Marie-Hélène Gbaguidi

Avec Sourire Et À L'écoute

Vincent Delos

Isa- Belle Valéro

Christophe Castellani

Thierry Raedersdorff

Azedine Mani

Arnaud Meu

Merci Éga- Lement

Chantal Guihenneuc-Jouyaux

Ioannis Nicolis

Patrick Deschamps

Virginie Lasserre

Jean R ; Gwennaëlle

Louis M ; Hélène

Nicolas I ; Clémentine D'

A ; Hervé

Loup L ; Bastien

Keywords: système de particules en interaction, modèle stochastique spatial, processus de contact, milieu aléatoire, attractivité, percolation, transition de phase, limite hydrodynamique, réservoirs interacting particle system, spatial stochastic model, contact process, random environment, attractiveness, percolation, phase transition, hydrodynamic limit, reservoirs multitype process . . . . . .

L'exercice des remerciements... loin d'être aussi facile que je ne l'imaginais, tant l'écosystème de la recherche est riche, tant les proches omniprésents sont nombreux. Il y a certainement des oublis et je prie les personnes concernées de m'en excuser.

Mes premiers mots vont naturellement à ma directrice de thèse, Ellen Saada, envers qui j'exprime ma plus profonde gratitude. Non seulement elle a été -incroyablementpatiente et humaine mais elle a su me mettre sur le chemin -malgré tous mes possibles déboires-avec les encouragements nécessaires. C'est une chance et un immense honneur d'avoir pu travailler avec elle, je ne serai que toujours friand de ses critiques et conseils tant ils sont chers et avisés.

À ceux qui me font l'honneur de composer le jury. Je remercie chaleureusement Rinaldo Schinazi, pour avoir initié ce modèle mathématique et pour les discussions (malgré le décalage horaire) que nous avons pu partager. Il a su attiser ma curiosité et être une source d'inspiration considérable. Je tiens également à remercier Mustapha Mourragui, sa disponibilité, les innombrables aller-retours entre Paris et Rouen mais par dessus tout, sa convivialité et ses mots d'encouragement ont été d'un grand apport. Rapporter une thèse est loin d'être une part de gâteau. Je tiens à remercier Thierry Bodineau et Thomas Mountford d'avoir accepté de rapporter cette thèse, pour l'attention et les commentaires prodigués à ce manuscrit. J'aimerais aussi remercier Frank Redig de prendre part à ce jury et pour son chaleureux accueil aux Pays-Bas.

Ayant été le premier à avoir mis un article entre mes mains, j'aimerais remercier Amaury Lambert pour m'avoir initié à cette voie mais aussi pour s'être montré disponible et m'avoir conforté dans des choix. D'autre part, le long de mes études, j'ai eu la chance d'avoir pu suivre des personnes qui m'ont fait avancer jusque là aujourd'hui, en particulier, j'aimerais remercier Jean Bertoin, Marc Yor et Olivier Zindy, pour avoir été d'une notable attention et des sources d'inspiration.

Après ces années passées à Paris Descartes. Je remercie Annie Raoult, pour le soin qu'elle a porté au laboratoire. La terrasse étant sujette à beaucoup de pauses et de discussions : Merci à l'équipe de Probabilités, pour leur accueil et leurs conseils ; Merci à Joan Glaunès pour sa sympathie et les parties de tennis ; Merci également à Avner Bar-Hen, à Mikael Falconnet pour leur gentillesse. Aux

Interacting particle systems are a class of Markov processes that arose in the early seventies due to pioneering works by F. Spitzer [START_REF] Spitzer | Random processes defined through the interaction of an infinite particle system[END_REF][START_REF] Spitzer | Interaction of Markov processes[END_REF] and R.L. Dobrushin [START_REF] Dobrushin | Prescribing a system of random variables by conditional distributions[END_REF]. They have provided a framework that describes the space-time evolution of an infinity of indistinguishable particles governed by a strong random and local interaction.

This particular class of stochastic processes comes up in various areas of applications : physics, biology, computer science, economics and sociology,... that dictate the nature of the randomness of the processes.

The setup

As a preparation, one first reviews some necessary background theory about interacting particle systems. For further contents on the topic, one refers the reader to T.M. Liggett's books [START_REF] Liggett | Interacting particle systems[END_REF][START_REF] Liggett | Stochastic interacting systems : contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF].

State spaces are of the form Ω " F S , where F is discrete and finite, S is a countable set of sites. Note that Ω is compact in the product topology. A configuration ζ P Ω is described by the state of each site x of the graph S, given at time t by ζ t pxq P F . For each ζ P Ω and T Ă S, the local dynamics of the system is depicted by a collection of transition measures c T pζ, dαq, assumed to be finite and positive on F T . Assume further that the mapping ζ Þ Ñ c T pζ, dαq is continuous from Ω to the space of finite measures on F T with the topology of weak convergence. If ζ is the current configuration, a transition of state or flip involving the coordinates in T occurs at rate c T pζ, F T q and c T pζ, dαq{c T pζ, F T q is the distribution of the resulted configuration restricted to T .

