a solucaoda equacao diferencial dada.

Mostraremos que os limites de escala dos processoos acima consideradoscorrespondem a solucao de um problema martingal bem posto e que a leiinduzida por ela trata-se da lei induzida pela solucao da equacao diferencialque tomamos como ponto de partida.

Palavras-chave: Modelos de reacao e difusao, Limites de escala de sis-temas de partıculas, Equacoes Diferenciais Estocasticas (EDE’s), problemasmartingais

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixResumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction 11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Solving a family of SDE’s . . . . . . . . . . . . . . . . . . . . 61.3 Strategy of proof . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 SDE’s and martingale problems . . . . . . . . . . . . . 71.3.2 From Scaling limits of particle systems to solutions of

Martingale problems . . . . . . . . . . . . . . . . . . . 101.3.3 From scaling limits of particle systems to solutions of

SDE’s . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Scheme of the proof . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Discrete models and uniform properties 152.1 Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Construction of the models . . . . . . . . . . . . . . . . . . . . 172.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Dynkin Martingales and useful computations . . . . . . . . . . 212.5 Discrete analogues of a and b for Ln . . . . . . . . . . . . . . . 222.6 Uniform results for the family ηn· n . . . . . . . . . . . . . . 23

3 The limit process 253.1 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Characterization of the limit . . . . . . . . . . . . . . . . . . . 28

3.2.1 Continuity of paths of the limit process . . . . . . . . . 283.2.2 Martingale problem . . . . . . . . . . . . . . . . . . . . 293.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Technical issues and computations 374.1 Non explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Lipschitz functions are in the domain of Ln and Qn . . . . . . 39

CONTENTS

4.3 Explicit formulae for Ln and Qn over coordinate functions . . 414.4 Uniform bounds on Lnh

mn . . . . . . . . . . . . . . . . . . . . 44

Chapter 1

Introduction

Philosophy is written in that great book which ever lies beforeour eyes — I mean the universe — but we cannot understand it ifwe do not first learn the language and grasp the symbols, in whichit is written. This book is written in the mathematical language,and the symbols are triangles, circles and other geometrical fig-ures, without whose help it is impossible to comprehend a singleword of it; without which one wanders in vain through a darklabyrinth.

Galileo–“Il Saggiatore” 1623

Somewhat surprisingly we have been discovering patterns in Nature andtranslating them into mathematical language. For this translation, differen-tial equations play a key role. To describe the evolution of a system withmany variables one often makes use of the theory of (partial) differential equa-tions. Differential equations are the common denominator of several modernexact sciences such as physics, chemistry, biology or engineering. Since thecentral aspect of various applications is the intrinsic randomness of thosesystems, one needs to take this into account. Probability theory has achievedimportant positive results in this direction: first it provided a theory for theBrownian motion1, then it opened the path for the theory of Stochastic differ-ential equations2, Finally it has develloped systematic thecniques to obtainfamily of processes that converge to solutions of an Stochastic DifferentialEquation (SDE).

1notably with the works of Einstein [Ein56] and Wiener [Wie21]2for instance with the works of Kolmogorov [KF31], Ito [Ito51] and Stroock-

Varadhan [SV69a, SV69b]

1

2 CHAPTER 1. INTRODUCTION

1.1 Motivations

At this point of our inquiry we look, from a distance, for what might bethe possible guiding lines for this work. It’s possible to single out a fewperspectives:

• To study reaction-diffusion models from the perspective ofparticle systems

Reaction-diffusion models have been interesting the scientific communityin general because of their wide range of applications. Indeed, reaction-diffusion models have been used to model a large class of phenomena inseveral fields of knowledge, for instance:

• Chemical Physics: Heat explosion, Chemical Kinetics, Polymerization,

• Biology: Morphogenesis, Epidemics, Population dynamics, and

• Physiology: Growth of tissues, Coagulation, Atherosclerosis

According to Volpert [Vol14], reaction-diffusion models originated withthe works of Semenov on thermal explosion in the 1930’s. Later, in 1952,came a notable work of Alan Turing: “The Chemical Basis of Morphogen-esis” [Tur52]. The idea was to capture a physical phenomenon through asuitable differential equation and then use the knowledge from its solutionsto understand the relevant features of the phenomenon.

More recent studies of reaction-diffusion models through Particle Systemsappear in [AT80], [Blo92], [Kot86] and [FG12]. In this context, we try tounderstand a physical phenomenon as a consequence of the interactions ofits underlying characters, the particles. To do so we construct a familyof probabilistic models that try to capture basic evolution patterns of thesystem. For Reaction-Diffusion models, we consider: motion, birth and deathof particles that occur at random times in a given environment.

The models of this family are meant to describe the same physical realitywith a different degree of precision. We hope that, as we increase precision,we will be able to extract some knowledge about this phenomenon. So ifwe prove convergence of a family of particle systems to the solution of anSDE we can use our knowledge about the particle systems to learn about thebehaviour of the solution of the SDE.

• To classify the behavior of fluctuations with respect to a con-stant solution of an SDE

As we deal with Reaction-Diffusion models in particle systems, we learnthat they give rise in the limit to solutions to equations of the following kind:

dζt(x) = (∆V ζt(x) + F (ζt(x))) dt+√G(ζt(x)) dBx

t . (1.1)

1.1. MOTIVATIONS 3

We are interested in working with functions F and G that behave locally aspolynomials. One reason for this is that, in the context of chemical reactions,the rate equation [Vol14, p.14] states that the rate of reaction is proportionalto integer powers of the relevant densities (depending on the stoichiometriccoefficients of the reaction under consideration). However, it is not truethat the same conditions hold independently of the density, this means thatreactions might have a polynomial behavior near a given density point butfor high densities this polynomial rule might no longer hold due to saturationand interactions. so one expect to be dealing with polynomial rates at a localscale and correcting terms for extreme densities.

Another reason for dealing with F and G as polynomials near a given den-sitity is because this gives rise to non linear SDE’s that have not been stud-ied so far and we hope to gain some knowledge about them using reaction-diffusion models in the particle systems approach.

To explain the square root on the term√G(ζ(x)) we turn to Stroock-

Varadhan theory of martingale problems (see [KS91] pgs 311 - 327). Forevery SDE there corresponds a second order differential operator. In ourcase for f ∈ C2(RV ) and ζ ∈ RV , we have L : C2(RV )→ C(RV ) defined by:

Lf(ζ) =∑x∈V

G(ζ(x))∂x∂xf(ζ) +∑x∈V

(∆V ζt(x) + F (ζ(x))) ∂xf(ζ)

The SDE corresponding to L is (1.1).If F and G are smooth functions and F (0) = G(0) = 0 we have that

ζt(x) = 0 for every x ∈ V and t ≥ 0 is a solution to (1.1). We would like tosee what happens in a vicinity of 0. The behaviour of the system will dependon the local behavior of the functions F and G. Our first interest will beto study the relation between solutions of the SDE and the derivatives of Fand G. The idea is that the first non-null derivative will determine the localbehaviour of the system, that is, we consider that there are integers k, l suchthat:

• F (k) 6= 0 and F (j) = 0 for every j < k and

• G(l) 6= 0 and G(j) = 0 for every j < l.

In order to obtain solutions that remain in the vicinity of 0 we assume thatF (k) < 0. Let’s say that F (k)(0) = −β < 0 and G(l)(0) = α > 0. then wewould like to find solutions to the following SDE:

dζt(x) =(∆V ζt(x)− β(ζt(x))k(1 + o(1))

)dt

+√α(ζt(x))l(1 + o(1)) dBx

t

(1.2)


To remove this undesired o(1) terms of the equation, we consider thesimpler equations:

dζt(x) =(∆V ζt(x)− β(ζt(x))k

)dt+

√α (ζt(x))l dBx

t (1.3)

It turns out that for this equations we allow for arbitrary initial condition.So we don’t need to worry about being in a vicinity of 0.

• To study nonlinear SPDE’s from the perspective of particlesystems

The study of differential equations is at the core of the major advancesof the sciences of the modern period. Mathematicians and physicists of thattime shared the idea that in some sense natural phenomena could be in-terpreted by mathematical equations. The differential equations could beunderstood in some vague sense as the ultimate laws of nature governing alarge class of phenomena. This motivation thrived and remains behind thestudy of differential equations. Indeed, in the words of Richard Feynmannone can hear the echoes of this conceptions (see epigraph or [Gow11]). Inthis spirit when we derive new equations we learn more about the world welive in, we uncover new laws of nature.

One way to study SDE’s is via scaling limits of particle systems (see[KL99]). In the same spirit as before when we described physical phenomenavia particle systems, now we describe with increasing precision some idealizedphysical reality and then find convenient scales to fit together those descrip-tions into a macroscopic object. When scales are well chosen, the familyof rescaled processes converge to a limiting object that solves an SDE. Thisprovides a way to use Reaction-Diffusion models to derive solutions to SDE’s.

In our case we consider Reaction-Diffusion models taking values on a finiteundirected graph (V,E). Typically one can think of (V,E) as a discrete toruson R3. This graph is a model for the space. As we obtain convergence of afamily of particle systems to the solution of an SDE on the graph, the ideais to make the distance between points smaller and smaller, by increasingthe number of vertices and edges and ajusting the scales to preserve thedescription of a macroscopic object, to obtain a solution in the continuousspace, that is, a solution for an SPDE.

Establishing the existence and uniqueness of solutions for SPDE’s, suchas the KPZ [KPZ86], given formally in [HS15] by:

∂th = ∂2xh+ λ (∂xh)2 + ξ (1.4)

as well as interpreting their meaning has challenged the community forover 30 years. It was not until recently that major progresses have been

1.1. MOTIVATIONS 5

made. Notable among those are: Martin Hairer’s theory of regularity struc-tures [Hai14], the notion of energy solutions introduced by Goncalves andJara [GJ13], and the proof by Gubinelli and Perkowski that those solutionsare unique [GP15].

An also interesting SPDE is the Parabolic Anderson Model:

∂tu = ∆u+ u · ξ (1.5)

Where u is a function of t ≥ 0, x ∈ R2 and ξ is a white noise on R2 see [HL15].The study of the PAM equation has also benefited from the theories dev-

elloped to solve the KPZ. The interesting (mathematically challenging) termin the KPZ equation is (∂xh)2 and in the PAM equation is u · ξ. Thoseequations are ill posed because in the continuous setting solutions are dis-tributions and the product of distributions is not in general a well definedobject. The recent efforts and advances towards solving those equations haveprovided grounds for making sense of those products.

One might combine these interesting features in a new equation:

∂tu = ∆u− u2 + u · ξ. (1.6)

It is worth noting that this equation is very different from both the KPZand the PAM. Since for instance we have the term u2 in the above equationinstead of (∂xu)2. However, from the our point of view, this equation shouldchallenge us in simmilar ways as the two above have. So there is an interestin solving it. One way to approach the problem is to consider a graph (V,E)as a discrete model for the space and deal with the spatially discrete analogof this equation:

du =(∆V u− u2

)dt+ u dBt (1.7)

In this setting the Laplacian is no longer continuous but discrete:

∆V u(x) =∑y:y∼x

u(y)− u(x)

and the noise term is driven by a vector valued Brownian motion. So, inthis case, instead of an SPDE we have an ordinary Stochastic DifferentialEquation (an SDE). The idea is to learn from the discrete setting and thento devise a strategy to move to the continuous case.

If for a moment we ignore the noise term, we get:

du =(∆V u− u2

)dt

Which is reminiscent of the family of Reaction-Diffusion differential equations(for an introduction see [Vol14])

This leads us to use Reaction-Diffusion models in particle systems tostudy solutions to equations such as (1.7).


1.2 Solving a family of SDE’s

We fix k, l ∈ N, k > l and α, β > 0, and consider the following SDE:dζt(x) =[∆V ζt(x)− β (ζt(x))k

]dt+

√α (ζt(x))l dBx

t ∀x ∈ Vζ0(x) = ρ0(x)

(1.8)

where (Bx)x∈V is a |V |-dimensional Brownian motion and ρ0 is an arbitraryinitial condition.

The main new result in this work is the explicit construction for eachof these SDE’s of a family of stochastic processes (ηn· )n = ( ηnt ; t ≥ 0 )nissued from Particle Systems. Under suitable initial conditions, this familyconverge after scaling to the solution of the given SDE with respect to afixed initial condition. Convergence here is meant as convergence in law ofthe probability measures with respect to the J1-Skorohod topology [Bil99,

chapter 3] and is denoted by Pn J1−−−→n→∞

P∗ where Pn and P∗ are probability

measures on D =(D([0,∞),RV ), J1

).

We consider the scaling ζn· = ηn·n

, and note that they correspond to proba-bility measures on

(D([0,∞),RV ), J1

). We can then state the main theorem

of this work:

Theorem 1. Let k, l ∈ N, k > l, let α, β > 0 (Bx)x∈V be a |V |-dimensionalBrownian motion and let ρ0 ∈ RV . There is a family of stochastic processes(ηn· )n induced by reaction-diffusion models such that for every x ∈ V ζn0 (x)→ρ0(x) and ζn· (x)

J1−−−→n→∞

ζ∗· (x) where ζ∗· = (ζ∗· (x))x∈V is the (unique) solution

of the stochastic differential equation (1.8) with ζ∗0 = ρ0.

An interesting consequence of our proof is the existence of solutions to aclass of non-linear SDE’s that do not admit solutions via the Picard method.From a technical point of view, the estimate we used for the uniform boundsused in the proof of the Theorem is a usefull tool and we hope it can be usedin other contexts.

In a sense, what is really new is the idea to use Reaction-Diffusion modelsin particle systems to this kind of problems. Not only they allow us to solvedifferential equations that have reaction and diffusion terms but also providea framework to deal with the fluctuations terms encoded in the function thatmultiplies the noise. Traditionnaly, the fluctuation terms were interpretedas the result of random catalytic activity. Our models are thus called auto-catalytic, since their randomness stems from the dynamic of the particlesystem instead of an external source.

1.3. STRATEGY OF PROOF 7

One interesting property left to understand in this model is whether thesolutions to the SDE’s will eventually die. More deeply, we could study thedistribution of the time it takes for the process to die. For this, one mighthope to get some insight from the discrete models and the simulations theyallow us to do. One last related question is to study the implementation ofthose simulations and compare them with other discretization techniques.

Theorem 1 also suggests investigating its generalizations. A first directionis to include the case k ≤ l. Another, is the case of infinite sites, |V | = ∞.Besides that, one can treat more general neighboring relations, modelingasymetrical environments.

In more broad directions, we could also study the case where the reactionterm is positive, which requires studying finite explosion times. Open stillis the case in which F has multiple zeros; this might lead to the study ofmultiple equilibria and metastability. Finally, the ultimate goal is to adaptour procedure to general functions F and G, and general graphs V in such away that it becomes a framework to obtain solutions to SPDE’s.

The proof of this theorem is now the subject of our concern.

1.3 Strategy of proof

To prove the theorem, for each given SDE (1.8) we have to build a particlesystem and then prove tightness for the scaled processes derived from it,according to conveniently chosen initial conditions, and finally characterizethe limit points as the solution of a well posed martingale problem. Beforemoving on, it is important to discuss somewhat vaguely the relation betweena few major ideas that will make the scheme of the proof more sensible.Those are:

• the relation between SDE’s and martingale problems

• the relation between scaling limits of particle systems and martingaleproblems

• a heuristic argument for how scaling limits of particle systems mightgive rise to solutions of an SDE.

1.3.1 SDE’s and martingale problems

Let b : Rd → Rd and σ : Rd×r → R be measurable functions. Let B· =(B1· , . . . , B

r· ) be an r-dimensional Brownian motion. Consider the following


abstract d-dimensional SDEdXt = b(Xt) dt+ σ(Xt) dBt

X0 = x0(1.9)

or in coordinates dX i

t = bi(Xt) dt+∑

j σi,j(Xt) dBjt

X i0 = xi0

A weak solution [KS91, p. 300] to this equation with initial condition x0is a triple [(X,B), (Ω,F ,P) ,Ft] where

(i) (Ω,F ,P) is a probability space and Ft t is a filtration satisfying theusual conditions, [KS91, p.10]

(ii) X = Xt,Ft; 0 ≤ t <∞ is a continuous adapted Rd-valued process,B = Bt,Ft; 0 ≤ t <∞ is an r-dimensional Brownian motion,

(iii) Almost surely∫ t0|bi(Xs)|+σi,k(Xs) ds <∞ for all 1 ≤ i ≤ d, 1 ≤ k ≤ r,

0 ≤ t <∞, and,

(iv) Almost surely

X it = xi0 +

∫ t

0

bi(Xs) ds+r∑

k=1

∫ t

0

σi,k(Xs) dBjs ; ∀t ≥ 0, ∀ 1 ≤ i ≤ d

The probability law induced by the weak solution [(X,W ), (Ω,F ,P) ,Ft] is theprobability measure P∗ on

(C[0,∞)d,B

(C[0,∞)d

)), obtained by

P∗(A) = P [X· ∈ A] .

Ito’s calculus provides a closed formula for the value of f(Xt) for every f ∈C2(Rd). The following equality holds almost surely:

f(Xt) = f(X0) +d∑i=1

(∫ t

0

∂if(Xs)bi(Xs) ds+r∑

k=1

∫ t

0

∂if(Xs)σi,k(Xs) dBks

+1

2

d∑j=1

∫ t

0

∂i∂jf(Xs)ai,j(Xs) ds

)where ai,j(x) =

∑rk=1 σi,k(x)σj,k(x) for 1 ≤ i, j ≤ d. Rearranging the terms

we obtain


f(Xt)− f(X0)−d∑i=1

∫ t

0

∂if(Xs)bi(Xs) ds−1

2

d∑i,j=1

∫ t

0

∂i∂jf(Xs)ai,j(Xs) ds

=d∑i=1

r∑k=1

∫ t

0

∂if(Xs)σi,k(Xs) dBks

which is a continuous local martingale.

Based on this we define the second order differential operator L by

Lf(x) =d∑i=1

bi(x)∂if(x) +1

2

d∑i,j=1

ai,j(x)∂i∂jf(x)

and the function of trajectories

M f,Lt (X·) = f(Xt)− f(X0)−

∫ t

0

Lf(Xs) ds

and we see that for each solution [(X,W ), (Ω,F ,P) ,Ft] we obtain a family

of continuous local martingalesM f,L

t (X·),Ft; t > 0

.

This illuminates the following definition:

Definition 1. A probability measure P on(C[0,∞)d,B

(C[0,∞)d

))under

whichM f,L

t ,Ft; t ≥ 0

is a continuous local martingale for every f ∈C2(Rd) is called a solution to the martingale problem associatedwith L.

Using the notation defined up to now, the next theorem [KS91, p. 315]completes the relation between Martingale problems and solutions to an SDE.

Theorem. Let P∗ be a probability measure on(C[0,∞)d,B

(C[0,∞)d

))un-

der which M f,L· is a continuous local martingale for the choices of f(x) = xi

and f(x) = xixj, 1 ≤ i, j ≤ d then there is an r-dimensional Brownianmotion W on (Ω,F ,P) an extension of

(C[0,∞)d,B

(C[0,∞)d

))such that

[(X,W ), (Ω,F ,P) ,Ft] is a weak solution of

dXt = b(Xt)dt+ σ(Xt)dWt

and P∗ is the law induced by this weak solution.


So we see that to each solution to a martingale problem, there corre-sponds a solution to an SDE. This theorem is interesting not only because itcompletes the relation between solutions of an SDE and solutions to a mar-tingale problem but also because it gives a simpler criteria to check whetheror not, a given measure is a solution to a martingale problem. We don’tneed to verify for every f ∈ C2(Rd) that the function ML,f

· is a continuouslocal martingale, now we only need to verify this for the coordinate functionsf(x) = xi and the product of two coordinates f(x) = xixj.

We conclude this section with a final remark:

Remark. The martingale problem that corresponds to a weak solution to (1.9)is the martingale problem associated with L given by

Lf(Xs) =d∑i=1

∂if(Xs)bi(Xs) +1

2

d∑i,j=1

∂i∂jf(Xs)ai,j(Xs)

where a = σσT or in coordinates, ai,j(x) =∑r

k=1 σi,k(x)σj,k(x).

1.3.2 From Scaling limits of particle systems to solu-tions of Martingale problems

Let’s assume for simplicity that we have a family of particle systems ζn· ,associated with the infinitesimal generator Ln. That ζn· converges to ζ∗· andthat it happens that for every f ∈ C2(Rd) and ζ ∈ Rd

Lnf(ζ)→ L∗f(ζ)

uniformly in compact sets, where L∗ is a second order differential operator

such thatd∑

i,j=1

zia∗i,j(ζ

∗)zj ≥ 0 .

Consider for each f ∈ C2(Rd) ∩n D(Ln) the Dynkin martingales [Lig99,Theorem 3.32]

M f,Lnt = f(ζnt )− f(ζn0 )−

∫ t

0

Lnf(ζns ) ds.

From the convergence of ζn· to ζ∗· we obtain

M f,Lnt = f(ζnt )− f(ζn0 )−

∫ t

0

Lnf(ζns ) ds

↓

M f,L∗t = f(ζ∗t )− f(ζ∗0 )−

∫ t

0

L∗f(ζ∗s ) ds.


Here convergence follows from the continuity of the function M f,L∗(ζn) withrespect to the Skorohod topology, plus the fact that uniform convergence incompact sets and tightness give us that

Pn[supt≤T

∣∣∣M f,Lnt (ζ)−M f,L∗

t (ζ)∣∣∣ > ε

]−−−→n→∞

0.

If we prove that M f,Ln· are uniformly integrable then we can conclude

that the limit M f,L∗· is also a martingale [Dur10, p. 259 Theorem 5.5.2].

This means that ζ∗· is a solution to the martingale problem associated withL∗

1.3.3 From scaling limits of particle systems to solu-tions of SDE’s

Consider the functions fi(ζ) = ζ(i). Rearranging the expressions of theDynkin Martingales, we get:

ζnt (i) = ζn0 (i) +

∫ t

0

bi(ζns ) ds+M fi,Ln .