We will use the notation P ζ for the distribution of the process pζ t q tě0 starting from the initial configuration ζ, and E ζ will denote the corresponding expectation. The infinitesimal description of a process ζ P Ω is given by its generator L, a linear unbounded operator defined on an appropriate dense domain DpΩq of the space of functions f : Ω Ñ R. For any cylinder function f , i.e. that depends only on finitely many coordinates, L is defined by

Lf pζq " ÿ T ż Ω c T pζ, dαq `f pζ α q ´f pζq ˘, (1.1.1)
where ζ α is obtained from ζ only by flipping the coordinates in T , that is, for α P F T ,

ζ α " " ζpxq if x R T, αpxq if x P T.
The series converges provided that c T p., .q satisfies natural summability conditions. Let CpΩq be the space of continuous real-valued functions on Ω equipped with the uniform norm. All the processes we consider here have the Feller property (i.e. strong Markov processes whose transition measures are weakly continuous in the initial state) so that the semigroup S t of the process on CpΩq is well defined : 2 1.1. Interacting Particle Systems Theorem 1.1.1. Suppose tS t , t ą 0u is a Markov semigroup on CpΩq. Then there exists a unique Markov process tP ζ , ζ P Ωu such that S t f pζq " E ζ f pζ t q for all f P CpΩq, ζ P Ω and t ě 0.

The link binding the infinitesimal description of the process (generator) to the timeevolution of the process (semigroup) is given by the Hille-Yosida theory set in the Banach space CpΩq.

Theorem 1.1.2 (Hille-Yosida).

There is a one-to-one correspondence between Markov generators on CpΩq and Markov semigroups on CpΩq. This correspondence is given by [START_REF] Andjel | Survival of multidimensional contact process in random environments[END_REF] 2. for t ě 0, S t f " lim nÑ8 pf ´t n Lf q ´n, f P CpΩq.

Relying on the Hille-Yosida theory, the following result states sufficient conditions for the existence of an infinite particle system. where } ¨}T stands for the total variation norm of a measure on F T . Then the closure L of L defined in (1.1.1) is the generator of a Feller Markov process pζ t q tě0 on Ω. In particular, if f is a cylinder function then,

Lf " lim tÑ0 S t f ´f t ,
LS t f " S t Lf and uptq " S t f is the unique solution to the evolution equation B t uptq " Luptq, up0q " f.

(1.1.2)

Let P be the set of probability measures on Ω equipped with the topology of weak convergence, i.e.

µ n Ñ µ in P if and only if

ż Ω f dµ n Ñ ż Ω f dµ
for all f P CpΩq. Note that the compactness of Ω implies the compactness of P in this induced topology.

Invariant measures

Study of interacting particle systems involves use of their invariant measures and ideally, convergence to them. If µ is a probability measure on Ω, the distribution of ζ t with initial distribution µ is denoted by µS t and is defined by ż Ω f dpµS t q "

ż Ω S t f dµ, f P CpΩq.

By the Riesz Representation theorem, this relation defines uniquely µS t . The measure µ is invariant with respect to the process if µS t " µ for all t ą 0. Denote by I the set of all invariant measures. Furthermore, Theorem 1.1.4 (Proposition 1.8 [START_REF] Liggett | Interacting particle systems[END_REF]). i. µ P I if and only if ż Ω Lf dµ " 0, for all cylinder functions f.

ii. I is compact, convex and non-empty.

iii. I is the closed convex hull of its extreme points. iv. Let µ P P. If µ :" lim tÑ8 µS t exists, then µ P I.

Remark that a process always has at least one invariant measure. This measure might satisfy a symmetry property called reversibility that allows simpler computations or even, further results. A probability measure µ on Ω is reversible for the process if ż Ω f S t gdµ " ż Ω gS t f dµ, for all f, g P CpΩq or equivalently, ż

Ω f Lgdµ " ż Ω
gLf dµ, for all cylinder functions f, g.

Coupling and stochastic order

A coupling is a construction of two (or even more) stochastic processes on a common probability space. To make use of this powerful tool, we will deal with several topics that are closely connected with coupling such as stochastic order relations between probability measures, monotone processes and correlation inequalities. These useful relations allow us to compare processes, so that one can deduce properties from one to another by domino effect.

Assuming that F is totally ordered, the state space Ω is a partially ordered set, with partial order given by ζ ď ζ 1 if for all x P S, ζpxq ď ζ 1 pxq,

(1.1.3)

Interacting Particle Systems

where this last inequality refers to the order on F . A function f P CpΩq is increasing if

ζ ď ζ 1 ñ f pζq ď f pζ 1 q.
This leads naturally to define the stochastic order between two probability measures µ 1 and µ 2 on Ω, that is, µ 2 is stochastically larger than µ 1 , written µ 1 ď µ 2 if :

ż Ω f dµ 1 ď ż Ω
f dµ 2 for any increasing function f on Ω.