Using the continuity of the above functions and the convergence of ζn· toζ∗· , we obtain

ζnt (i) = ζn0 (i) +

∫ t

0

(Lnfi)(ζns ) ds+M fi,Ln

t

↓

ζ∗t (i) = ζ∗0 (i) +

∫ t

0

(L∗fi)(ζ∗s ) ds+M fi,L∗

t

In the case our family of martingales is uniformly integrable and the limitζ∗· is almost surely continuous, then M fi,L∗

· is a continuous local martingale.If we do some more calculations which involve the product of two coordinatesfi,j(ζ) = ζ(i)ζ(j) we learn about the covariance of M fi,L∗ and M fj ,L∗ . Bythe representation of martingales theorem [KS91, pp.315-316 pp.170-172] weobtain [(ζ∗, B), (Ω,F ,P) ,Ft] such that:

M fi,L∗t =

d∑j=1

∫ t

0

σ∗i,j(ζ∗s ) dBj

s

with

P[∫ t

0

(σ∗i,j(ζ

∗s ))2ds <∞

]= 1; 1 ≤ i, j ≤ d; t ≥ 0


and defining b∗i (ζ∗s ) = L∗fi(ζ

∗s ) since M fi,L∗ is a continuous local martingale

P[∫ t

0

|bi(ζ∗s )|2 ds <∞]

= 1; 1 ≤ i, j ≤ d; t ≥ 0

So, to sum up:

(i) (Ω,F ,P) is a probability space and Ft t is a filtration satisfying theusual conditions,

(ii) ζ∗ = ζ∗t ,Ft; 0 ≤ t <∞ is a continuous adapted Rd-valued process,B = Bt,Ft; 0 ≤ t <∞ is an d-dimensional Brownian motion,

(iii) Almost surely,∫ t0|b∗i (ζ∗s )| + σ∗i,j(ζ

∗s ) ds < ∞ for every 1 ≤ i, j ≤ d

0 ≤ t <∞, and,

(iv) Almost surely,

ζ∗t (i) = ζ∗0 (i) +

∫ t

0

bi(ζ∗s ) ds+

d∑j=1

∫ t

0

σ∗i,j(ζ∗s ) dBj

s ; ∀t ≥ 0 ∀ 1 ≤ i ≤ d

Once we have this and convergence on the initial condition ζn0 → x0 we havea weak solution to the SDE

dζ∗t = b∗(ζ∗t ) dt+ σ∗(ζ∗t ) dBt

ζ∗0 = x0(1.10)

1.4 Scheme of the proof

Now that we’ve put together the relation between the major concepts (SDE’s,martingale problems and scaling limit of particle systems) and that we haveseen a little bit how tightness and uniform integrability come into play, wecan present the scheme of the proof of the theorem and then the steps wewill need to complete the proof:

1.4. SCHEME OF THE PROOF 13

GeneratorsLn

Processes ζnDynkin

MartingalesM f,Ln

“Generator”L∗

Process ζ∗Limit

MartingalesM f,L∗

SDE

SDE

Discrete models

Rescaling

Tightness Unif. Integrability

Martingale Problem

uniqueness

The idea is that given one SDE, we would like to construct a family of particlesystems3, inspired by the relations between martingale problems and SDE’s.

Chapter 2 is devoted to the study of the discrete models, from the con-struction to the analysis of the scaled processes ζn· = ηn·

nand the Dynkin

martingales associated to them. To do this we consider fn(η) = f(ηn

)and

then:

M fn,Lnt := fn(ηnt )− fn(ηn0 )−

∫ t

0

(Lnfn)(ηns ) ds.

We then prove uniform bounds that will be useful in the proof of convergence,since those uniform bounds are fundamental in the proof of tightness on theone hand and uniform integrability on the other.

Chapter 3 deals with the passage to the limit. To do this we prove that:

(a) the family of processes (ζn· )n is tight, thus we can consider a subse-quence of ζn· converging to a limit point ζ∗· ;

(b) the martingales M fn,Ln (for convenient fn) are uniformly integrable,and the expression of the limit martingales M f,L∗ can be written interms of the limit trajectories ζ∗· and a second order operator L∗.

3we refer to the family of particle systems as discrete models due to some vague resem-blance with Euler discretization scheme used to solve differential equations in the earlydays.


Then we need to show that the limit is unique. To do so we prove that:

(c) the probability measure corresponding to ζ∗· is the unique solution tothe martingale problem associated with L∗;

The question of uniqueness deserves some comments. The diffusion co-efficient G(ζ(x)) vanishes if ζ(x) = 0, so we don’t have uniform ellipticity,which is a usual criterion for uniqueness of martingale problems.

However, since there is a correspondence between martingale problemsand SDE’s, if we prove uniqueness of solutions to the SDE it will implyuniqueness of solutions to the corresponding martingale problem. We thentry to prove pathwise uniqueness for the SDE’s. There are two cases toanalyse: l ≥ 2 and l = 1. The case l ≥ 2 is simpler, since the coefficientsof the SDE (b, σ) satisfy locally Lipschitz conditions, from which a standardargument shows that pathwise uniqueness follows.

For the case l = 1, the dispersion coefficients vanish as√x near the

boundary of the positive region, so the Lipschitz condition fails near x = 0.Moreover, we know that if the dispersion matrix vanished with rate xα withα < 1

2we would no longer have uniqueness. Therefore, the fact that we can

also prove uniqueness here indicates that this a limit situation.From an intuitive point of view one believes that uniqueness holds since

the drift term in the boundary away from zero is a positive vector driving thetrajectory towards the interior of the region. So one expects that the localtime in the boundary away from is zero is zero and that there is no morethan one solution before reaching zero. Since zero is an absorbing state thatwould give us a global uniqueness result.

We found well established criteria in the papers of Yamada and Watanabe(1971) and in the book of Karatzas that suit the case l = 1. So even thoughwe prove uniqueness of solutions in a quite straightforward way, it might bethat for similar problems one such criterion is not available. So there is ageneral interest in understanding more about boundary conditions and othercriteria for uniqueness as for instance checking that the limit second orderdifferential operator L∗ is a probability generator as in [Lig99, p.312]. Thisleads us to investigate among other properties, whether R(I − λL∗) is densein C(RV ) for all sufficiently small λ.

Fortunately, for the moment, we don’t need to worry about those issues.

Chapter 2

Discrete models and uniformproperties

The purpose of this chapter is to complete the first part of the scheme ofthe proof in (1.4). We first discuss how to build general reaction-diffusionmodels. Then, inspired by a fixed SDE as in (1.8) we construct a family ofparticle systems that, after scaling, converges to a solution to the given SDE.Afterwards, we discuss briefly the Dynkin Martingales with respect to thecoordinate functions, and introduce the notion of discrete coefficients for Ln.Finally, we derive uniform properties for the family of particle systems.

To sum up, in this chapter we prepare the key ingredients for the passageto the limit that takes place in chapter 3.

2.1 Set up

To construct a particle system, one needs to define its configurations, specifythe admissible transitions and the rates at which they occur. This is whatwe shall do next for a general reaction-diffusion model. In this work, allreaction-diffusion models are going to be particular instances of this generalone.

Fix a finite set V . The points of this set shall be denoted by x, y, z, . . .and are called sites. We consider a neighboring relation ∼ on V such thatx x, x ∼ y ⇒ y ∼ x, and we put E = x, y | x ∼ y . The resultinggraph G = (V,E) is a model for the space on which our dynamics takesplace, where the points in V represent the loci, and the edges of E indicatewhich loci are close to one another. To keep track of the number of particlesat each site, we denote by η(x) the number of particles at site x and byη = (η(x))x∈V the configuration of the system.

15

16 CHAPTER 2. DISCRETE MODELS AND UNIFORM PROPERTIES

The dynamics we consider is of a probabilistic nature. There are 3 typesof transition that occur independently of one another and at random times.Particles on site x may jump to each neighboring site y ∼ x according toexponential times with rate 1; moreover, at each site x, a particle can becreated with rate F+(η(x)); and finally at each site x a particle can beannihilated with rate F−(η(x)).

We need a notation for the resulting configuration for each transition:

• ηx,y is the configuration obtained from η by moving one particle fromx to y, if possible. That is, if η(x) > 0:

ηx,y(z) =

η(y) + 1 if z = y

η(x)− 1 if z = x

η(z) otherwise

while if η(x) = 0 then ηx,y = η.

• ηx,+ corresponds to the creation of one particle at site x:

ηx,+(z) =

η(x) + 1 if z = x

η(z) otherwise

• ηx,− corresponds to the annihilation of one particle at site x if possible.That is, if η(x) > 0:

ηx,y(z) =

η(x)− 1 if z = x

η(z) otherwise

while if η(x) = 0 then ηx,− = η.

Thanks to Hille-Yoshida’s Theorem [Lig99, pp.102-103], we can describethis class of models more briefly by encoding all this information on theoperator L:

Lf(η) =∑x∼y∈V

η(x) [f(ηx,y)− f(η)] +∑x∈V

F+(η(x))[f(ηx,+)− f(η)

]+∑x∈V

F−(η(x))[f(ηx,−)− f(η)

].

Transition rates can be read as the factors multiplying the differences betweenthe transitions f(η•) and the original state f(η). Note that since all particles

2.2. CONSTRUCTION OF THE MODELS 17

at a given site x can move to a neighboring y, the transition rate for jumpsfrom any particle in x to y is η(x), and not 1.

Our processes will be well-defined once we rule out explosions. Explosionsoccur when, in a finite time, an infinite number of transitions occurs. In ourcase, F+(u) ≤ C(1 + u) and there can be no explosions; we postpone theproof of this result to the appendix 4.1.

On these conditions, L is a probability generator [Lig99, p. 97]. So, wecan construct the particle system by first defining a probability semigroup(TL(t)

)t≥0 by the following formula:

∀ f ∈ Cb(RV ) : TL(t)f = limn

(I − t

nL)−n

f. (2.1)

Then, for each ξ ∈ RV , we define Pξ as the only measure whose finite-dimensional evaluations are compatible with the given transition semigroupTL. That is, for every k ∈ N, t1 < t2 < . . . < tk and η1, . . . , ηk ∈ NV :

Pξ [η0 = η, ηt1 = η1 . . . , ηtk = ηk] =

1η=ξ TLt1

(1η1)(η) TLt2−t1(1η2)(η1) · · · TLtk−tk−1

(1ηk)(ηk−1) (2.2)

Finally, we fix µ0, a measure in RV to stand for an initial condition, anddefine for a bounded and measurable function f : D([0,∞),RV )→ R:

Eµ0 [f ] :=

∫Eξ [f ] dµ0(ξ). (2.3)

This defines the stochastic process associated with the generator L and theinitial condition µ0 which is the measure Pµ0 on D([0,∞),RV ) such thatPµ0(A) = Eµ0 [1A]. We will denote it by

ηLt ; t ≥ 0, ηL0 = µ0

or simply

η· = ηt; t ≥ 0 leaving the generator and the initial condition implicit.At this point, we can show that

Lf(η) = limh→0

1

hEη [f(Xh)− f(X0)]

for all functions f for which the above limit exists. We say that these func-tions belong to the domain of L.

2.2 Construction of the models

We will consider a family of stochastic processes (ηn· = ηnt ; t ≥ 0 )n∈N cor-responding to reaction-diffusion models on the graph (V,E).


The goal in constructing these models is to find a family of processesconverging to the solution of the following SDE:

dζt(x) =[∆V ζt(x)− β (ζt(x))k

]dt+ (αζt(x))l/2 dBx

t ∀x ∈ Vζ0(x) = ρ0(x)

(2.4)

where B = (Bx· )x∈V is a vector of independent Brownian motions.

The macroscopic parameter for this family is the “density” of particles

ζnt (x) =ηnt (x)

n. In the spirit that particle systems describe a same physical

reality with different degrees of precision, we can imagine that our modelbecomes more precise as n gets larger. and by taking n→∞ we expect thatif the initial density of particles converges (ζn0 (x) → ρ0(x) for every x ∈ V )then ζn· → ζ∗· where ζ∗· solves (2.4).

The reaction-diffusion models can be described by their infinitesimal gen-erators Ln. In our case they have the following expression:

Lnf(η) =∑x∼y∈V

η(x) [f(ηx,y)− f(η)] +∑x∈V

F+n (η(x))

[f(ηx,+)− f(η)

]+∑x∈V

F−n (η(x))[f(ηx,−)− f(η)

](2.5)

where the rates F+n (η(x)) and F−n (η(x)) now depend on n to account for the

macroscopic compatibility after scaling. For this, we consider the simple casewhere V = x as in this case no diffusion occurs and the notation simplifiesas η = η(x).

The relevant quantities observed from these models due to birth and deathof particles are the average incremental1 net result of birth and death in theprocess ζnt , and the incremental deviation to be expected from this average.These are given by:

Fn(nζn) = 1n

[F+n (nζn)− F−n (nζn)]

Gn(nζn) = 1n2 [F+

n (nζn) + F−n (nζn)](2.6)

After introducing suitable scalings, we want that Fn and Gn converge tothe macroscopic rates −βζk and αζk. So we can see how the parametersthat appear in our SDE (2.4), the integers (k, l) and the positive reals (α, β),influence the discretized models.

1average incremental values are obtained by limh→0 h−1E [f(Xh)− f(X0)] = Lf(X0)

and incremental deviations by limh→0 h−1E

[(f(Xh)− f(X0))

2]

= Qf(X0)

2.2. CONSTRUCTION OF THE MODELS 19

This can be done by defining:

F (u) = −βuk + f(u)uk

G(u) = αul + g(u)ul.

Here, the functions f and g are correcting factors such that

• F+ = 12(F + G) and F− = 1

2(G − F ) are positive, since both will be

related to rates F+n and F−n in our process;

• F+n (u) ≤ Cn(1 + u); and

• limu→0 f(u) = 0 and limu→0 g(u) = 0.

For concreteness, as one example to have in mind, one can take f(u) = 0and g(u) =

(βuk−l − α

)1βuk≥αul(u).

We introduce the scalings

F+n (u) = nλF+

( unν

)(2.7)

F−n (u) = nλF−( unν

)(2.8)

and go back to the macroscopic quantities (2.6) to obtain

Fn(nζn) = nλ−1(ζnn1−ν)k (−β + f

(nζn

nν

))Gn(nζn) = nλ−2

(ζnn1−ν)l(α + g

(nζn

nν

)).

Equating the exponents of n to zero, this gives usλ− 1 + k(1− ν) = 0

λ− 2 + l(1− ν) = 0.

This determines the parameters ν = 1 + 1k−l and λ = 1 + k

k−l . Since ν > 1and assuming that ζn stays bounded (tightness gives us boundedness witharbitrarily high probability), we have that nζn

nν→ 0. Therefore

Fn(nζn) = (ζn)k(−β + f

(nζn

nν

))= (ζn)k (−β + f(o(1)))

Gn(nζn) = (ζn)l(α + g

(nζn

nν

))= (ζn)l (α + g(o(1))) .

This shows that Fn and Gn do converge to the desired macroscopic quantities.To sum up,

((ηn· (x))x∈V = ηn·

)n∈N will be the family of Markov processes

corresponding to the reaction-diffusion process with birth rate F+n as in (2.7)

death rate F−n as in (2.8) and initial conditions ηn0 satisfying ζn0 → ρ0.


2.3 Notation

Since the process ηn· corresponds to a probability measure Pn on the spaceD it will be convenient to write for an event A:

P [ηn· ∈ A] = Pn (A) .

In the same spirit, we let En denote the expectation with respect to Pn andwe write for f : D → R

E [f(ηn· )] = En [f ] .

From the definitions of the previous section, we can write

F+n (ηn(x)) = nλF+

(ζn(x)

nν−1

)=

1

2nλ[G

(ζn(x)

nν−1

)+ F

(ζn(x)

nν−1

)]=

1

2n2ζn(x)l

[α + g

(ζn(x)

nν−1

)]− nζn(x)k

[β − f

(ζn(x)

nν−1

)]and analogously

F−n (ηn(x)) =1

2n2ζn(x)l

[α + g

(ζn(x)

nν−1

)]+ nζn(x)k

[β − f

(ζn(x)

nν−1

)].

This motivates the following notation:

gn,α (ζn(x)) =1

2(ζn(x))l

(α + g

(ζn(x)

nν−1

))fn,β (ζn(x)) =

1

2(ζn(x))k

(β − f

(ζn(x)

nν−1

))which simplifies the expressions of F+

n and F−n to

F+n (ηn(x)) = n2gn,α(ζn(x))− nfn,β(ζn(x)) (2.9)

F−n (ηn(x)) = n2gn,α(ζn(x)) + nfn,β(ζn(x)). (2.10)

We now define the function Sn(η) :=∑

x∈Vη(x)n

. When applied to ηntwe denote it by Snt =

∑x∈V ζ

nt (x) which is a measure of the total mass of

particles in the system for the process ηn· at time t. This allows us to definethe following stopping times:

τnK := inf t > 0;Snt > K

τn := inft > 0;Snt > w n

ν−12

τn := inf

t > 0;Snt > w nν−1

2.4. DYNKIN MARTINGALES AND USEFUL COMPUTATIONS 21

where w is such that sup|z|≤w |f(z)| ≤ β and sup|z|≤w |g(z)| ≤ α. So forSn(η) ≤ wnν−1:

−2fn,β(ζn(x)) = (ζn(x))k(−β + f

(ζn(x)

nν−1

))≥ −2β (ζn(x))k (2.11)

2gn,α(ζn(x)) = (ζn(x))l(α + g

(ζn(x)

nν−1

))≤ 2α (ζn(x))l (2.12)

which means that for t < τn (and consequently for t < τn) we have a goodestimate on the increment rate of the relevant macroscopic quantities of theprocess.

2.4 Dynkin Martingales and useful computa-

tions

Let f : NV → R be a function in the domain of Ln in the sense of Section 2.2.We omit the dependence on ω and denote by M f,Ln

t the Dynkin martingaleassociated with Ln and f :

M f,Lnt = f(ηnt )− f(ηn0 )−

∫ t

0

Lnf(ηns ) ds. (2.13)

It basically says that the Dynkin martingale is an error term with respect tothe prediction f(ηn0 ) +

∫ t0(Lnf)(ηns ) ds.

The variance at time t of the Dynkin martingale is

E[(M f,n

t

)2]=

∫ t

0

(Lnf2)(ηns )− 2f(ηns )Lnf(ηns ) ds =

∫ t

0

(Qnf)(ηns ) ds

where we define Qnf(η) := (Lnf2)(η)− 2f(η)Lnf(η).

In the proof of the main theorem, we will need to consider the expressionof the Dynkin martingales associated to the functions fx,n(η) = η(x)

nand

fx,y,n = (fx,n · fy,n) (η). We give here the corresponding values of Lnf andQnf to have them all ready when needed.

For f in the domain of Ln and Qn, Lnf is given by (2.5) and Qnf is givenby:

Qnf(η) =∑x∼y∈V

η(x) [f(ηx,y)− f(η)]2 +∑x∈V

F+n (η(x))

[f(ηx,+)− f(η)

]2+∑x∈V

F−n (η(x))[f(ηx,−)− f(η)

]2(2.14)


The Lipschitz function fx,n belongs to the domain of Ln and Qn; thefunction fx,y,n is the product of Lipschitz functions and therefore also belongsto the domain of Ln and Qn (see 4.2). From 4.3, we get

Ln(fx,n)(ηns ) = ∆V ζns (x) + (ζns (x))k

(−β + f

(ζns (x)

nν−1

))(2.15)

Qn(fx,n)(ηns ) =∑y∼x

ζns (y) + ζns (x)

n+ (ζns (x))l

(α + g

(ζns (x)

nν−1

)). (2.16)

Also from 4.3, we see that Lnfx,y,n(ηns ) can be written as:

Lnfx,n(ηns ) · fy,n(ηns ) + Lnfy,n(ηns ) · fx,n(ηns )− 1x∼y

(ζns (x) + ζns (y)

n

).

As for Qnfx,y,n(ηns ), we only need to know it is polynomialy bounded withrespect to the total mass of the system.

2.5 Discrete analogues of a and b for Ln

We would like to show that the limit process is a solution to the martingaleproblem associated with a second order differential operator. These operatorshave the general form:

(L∗f)(ζ) =∑x,y∈V

a∗x,y(ζ)∂x,yf(ζ) +∑x∈V

b∗x(ζ)∂xf(ζ).

To recover a∗ and b∗, we use the coordinate functions fx(ζ) = ζ(x):

b∗x(ζ) = L∗fx(ζ)

a∗x,y(ζ) = L∗fx · fy(ζ)− fx(ζ)L∗fy(ζ)− fy(ζ)L∗fx(ζ)

a∗x,x(ζ) = L∗(ζ(x)ζ(y))− ζ(x)L∗ζ(y)− ζ(y)L∗ζ(x)

By analogy, we define the coefficients for Ln:

bnx(ζns ) := Lnfx,n(ηns )

anx,y(ζns ) := Ln(fx,n · fy,n(ηns ))− fx,n(ηns )Lnfy,n(ηns )− fy,n(ηns )Lnfx,n(ηns )

anx,x(ζns ) := Ln(fx,n · fx,n(ηns ))− 2fx,n(ηns )Lnfx,n(ηns ) = Qn(fx,n(ηn))

2.6. UNIFORM RESULTS FOR THE FAMILYηN·N

23

In light of our computations above we have that:

bnx(ζ) = ∆V ζn(x)− (ζns (x))k

(−β + f

(ζns (x)

nν−1

))anx,y(ζ) = 1x∼y

[−ζ

n(x) + ζn(y)

n

]anx,x(ζ) =

∑y∼x

ζn(x) + ζn(y)

n+ (ζns (x))l

(α + g

(ζns (x)

nν−1

))

2.6 Uniform results for the family ηn· nNow we can state the first technical results that will be crucial to obtaintightness and uniform integrability.

Proposition 1 (uniform bound on moments). For every m ∈ N, there is aconstant C(m), independent of n, such that:

E [(Snt∧τn)m] ≤ C(m)(1 + t).

Remark. We have omitted the dependency of C on the terms k, l, α and βbecause they are fixed parameters of our SDE. The really important part isthat C does not depend on n. Moreover, we have made explicit the dependencyof C on m to emphasize this dependence when the estimate is used.

Proof. Define hmn (η) = min

(Sn(η))m , (wnν−1 + 1)m

. We have:

hmn (ηnt ) = hmn (ηn0 ) +

∫ t

0

(Lnhmn )(ηns ) ds+M

hmn ,nt ,

We note that while Sn(η) ≤ wnν−1 we have Lnhmn (η) ≤ C0(m) (see 4.4).

Therefore, using the stopping time τn := inf t > 0;Snt > w nν−1 on theabove equation we obtain:

(Snt∧τn)m = (Sn0 )m +

∫ t∧τn

0

(Lnhmn )(ηns ) ds+M

hmn ,nt∧τn ,

and taking expectations on both sides suffices to make the martingale termdisappear and we can find the desired bound:

E [(Snt∧τn)m] ≤ (Sn0 )m + tC0(m) < C(m)(1 + t).


The final result we will prove in this section is the following

Proposition 2 (Uniform non-explosion). For all T > 0,

limA→∞

supnP[sups≤T

Sn(ηns ) > A

]= 0.

Proof. If Sn < wnν−1 then LnSn(η) < 0, indeed:

LnSn(η) =

∑x∈V

(ζns (x))k(−β + f

(ζns (x)

nν−1

))< 0.

This yieldsSn(ηnt∧τn) ≤ Sn(ηn0 ) +MSn,n

t∧τn .

For wnν−1 > A we have that P[supt≤T S

nt > A

]= P

[supt≤T S

nt∧τn > A

]and so:

P[supt≤T

Snt > A

]≤ P

[Sn0 >

A

2

]+ P

[supt≤T

MSn,nt∧τn >

A

2

].

Since supn P[Sn0 >

A2

]−−−→A→∞

0 we need to estimate the second term. To do so

we use Chebyshev inequality and Doob’s inequality along with the estimates

used in (2.12) for the g function plus the fact that∑

x∈V ζs(x)l ≤(SNs)l

P[supt≤T

MSn,nt∧τn >

A

2

]≤ 4

A2E

[(supt≤T

MSn,nt∧τn

)2]≤ 16

A2E[(MSn,n

T∧τn

)2]

≤ 16

A2E

[∫ T∧τn

0

∑x∈V

(ζns (x))l(α2 + g

(ζns (x)

nβ−1

))ds

]

≤ 16

A2E[∫ T∧τn

0

2α2 (Sns )l ds

]≤ 16

A22α2C(l)(1 + T )T −−−→

A→∞0

Which gives us that limA→∞ supn P[sups≤T S

n(ηnt∧τn) > A]

= 0.

Chapter 3

The limit process

After having established the first results in the previous chapter, the finalarguments in the proof are of a qualitative nature and involve much lesscomputation. In this chapter, we prove that:

(a) the sequence (ζn· )n∈N is tight,

(b) limit points of (ζn· )n∈N solve a martingale problem, and that

(c) this martingale problem is well-posed.

Throughout this chapter, we use a notation in the same spirit as in theprevious one. We write Pn to refer to the probability measure induced by ζn·and similarly we will denote (for A an event in D):

Pn (A) = P [ζn· ∈ A]

Analogously, we make En denote the expectation with respect to the proba-bility law Pn, so that

En [f(ω)] = E [f(ζn· )] .

3.1 Tightness

In our context, a tight family of processes (ζn· )n∈N is pre-compact by Pro-horov’s Theorem [Bil99, p. 51 and 138]. This means that every sequence ofprocesses from this family admits a limit point ζ∗· . That is, a subsequence of(ζn· )n converges to some ζ∗· , which means that the corresponding probabilities(Pn)n converge weakly (as measures over D with the Skorohod topology) tosome probability law P∗.

25

26 CHAPTER 3. THE LIMIT PROCESS

By the Portemanteau theorem [Bil99, p. 16], this is equivalent to say thatfor every (Skorohod) open set A ⊂ D

([0,∞),RV

)we have

P∗(A) ≤ lim infn

Pn(A).

Remember a family Pn n of probability measures is tight when for everyε > 0, there is a (Skorohod) compact set K(ε) such that

infnPn[K(ε)] ≥ 1− ε.

To prove tightness for the vector-valued process ζn· n∈N it suffices toverify that for every x ∈ V the sequence of paths ζnt (x) ; t ∈ [0,∞)n∈N inD([0,∞);R) is tight.

Indeed, if each ζnt (x) is tight, denote by K(x, ε) the compact set for which

infnPn [ζn· (x) ∈ K(x, ε)] ≥ 1− ε.

Now take K(ε) = ∪x∈VK(x, ε|V |). This gives us, for all n:

P[ζn· /∈ K (ε)] ≤∑x∈V

P[ζn· (x) /∈ K

(x,

ε

|V |

)]≤ ε.

So

infnP [ζn· ∈ K(ε)] ≥ 1− ε.

Proposition 3. The sequence ζnt (x) ; t ∈ [0,∞)n∈N satisfies Aldous’s Cri-terion [Bil99, p. 178], and therefore it is tight. More precisely, given anyT > 0, it satisfies the following two conditions:

i)

limA→∞

supnP[supt≤T|ζnt (x)| > A

]= 0

ii) ∀ ε > 0 :

limδ0→0

supn∈N

supδ≤δ0

supτ∈TT

P(∣∣ζnτ+δ(x)− ζnτ (x)

∣∣ > ε)

= 0

where TT = τ | τ is a stopping time bounded by T 1

1 The abuse notation τ + δ = min τ + δ, T for convenience.

3.1. TIGHTNESS 27

Proof. Condition i) follows from Proposition 2:

limA→∞

supnP[supt≤T|ζnt (x)| > A

]≤ lim

A→∞supnP[supt≤T

Snt > A

]= 0

To see condition ii) let Mx,n denote the Dynkin martingale associated

with the function fx,n(η) = η(x)n

. Upon rewriting its defining equation (2.13),we get:

ζnt (x) = ζn0 (x) +

∫ t

0

∆V ζns (x)− fβ,n (ζns (x)) ds+Mx,n

t .

So, for δ ≤ δ0:

ζnτ+δ(x)− ζnτ (x) =

(∫ τ+δ

τ

∆ζns (x)− fβ,n (ζns (x)) ds

)+

(Mx,n

τ+δ −Mx,nτ

).

Since we want a bound on the left-hand side, we will bound each term onthe right by ε/2. We start with an estimate of the integral term.

Observe that

P[ ∣∣∣∣∫ τ+δ

τ

∆ζns (x)− fβ,n(ζn(x)) ds

∣∣∣∣ > ε

2

]≤ P

[ ∣∣∣∣∣∫ (τ+δ)∧τnA

τ∧τnA∆ζns (x)− fβ,n(ζn(x)) ds

∣∣∣∣∣ > ε

2

]+ P [τnA < T ] .

Using again Proposition 2, it follows that

supnP [τnA < T ] = sup

nP[supt≤T

Snt (x) > A

]−−−→A→∞

0.

The stopping time τnA ensures that the absolute value of the integrated termremains bounded, since∣∣∆ζns (x)− fβ,n(ζn(x))

∣∣ ≤∑y∼x

(ζns (y) + ζns (x))+2β (ζns (x))k ≤ 2 |V |A+2βAk.

Coming back to the stopped integral, by Markov’s inequality:

P

[ ∣∣∣∣∣∫ (τ+δ)∧τnA

τ∧τnA∆ζns (x)− fβ,n(ζn(x)) ds

∣∣∣∣∣ > ε

2

]

≤ 2

εE

[∫ (τ+δ)∧τnA

τ∧τnA

∣∣∆ζns (x)− fβ,n(ζn(x))∣∣ ds]

≤ 2

εδ(2 |V |A+ 2βAk

)≤ 2

εδ0(2 |V |A+ βAk

)−−−→δ0→0

0.


We estimate P[ ∣∣Mx,n

τ+δ −Mx,nτ

∣∣ > ε2

]by a similar stopping-time argu-

ment, so the remaining term is

P[ ∣∣∣Mx,n

(τ+δ)∧τnA−Mx,n

τ∧τnA

∣∣∣ > ε

2

].

By Chebychev’s inequality and the martingale property of Mx,n· ,

P[ ∣∣∣Mx,n

(τ+δ)∧τnA−Mx,n

τ∧τnA

∣∣∣ > ε

2

]≤ 4

ε2E[(Mx,n

(τ+δ)∧τnA

)2−(Mx,n

τ∧τnA

)2]=

4

ε2E

[∫ (τ+δ)∧τnA

τ∧τnAQn(fx,n)(ηns )

]and if nν−1w > A, then we can again bound the integrated term by:

4

ε2E[∫ t

0

1[τ∧τnA, (τ+δ)∧τnA]

(t)

(2 |V |An

+ 2αAl)ds

]≤ 4

ε2δ(2 |V |A+ 2αAl

)≤ 4

ε2δ0(2 |V |A+ 2αAl

)−−−→δ0→0

0.

This shows that the family ζn· (x) n satisfies condition ii) of Aldous’sCriterion, and concludes the proof of its tightness.

3.2 Characterization of the limit

Now that we proved that the sequence (ζn· )n is tight, it remains to charac-terize the limit points of this sequence. For this, we will show that:

• the limit points are continuous,

• the limit points solve the martingale problem for some second-orderelliptic differential operator L∗, and that

• there is a unique solution to the martingale problem for L∗

3.2.1 Continuity of paths of the limit process

We claim that any limit measure of the tight family (Pn)n∈N gives measure 1to continuous trajectories. This follows from an analysis of the jump function:

J : D([0, T ], E)→ Rx 7→ sup

t∈(0,T ]dE(x(t−), x(t)).

3.2. CHARACTERIZATION OF THE LIMIT 29

Since every x ∈ D is right continuous, the condition that J(x) = 0 impliesthat x is a left-continuous path in [0, T ] and therefore it implies that x is acontinuous path in [0, T ].

Let P∗ be a limit point of the sequence Pn. That is, for some subsequence

Pn′ , we have Pn′ J1−−−→n′→∞

P∗. We claim that P∗ [J(x) = 0] = 1. The key

observation to proof this is to note that J is continuous with respect to theJ1-Skorohod topology [Bil99, p. 125]. This means that

[J(x) > a] = x | J(x) > a = J−1 ((a,∞))

is an open set. Then, since for n > K, Pn[J(x) > 1

K

]= 0 (because all jumps

of ζn have magnitude 1/n), by the Portemanteau Theorem [Bil99, p. 16]:

P∗[J(x) >

1

K

]≤ lim inf

n′Pn′[J(x) >

1

K

]= 0.

This implies that

P∗ [J(x) > 0] = P(∪K∈N

[J(x) >

1

K

])≤∑K∈N

P∗[J(x) >

1

K

]= 0

or in other words, that P∗ [J(x) = 0] = 1.

3.2.2 Martingale problem

In this section, we construct a second-order differential operator L∗ follow-ing the heuristics outlined in the Introduction (more precisely, in subsec-tion 1.3.2), and prove that any limit point of Pn is a solution to the martingaleproblem associated with it.

The candidate for L∗ is obtained by analysing the expressions of the(discrete analogues of the) coefficients a and b of Ln:

bnx(ζ) = ∆V ζn(x)− fβ,n (ζn(x))

anx,y(ζ) = 1x∼y

[−ζ

n(x) + ζn(y)

n

]anx,x(ζ) =

∑y∼x

ζn(x) + ζn(y)

n+ gα,n (ζn(x)) .

Since ζn is tight, we can assume that ζnJ1−−−→

n→∞ζ∗. So we define:

b∗x(ζ) = ∆V ζ(x)− β (ζ(x))k

a∗x,y(ζ) = 0

a∗x,x(ζ) = α (ζ(x))l ,


which determine the coefficients of the differential operator L∗.

Remember that we don’t need to verify thatM f,L∗

t ,Ft; t ≥ 0

is a

continuous local martingale for every f ∈ C2(Rd) [KS91, p. 318]. Fromthe discussion in Subsection 1.3.1, it suffices to verify this property for thecoordinate functions fx(ζ) = ζ(x) and fx,y(ζ) = ζ(x)ζ(y). So we only needto show that

Proposition 4.

Mx,L∗t := ζ∗t (x)− ζ∗0 (x)−

∫ t

0

b∗x(ζ∗s ) ds

Mx,y,L∗t := ζ∗t (x)ζ∗t (y)− ζ∗0 (x)ζ∗0 (y)

−∫ t

0

b∗x(ζ∗s )ζ∗s (y) + b∗y(ζ

∗s )ζ∗s (x) + a∗x,y(ζ

∗s ) ds

are continuous local martingales.

Proof. This will follow from the study of the family of stopped martingalesMx,Ln

t∧τn and Mx,y,Lnt∧τn , where

τn := inft > 0;Snt > w n

ν−12

. (3.1)

We will first prove that these are uniformly integrable martingales, whichimplies that their limits are martingales as well. Then, we show that thelimits are the martingales Mx,L∗

t and Mx,y,L∗t .

Let’s see the first martingale. It suffices to show that, for every t ≤ T ,

supnE[(Mx,Ln

t∧τn

)2]<∞.

The quadratic variation of the stopped Dynkin martingales can be cal-culated with the operator Qn. Using the expansion given in equation (2.16),and that before τ the function g is small by (2.12), we have:

supnE[(Mx,Ln

t∧τn

)2]≤ sup

nE

[∫ t∧τn

0

∑y:y∼x

ζns (y) + ζns (x)

n+ 2α (ζns (x))l ds

]≤ T (2 |V |C(1)(1 + T ) + 2αC(l)(1 + T )) <∞

where we bounded |ζ| and |ζ|l by the uniform estimates of Proposition 1.


Now, we claim that Mx,Lnt∧τn

J1−−−→n→∞

Mx,L∗t . Indeed, define by analogy with

Mx,Lnt∧τn = ζnt∧τn(x)− ζn0 (x)−

∫ t∧τn

0

bnx(ζns ) ds

the processes (not stopped, and with the limit function b∗ in place of bn):

Mx,nt = ζnt (x)− ζn0 (x)−

∫ t

0

b∗x(ζns ) ds

Note that since ζn·J1−−−→

n→∞ζ∗· , the continuity of the functions Mx,n

t implies

that Mx,nt

J1−−−→n→∞

Mx,L∗t .

Moreover, we have

limnP(

supt≤T

∣∣∣Mx,Lnt∧τn − M

x,nt

∣∣∣ > ε

)= 0,

Indeed, because τn > τA for n large enough:

limnP(

supt≤T


x,nt

∣∣∣ > ε

)≤ lim sup

nP(

supt≤T

∣∣∣Mx,Lnt∧τnA− Mx,n

t∧τnA

∣∣∣ > ε

)+ lim sup

nP [τnA < T ] .

Define εn(A) = supz≤ Anν−1|f(z)| and note that εn −−−→

n→∞0, so:

∣∣∣Mx,Lnt∧τnA− Mx,n

t∧τA

∣∣∣ ≤ ∫ t∧τnA

0

∣∣∣β (ζns )k − fn,β(ζns )∣∣∣ ds

=

∫ t∧τnA

0

∣∣∣∣f ( ζnsnν−1

)(ζns )k

∣∣∣∣ ds ≤ ∫ t∧τnA

0

εnAk ≤ εnA

kT −−−→n→∞

0.

We then conclude that:

limnP(

supt≤T


x,nt

∣∣∣ > ε

)≤ lim sup

nP [τnA < T ] −−−→

A→∞0.

We have proved that d(Mx,n,Mx,Ln·∧τn ) −−−→

n→∞0 in probability, and also that

Mx,n →Mx,L∗ , so Mx,Ln·∧τn

J1−−−→n→∞

Mx,L∗ [Bil99, p. 27]. Since the family Mx,Lnt∧τn

is uniformly integrable, we conclude that the limit Mx,L∗ is a martingale.


We do the same for the martingales Mx,y,Ln·∧τn . For this purpose we remem-

ber that the quadratic variations of these martingales are given by

En[(Mx,y,Ln

T∧τn

)2]= En

[∫ T∧τn

0

Qnfx,y,n(ηns ) ds

].

Since the expression of Qnfx,y,n(ηns ) is given by a finite sum of products ofζns (x) and ζns (y), we can again find a convenient constant C > 0 such that:

Qnfx,y,n(ηns ) ≤ C(

1 + (Sns )l+2)

Therefore:

En[(Mx,y,Ln

t∧τn

)2]= En

[∫ T∧τn

0

Qnfx,y,n(ηns ) ds

]< T (1 + T )C

(1 + (Sns )l+2

)<∞.

The proof that Mx,y,Ln·∧τn

J1−−−→n→∞

Mx,y,L∗ follows an analog construction of

martingales

Mx,y,nt = ζnt (x)ζnt (y)−ζn0 (x)ζn0 (y)−

∫ t

0

ζns (x)b∗x(ζns )+ζns (y)b∗x(ζ

ns )+a∗x,y(ζ

ns ) ds.

3.2.3 Uniqueness

To conclude that the tight sequence of processes ζn· converges, it sufficesto prove that it has a unique limit point. This is the final step neededfor the characterization of the limits of the particle systems associated withLn. Previously we have shown that the limit measures are concentrated oncontinuous paths, then we have shown that every limit point is a solution tothe martingale problem associated with L∗ and now we shall show that thereis only one such solution.

The correspondence between solutions to martingale problems and solu-tion to SDE’s is given in [KS91, Corollaries 4.8 and 4.9, p. 317]. Existenceand uniqueness of solutions [(X,W ), (Ω,F ,P) ,Ft] in the sense of probabilitylaw to an SDE with a fixed but arbitrary initial distribution

P [X0 ∈ Γ] = µ(Γ)


is equivalent to existence and uniqueness of solutions P to the correspondingmartingale problem with the initial condition

P[y ∈ C([0,∞),Rd | y(0) ∈ Γ

]= µ(Γ) Γ ∈ B(Rd).

The two solutions are related by P(X ∈ A) = P (A), that is, the solution tothe martingale problem is the law induced by the weak solution to the SDE.

Also, uniqueness in the sense of probability law follows from pathwiseuniqueness [YW71, KS91, p. 301 and 331]. So, we only need to verify thatpathwise uniqueness holds for our equations. Remember from the discussionat the end of Section 1.4 that we need to treat two cases separately.

For the case l ≥ 2, since both b and σ are C1, we can apply:

Theorem ([KS91, p.287]). Suppose that the coefficients b(t, ζ), σ(t, ζ) arelocally Lipschitz continuous in the space variable; i.e., for every n ≥ 1 there

exists a constant Kn > 0 such that for every t ≥ 0, ‖ζ‖ ≤ n and∥∥∥ζ∥∥∥ ≤ n:∥∥∥b(t, ζ)− b(t, ζ)

∥∥∥+∥∥∥σ(t, ζ)− σ(t, ζ)

∥∥∥ ≤ Kn

∥∥∥ζ − ζ∥∥∥ .Then pathwise uniqueness holds for (b, σ), that is, for the vector valued SDE

dζt = b(t, ζ)dt+ σ(t, ζ) dBt.

For the case l = 1 we note that the local Lipschitz condition is not truefor σ, since it behaves as a square-root near the boundary ζ(x) = 0. At thispoint, the first two criteria we could use for uniqueness have failed, namelyuniform elipticity of the diffusion matrix and local Lipschitz condition forthe coefficients of the corresponding SDE.

Before looking for alternative methods to prove uniqueness, one mightsuspect that this case l = 1 does not admit a unique solution. Let’s exam-ine some cases where uniqueness fails due to multiple behaviour at singularpoints.

First, in the context of martingale problems, multiplicity might be aconsequence of incomplete definition of the domain of the operator. Forinstance, suppose we consider L = ∆ in the half-line (0,∞). If we specifythe domain of L as

Dρ =

f ∈ C2

0(0,+∞)

∣∣∣∣ f ′(0+)

ρ= lim

x→0

1

2f ′′(x)

the resulting solution of the martingale problem is the Sticky Brownian mo-tion with parameter ρ > 0. This shows that the resulting random walk


depends not only on the differential operator formula, but also on the choiceof the functions it’s applied to.

Another way to prove uniqueness to the martingale problem is to considera suitable Green function that helps showing R(I − λL∗) = C0(RV ) forall small λ > 0. Finding a Green function is not an easy task but thismethod proved succesfull in the case of the reflected Brownian motion on awedge [Wil83].

From the point of view of SDE’s, there are also examples where uniquenessfails because trajectories can remain at the singular point X = 0 for anarbitrary time [KS91, pp.292]. First, from the family

dXt = |Xαt | dBt, α <

1

2

it becomes clear that our case α = 12

is a limit situation. Moreover, the SDE

dXt = 3X1/3t dt+ 3X2/3 dBt.

shows that it is not enough to have regularity in the noise term (note that2/3 > 1/2) but we also need a good behaviour in the drift term.

In fact, in our case, both drift and dispersion terms have the regularitiesneeded to assure uniqueness. Coming back to our problem, labeling the sitesV of our graph by x1, . . . , xn , we observe that the associated dispersionmatrix σ fits the particular form required by the following criterion, whichwill again give pathwise uniqueness in this case, concluding the last case forour theorem.

Theorem ([YW71, Theorem 1]). Let

dζt = σ(ζt) dBt + b(ζt) dt (3.2)

where

σ(ζ) =

σ1(ζ(x1)) 0

σ2(ζ(x2)). . .

0 σn(ζ(xn))

, b(ζ) =(b1(ζ), b2(ζ), . . . , bn(ζ)

)

such that

(i) there exists a positive increasing function ρ(u), u ∈ (0,∞) such that∫ ε0

1ρ2(u)

du = +∞ for every ε > 0

|σi(u)− σi(v)| ≤ ρ(|u− v|), ∀u, v ∈ R, , 1 ≤ i ≤ n


(ii) for every n ≥ 1 there exists a constant Kn > 0 such that for every

t ≥ 0, ‖ζ‖ ≤ n and∥∥∥ζ∥∥∥ ≤ n:∥∥∥bi(ζ)− bi(ζ)

∥∥∥ ≤ Kn

∥∥∥ζ − ζ∥∥∥Then pathwise uniqueness holds.

While condition ii) is again satisfied, since b remains polynomial, usingρ(u) =

√u gives us condition i).

And this concludes the proof of the uniqueness. So for every SDE weconsidered there is a family of Reaction-Diffusion models that converge afterscaling to the law induced by unique solution of the given SDE.

Chapter 4

Technical issues andcomputations

In this chapter we discuss some results that we believe are better treated herethan elsewhere. To deal with them in the middle of the text would interruptthe flow of ideas and furthermore would obscure the general understandingof the proof.

The first two sections deal with standard results of the construction ofparticle systems. In Section 4.1 we rule out explosions and in Section 4.2 weprove that polynomial functions of the number of particles in each site arein the domain of the generator.

Section 4.3 is mostly computations that will be used throughout the texta couple of times. We encourage the reader to calculte these quantities on hisown but if trouble arises we hope he finds help in the explicit computations.

Section 4.4 is the gem of this Chapter. If we should single out one resultthat is key and original in the proof of tightness it is the fact that we canuniformly bound Ln

(Sns∧τn

)m. That is, that we can bound uniformly the

infinitesimal increase of polynomials of the total mass of the system. Theseuniform estimates are used indirectly for the proof of tightness since they arerequired to prove Proposition 2. This result will be used once again for theproof of uniform integrability of the Martingales in Proposition 4.

4.1 Non explosion

Let’s consider the particle system defined in the set up 2.1. Since we allowfor an infinite number of configurations η ∈ NV , to complete the definitionof the process we need to rule out explosions, that is, an infinite numberof transitions in a finite time. We will see that it suffices to require that

37

38 CHAPTER 4. TECHNICAL ISSUES AND COMPUTATIONS

F+(u) ≤ C(1 + u).Indeed, let St =

∑x∈V ηt(x) be the total number of particles in the system

at time t. Summing over all rates F+(ηt(x)), we see that the rate at which aparticle is born at time t is less than Λt = C(|V | + St). Thus, the expectedtime for one such transition is greater than 1/Λt. Let Tn be the time ittakes for the n-th particle to be born, therefore by independence of thosetransitions we have that

E [T0] ≥ 1/(C(|V |+ S0)

)E [T1] ≥ 1/

(C(|V |+ S0 + 1)

)...

...

E [Tn] ≥ 1/(C(|V |+ S0 + n)

).

Therefore

E

[∞∑n=0

Tn

]≥

∞∑n=0

(C(|V |+ S(η0) + n)

)−1=∞.

This implies that∑∞

k=1 Tn =∞ almost surely [Nor97, theorem 2.3.2]. Since

P[infinite birth transitions before time T

]= P

[∞∑n=1

Tn < T

]= 0,

we conclude that almost surely there can’t be an infinte number of particlesbefore any arbitrary time T .

This means that the number of transitions (birth, death or jump) is al-most surely finite as well. Indeed, define λ(N) to be supremum of all transi-tions rates admissible between configurations with at most N particles.

λ(N) = sup

F+(η(x)), F−(η(x)), η(x)

∣∣∣∣∣∑x∈V

η(x) ≤ N

<∞. (4.1)

Consider the event that the total number of particle births up to time T staybounded by n and that the number of particles at the begining is less thenn. We need to rule out an infinite number of jumps or deaths of particlesbefore time T as well. Since those transitions are independent of the birthtransitions, the time it takes for each occurence depends on exponencial timesTi with rate less that λ(2n). By the same argument as before,

P[infinite jumps or deaths before time T

]=

[∞∑i=1

Ti < T

]= 0

and this rules out explosions.

4.2. LIPSCHITZ FUNCTIONS ARE IN THE DOMAIN OF LN AND QN39

4.2 Lipschitz functions are in the domain of

Ln and Qn

A function f : NV → R is Lipschitz if

supx∈V

supη∈NV

|f(η + δx)− f(η)| = Cf <∞.

We claim that a function satisfying the Lipschitz condition is on thedomain of Ln. Indeed, we compute

Lnf(η) = limh→0

1

hE [f(ηnh)− f(η)]

= limh→0

1

h

(E[f(ηnh)− f(η) | at most 1 birth before time h

]+ E

[f(ηnh)− f(η) | 2 births before time h

]+ . . .

+ E[f(ηnh)− f(η) | k births before time h

]+ . . .

)We claim that

limh→0

1

h

(E[f(ηnh)− f(η) | 2 births before time h

]+ . . .

+ E[f(ηnh)− f(η) | k births before time h

]+ . . .

)= 0

As f is Lipschitz, |f(ηnh)− f(η)|1 k births before time h ≤ Cfk1 k births before time h ,and then the limit above is less than

limh

1

hCf∑k≥2

P (k or more birth transitions before time h) .

Now, pk(h) = P(k births or more before time h) is given by∫. . .

∫λ1e

−λ1x1 . . . λke−λkxk1x1+···+xk≤h dx1 . . . dxk.

Bounding all exponentials by 1 and observing that we’re integrating over thek-dimensional simplex with side h, whose volume is hk(k!)−1, we see that

pk(h) ≤∏k

i=1 λi∏ki=1 i

hk.


Now, since λk ≤ C(|V |+ S0 + k) and supkC(|V |+S0+k)

k= C <∞ we have:

pk(h) ≤k∏i=1

λiihk ≤ Ckhk.

So for Ch ≤ 12:

Cfh

∑k≥2

pk(h) ≤ Cfh

∑k≥2

(Ch)k ≤ Cfh

2(Ch)2 −−→h→0

0

as we claimed.Therefore

Lnf(η) = limh→0

1

hE [f(ηnh)− f(η)]

= limh→0

1

h

(E[f(ηnh)− f(η) | at most 1 birth before time h

])=∑x∼y∈V

η(x) [f(ηx,y)− f(η)] +∑x∈V

F+n (η(x))

[f(ηx,+)− f(η)

]+∑x∈V

F−n (η(x))[f(ηx,−)− f(η)

]so we see that f is in the domain of Ln.

To see that f is in the domain of Qn it suffices to prove that

limh→0

1

hEn [(f(Xh)− f(X0))]

exists pointwisely. In this case we setQnf = limh→01hEn[(f(Xh)− f(X0))

2] .To prove this we follow analogous steps as above. We just note that if f isLipschitz, (f(ηnh)− f(η))2 1 k births before time h ≤ (Cfk)21 k births before time h ,and then

limh→0

1

h

(E[(f(ηnh)− f(η))2 | 2 births before time h

]+ . . .

+E[(f(ηnh)− f(η))2 | k births before time h

]+ . . .

)≤ 1

h

∑k≥2

(Cfk)2pk(h) (4.2)

so for Ch < 1/2 we have

Cfh

∑k≥2

k2pk(h) ≤ Cfh

∑k≥2

k2(Ch)k ≤ Cfh

(Ch)2∑k≥2

k2(Ch)k−2 −−→h→0

0 (4.3)

4.3. EXPLICIT FORMULAE FOR LN ANDQN OVER COORDINATE FUNCTIONS41

since∑

k≥2 k2(Ch)k−2 is uniformly bounded. This concludes the proof that

f is in the domain of Qn.An analogous argument holds for functions that are products of Lipschitz

functions such as H = f · g. Let ξ ∈ NV and let ‖ξ‖1 =∑

x∈V ξ(x) and usethe triangle inequality to obtain

|H(η + ξ)−H(η)| ≤ Cf ‖ξ‖1 (g(η) + Cg ‖ξ‖1) + f(η)Cg ‖ξ‖1

We want to see now that H is in the domain of Ln. The key estimatenow is |H(ηnh)−H(η)|1 k births before time h ≤ CH(η)(k)21 k births before time h where CH(η) is some constant that depends on Cf , Cg, f(η) and g(η).

With this we can follow the same arguments as in (4.2) and (4.3) toconclude that H is in the domain of Ln. Analogously we can prove that His in the domain of Qn.

4.3 Explicit formulae for Ln and Qn over co-

ordinate functions

We put fx,n(η) = η(x)n

= ζn(x) for the normalized projection on the coordinatex of the process ηn. Then, we have:

Ln(fx,n)(η) =∑

w∼z∈V

η(w) [fx,n(ηw,z)− fx,n(η)]

+∑x∈V

F+,k,ln (η(x))

[fx,n(ηx,+)− fx,n(η)

]+∑x∈V

F−,k,ln (η(x))[fx,n(ηx,−)− fx,n(η)

]and summing the terms with F+ together with F−, we can replace them

with the net result F and obtain:

=

(∑y∼x

η(y)− η(x)

n

)+

(η(x)

n

)k (−β + f

(η(x)

nν

)).

Passing to the macroscopic quantities ζn, and using the discrete laplacian∆V , this becomes

Ln(fx,n)(ηns ) =∑y∼x

(ζns (y)− ζns (x)) + (ζns (x))k(−β + f

(ζns (x)

nν−1

))= ∆V ζ

ns (x) + (ζns (x))k

(−β + f

(ζns (x)

nν−1

)).


Analogously, for the quadratic variation operator:

Qn(fx,n)(η) =∑y∼x

η(y) + η(x)

n2+

(η(x)

n

)l(α + g

(η(x)

nν

))Qn(fx,n)(ηns ) =

∑y∼x

ζns (y) + ζns (x)

n+ (ζns (x))l

(α + g

(ζns (x)

nν−1

))

Now, we move to the functions fx,y,n(η) = η(x)n

η(y)n

= ζn(x)ζn(y). Pro-ceeding in the same way as above, we now must take care of the case whenx and y are neighbors, so we split the sum comming from the jumps in threepieces: one to x, one to y, and one to handle the case when x ∼ y.

Lnfx,y,n(η)

=∑w∼xw 6=y

η(w) [fx,y,n(ηw,x)− fx,y,n(η)] + η(x) [fx,y,n(ηx,w)− fx,y,n(η)]

+∑w∼yw 6=x

η(w) [fx,y,n(ηw,y)− fx,y,n(η)] + η(y) [fx,y,n(ηy,w)− fx,y,n(η)]

+ 1x∼y (η(x) [fx,y,n(ηx,y)− fx,y,n(η)] + η(y) [fx,y,n(ηy,x)− fx,y,n(η)])

+ F+n (η(x))

[fx,y,n(ηx,+)− fx,y,n(η)

]+ F−n (η(x))

[fx,y,n(ηx,−)− fx,y,n(η)

]+ F+

n (η(y))[fx,y,n(ηy,+)− fx,y,n(η)

]+ F−n (η(y))

[fx,y,n(ηy,−)− fx,y,n(η)

]Now we compute the values of the differences inside brackets:

=∑w∼xw 6=y

η(w)− η(x)

n

(η(y)

n

)+∑w∼yw 6=y

η(w)− η(y)

n

(η(x)

n

)

+ 1x∼y

(η(x)

[η(x)

n2− η(y)

n2− 1

n2

]+ η(y)

[−η(x)

n2+η(y)

n2− 1

n2

])− 2fβ,n

(η(x)

n

)(η(y)

n

)− 2fβ,n

(η(y)

n

)(η(x)

n

)Finally, we recombine the terms from the different cases we end up with

a simpler expression:

4.3. EXPLICIT FORMULAE FOR LN ANDQN OVER COORDINATE FUNCTIONS43

=∑w∼x

η(w)− η(x)

n

(η(y)

n

)+∑w∼y

η(w)− η(y)

n

(η(x)

n

)− 1x∼y

(η(x)

n2+η(y)

n2

)− 2fβ,n

(η(x)

n

)(η(y)

n

)− 2fβ,n

(η(y)

n

)(η(x)

n

).

Again we translate in terms of the ζn:

Lnfx,y,n(ηns ) = Lnfx,n(ηns ) · fy,n(ηns ) + Lnfy,n(ηns ) · fx,n(ηns )

− 1x∼y

(ζns (x) + ζns (y)

n

)To conclude, we procede in the same manner when it commes to the term

Qnfx,y,n(η) this ammounts first to split in cases the jump terms:

Qnfx,y,n(η)

=∑w∼xw 6=y

η(w) [fx,y,n(ηw,x)− fx,y,n(η)]2 + η(x) [fx,y,n(ηx,w)− fx,y,n(η)]2 +

+∑w∼yw 6=x

η(w) [fx,y,n(ηw,y)− fx,y,n(η)]2 + η(y) [fx,y,n(ηy,w)− fx,y,n(η)]2

+ 1x∼y(η(x) [fx,y,n(ηx,y)− fx,y,n(η)]2 + η(y) [fx,y,n(ηy,x)− fx,y,n(η)]2

)+ F+

n (η(x))[fx,y,n(ηx,+)− fx,y,n(η)

]2+ F−n (η(x))

[fx,y,n(ηx,−)− fx,y,n(η)

]2+ F+

n (η(y))[fx,y,n(ηy,+)− fx,y,n(η)

]2+ F−n (η(y))

[fx,y,n(ηy,−)− fx,y,n(η)

]2then we compute the differences inside brackets:

=∑w∼xw 6=y

η(w) + η(x)

n2

(η(y)

n

)2

+∑w∼yw 6=y

η(w) + η(y)

n2

(η(x)

n

)2

+ 1x∼y

(η(x)

[η(x)

n2− η(y)

n2− 1

n2

]2+ η(y)

[−η(x)

n2+η(y)

n2− 1

n2

]2)

+ 2gα,n(η(x)

n

)(η(y)

n

)2

+ 2gα,n(η(y)

n

)(η(x)

n

)2

and finally we recombine the terms from the different cases to obtain a

simpler expression:


=∑w∼x

η(w) + η(x)

n2

(η(y)

n

)2

+∑w∼y

η(w) + η(y)

n2

(η(x)

n

)2

+ 1x∼y

([−2η(x)

η(x)

n2

η(y)

n2+ 2η(x)

(η(y)

n2− η(x)

n2

)1

n2+ η(x)

1

n4

][−2η(y)

η(y)

n2

η(x)

n2+ 2η(y)

(η(x)

n2− η(y)

n2

)1

n2+ η(y)

1

n4

])+ 2gα,n

(η(x)

n

)(η(y)

n

)2

+ 2gα,n(η(y)

n

)(η(x)

n

)2

Again we translate in terms of the ζn:

Qnfx,y,n(ηns )

=∑w∼x

ζns (w) + ζns (x)

n(ζns (y))2 +

∑w∼y

ζns (w) + ζns (y)

n(ζns (x))2

+ 1x∼y

([−2ζns (x)ζns (x)

ζns (y)

n+ 2ζns (x)

(ζns (y)

n− ζns (x)

n

)1

n+ ζns (x)

1

n3

][−2ζns (y)ζns (y)

ζns (x)

n+ 2ζns (y)

(ζns (x)

n− ζns (y)

n

)1

n+ ζns (y)

1

n3

])+ 2gα,n (ζns (x)) (ζns (y))2 + 2gα,n (ζns (y)) (ζns (x))2

and these expressions written in terms the macroscopic quantities ζns will beused throughout the thesis.

4.4 Uniform bounds on Lnhmn

From Proposition 1, we recall that hmn (η) = min

(Sn(η))m , (wnν−1 + 1)m

.We needed to show that while Sn(η) ≤ wnν−1, Lnh

mn (η) ≤ C0(m). This

requires a rather long computation, so in order to make it more direct wesingle out the following inequalities:∑

x∈V

fn,β (ζn(x)) ≥∑x∈V

β

2(ζn(x))k ≥ β

2c (Sn(ηn))k (4.4)∑

x∈V

gn,α (ζn(x)) ≤∑x∈V

2α (ζn(x))l ≤ 2α (Sn(ηn))l (4.5)

4.4. UNIFORM BOUNDS ON LNHMN 45

where c is some constant that depends on |V | and m.Since a jump transition conserves the total number of particles of the

system, in the expression of Ln (Sn(η))m we only need to take into accountthe birth/death transitions at each site, so for Sn(η) ≤ wnν−1:

Ln (Sn(η))m =∑x∈V

F+n (η(x))

[(Sn(ηx,+)

)m − (Sn(η))m]

∑x∈V

F−n (η(x))[(Sn(ηx,−)

)m − (Sn(η))m]

Then expanding the expressions for F+n and F−n according to (2.9) (2.10) and

using the binomial expression for the terms in brackets, we get:


(n2gn,α (ζn(x))− nfn,β (ζn(x))

) [∑j≥1

(m

j

)(Sn)m−j

(1

n

)j]

+(n2gn,α (ζn(x)) + nfn,β (ζn(x))

) [∑j≥1

(m

j

)(Sn)m−j

(−1

n

)j]

Now note that, depending if j is either odd or even, the terms in the aboveexpression simplify:


−2nfn,β ((ζn(x)))

[ ∑1≤j odd

(m

j

)(Sn)m−j

1

nj

]∑x∈V

(2n2gn,α (ζn(x))

) [ ∑1≤j even

(m

j

)(Sn)m−j

1

nj

]

So writing all in terms of even numbers we obtain for the right-hand side:

2∑

1≤j odd

(Sn)m−j−1

nj−1

[∑x∈V

(−fn,β ((ζn(x)))

(m

j

)Sn)

+ gn,α (ζn(x))

(m

j + 1

)]Since Sn(η) ≤ wnν−1 we obtain estimates (2.12) for fn,β and gn,α:

≤ 2∑

1≤j odd

(Sn)m−j−1

nj−1

[∑x∈V

(2α (ζn(x)))l(

m

j + 1

)− 2β (ζn(x))k

(m

j

)Sn

]From the inequalities (4.4) (4.5) we obtain:


≤ 2∑

1≤j odd

(Sn)m−j−1

nj−1

[(2α2 (Sn)

)l( m

j + 1

)− 2βc (Sn)k

(m

j

)Sn]

≤ 2∑

1≤j odd

(Sn)m+l−j−1

nj−1

[2α2

(m

j + 1

)− 2βc (Sn)k+1−l

(m

j

)]≤ C0(m)

Where the last inequality follows from the fact that the terms[2α2

(m

j + 1

)− 2βc (Sn)k+1−l

(m

j

)]are decreasing in Sn

Bibliography

[AT80] Ludwig Arnold and M. Theodosopulu. deterministic limit of thestochastic model of chemical reactions with diffusion. Advances inApplied Probability, 12(2):367–379, 1980.

[Bil99] Patrick Billingsley. Convergence of probability measures. Wileyseries in probability and statistics. Wiley-Interscience, 2ed edition,1999.

[Blo92] Douglas Blount. Law of large numbers in the supremum norm for achemical reaction with diffusion. The annals of applied probability,2(1):131–141, 1992.

[Dur10] Rick Durrett. Probability Theory and Examples. Cambridge Uni-versity press, 2010.

[Ein56] Albert Einstein. INVESTIGATIONS ON THE THEORY OF THEBROWNIAN MOVEMENT, volume d. Dover, New York, 1956.

[FG12] Tertuliano Franco and Pablo Groisman. A particle system withexplosions: law of large numbers for the density of particles and theblow-up time. Journal of Statistical Physics, 4:629–642, November2012.

[GJ13] Patrıcia Goncalves and Milton Jara. Nonlinear fluctuations ofweakly asymmetric interacting particle systems. Archive for Ra-tional Mechanics and Analysis, 212(2):597–644, Dec 2013.

[Gow11] Reid Gower. THE FEYNMAN SERIES - Curiosity. https://www.youtube.com/watch?v=lmTmGLzPVyM, 2011. [Online; accessed 27-Jan-2016].

[GP15] Massimiliano Gubinelli and Nicholas Perkowski. Energy solutionsof KPZ are unique. arXiv preprint: 1508.07764, Aug 2015.

47

https://www.youtube.com/watch?v=lmTmGLzPVyM

https://www.youtube.com/watch?v=lmTmGLzPVyM

48 BIBLIOGRAPHY

[Hai14] Martin Hairer. A theory of regularity structures. InventionnesMathematica, 198(2):269–504, 2014.

[HL15] Martin Hairer and Cyril Labbe. A simple construction of the con-tinuum parabolic anderson model on R2. arXiv:1501.00692, Jan2015.

[HS15] Martin Hairer and Hao Shen. A central limit theorem for the KPZequation. arXiv:1507.01237, Jul 2015.

[Ito51] Kiyoshi Ito. On a formula concerning stochastic differentials.Nagoya Mathematical Journal, 3(1):55–65, 1951.

[KF31] Andrey Nikolaevich Kolmogorov and Sergei Vasilyevich Fomin. Onanalytical methods in the theory of probability. MathematischeAnnalen, 104(1):415–458, 1931.

[KL99] Claude Kipnis and Claudio Landim. Scaling Limits of InteractingParticle Systems. Springer, 1999.

[Kot86] Peter Kotelenez. Law of large numbers and central limit theorem forlinear chemical reactions with diffusion. The Annals of Probability,14(1):173–193, 1986.

[KPZ86] Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. Dynamicscaling of growing interfaces. Physics Reviews Letters, 56(9):889–892, Mar 1986.

[KS91] Ioannis Karatzas and Steven E. Shreve. Brownian Motion andStochastic Calculus. Graduate Texts in Mathematics. Springer Ver-lag, 2ed edition, 1991.

[Lig99] Thomas Milton Liggett. Continuous Time Markov Processes, vol-ume 113 of Graduate Texts in Mathematics. The American Math-ematical Society, 1999.

[Nor97] J.R. Norris. Markov Chains. Cambridge Series in Statistical andProbabilistic Mathematics. Cambridge University press, 1997.

[SV69a] Daniel Wyler Stroock and S. R. I. S. Varadhan. Diffusion processeswith continuous coefficients, i. Communications on Pure and Ap-plied Mathematics, 22(3):345–400, 1969.

BIBLIOGRAPHY 49

[SV69b] Daniel Wyler Stroock and S. R. I. S. Varadhan. Diffusion processeswith continuous coefficients, ii. Communications on Pure and Ap-plied Mathematics, 22(3):479–530, 1969.

[SV71] Daniel Wyler Stroock and S. R. I. S. Varadhan. Diffusion processeswith boundary conditions. Communications on Pure and AppliedMathematics, 24:147–225, 1971.

[Tur52] Alan Turing. The chemical basis of morphogenesis. PhilosophicalTransactions of the Royal Society of London, 237(641):37–72, 1952.

[Vol14] Vitaly Volpert. Elliptic Partial Differential Equations : Volume2: Reaction-Diffusion Equations. Monographs in Mathematics.Springer Basel, 2014.

[Wie21] Norbert Wiener. The average of an analytic functional and thebrownian movement. Proceedings of the National Academy of Sci-ences of the United States of America, 7(10):294–298, 1921.

[Wil83] Ruth Jeannette Williams. Brownian Motion in a Wedge withOblique Reflection at the Boundary. Stanford University, 1983.

[YW71] Toshio Yamada and Shinzo Watanabe. On the uniqueness of so-lutions of stochastic differential equations. J. Math. Kyoto Univ.,11(1):155–167, 1971.

doctoral thesis · the world is strange, the whole universe is very strange, but see when you look...

Documents