A necessary and sufficient condition for a semigroup, acting on measures, to preserve the ordering on Ω is given by Theorem 1.1.5 (Theorem 2.2 [START_REF] Liggett | Stochastic interacting systems : contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]). For a Feller process on Ω with semigroup S t , the following two statements are equivalent :

a. If f is an increasing function on Ω then S t f is an increasing function of Ω for all t ě 0.

b. If µ 1 ď µ 2 then µ 1 S t ď µ 2 S t for all t ě 0.

Stochastic order between two particle systems pζ t q tě0 and pζ 1 t q tě0 is given by the existence of a coupled process pζ t , ζ 1 t q tě0 on the probability space Ω ˆΩ that preserves the order between their initial configurations, that is, if ζ 0 ď ζ 1 0 then ζ t ď ζ 1 t a.s. for all t ą 0. Such a coupling is said to be increasing and ζ 1 t is said to be stochastically larger than ζ t . When pζ t q tě0 and pζ 1 t q tě0 are two copies of the same process, we say the process is attractive.

The following result gives the connection between coupling and stochastic order. Theorem 1.1.6 (Theorem 2.4 [START_REF] Liggett | Interacting particle systems[END_REF]). Let µ 1 and µ 2 be probability measures on Ω. Then µ 2 is stochastically larger than µ 1 if and only if there exists a coupling pζ, ζ 1 q such that ζ has distribution µ 1 , ζ 1 has distribution µ 2 and ζ ď ζ 1 almost surely, that is, there exists a measure ν on Ω such that νtpζ, ζ 1 q : ζ P Au " µ 1 pAq νtpζ, ζ 1 q : ζ 1 P Au " µ 2 pAq νtpζ, ζ 1 q : ζ ď ζ 1 u " 1 Furthermore, we will consider different types of stochastic processes : pξ t q tě0 (basic) contact process pξ t , ω t q tě0 contact process in dynamic random environment pη t q tě0 multitype contact process

A short story of the contact process

Introduced by T.E. Harris in 1974 [START_REF] Harris | Contact interactions on a lattice[END_REF], the contact process on the graph S with growth rate λ 1 is an interacting particle system pξ t q tě0 on t0, 1u S , whose dynamics is given by the following transition measure : the involved sets T are singletons T " txu and, c T pξ, dαq "

" λ 1 n 1 px, ξqδ t1u if ξpxq " 0, δ t0u if ξpxq " 1, (1.2.1)
where n i px, ξq " ř yPS:|y´x|"1

1tξpyq " 1u stands for the number of neighbours of site x that are in state i. Here | ¨| refers to the maximum norm : |x| " max 1ďjďd |x j |, for x P R d . Denote by P λ 1 the law of the contact process with growth rate λ 1 .

It is usually interpreted as the spread of a population, an infection or a rumour. Regarded as an infection, infected sites (in state 1) become healthy spontaneously after a unit exponential time while healthy sites (state 0) become infected at some rate, proportional to the number of their infected neighbours.

General theory about the contact process is finely exposed by T.M Liggett [START_REF] Liggett | Interacting particle systems[END_REF] for results from 1974 to 1985, [START_REF] Liggett | Stochastic interacting systems : contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] for results after 1985 and by R. Durrett [START_REF] Durrett | Lecture notes on particle systems and percolation[END_REF] as well.

Construction of the process

Let A be a subset of S. Define ξ A t as the process starting from the initial configuration ξ 0 " 1 A . Configurations ξ P t0, 1u S are commonly identified with subsets of S via Ξ A t " tx P S : ξ A t pxq " 1u, regarded as the set of occupied sites at time t. When A " t0u, we will omit the exponent. As a consequence of Theorem 1.1.3, the transition measure c T pξ, dαq uniquely defines a Markov process, so that the infinitesimal generator of the contact process is defined for any cylinder function f on t0, 1u S by

L 1 f pξq " ÿ xPS ż Ω c T pξ, dαqrf pξ α q ´f pξqs (1.2.2)

Graphical representation

The graphical construction of the contact process is due to T.E. Harris [START_REF] Harris | Additive set-valued Markov processes and graphical methods[END_REF]. The idea is to construct a percolation structure on which to define the process, lending itself to the use of the theory of percolation (see G. Grimmett [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]).

To carry out this representation, for each pair px, yq P S 2 that are joined by an edge in S, let tT x,y n , n ě 1u be the arrival times of independent rate λ 1 Poisson processes and for each x P S, let tD x n , n ě 1u be the arrival times of independent rate 1 Poisson processes.

A short story of the contact process

Both families of Poisson processes are mutually independent. Now, think of the spacetime diagram S ˆr0, 8q. At time t " D x n , put a death symbol "x" at px, tq P S ˆr0, 8q. At time t " T x,y n , draw an arrow from px, tq to py, tq. By way of illustration, see Figure 1.1. or s ď t, there exists an active path in the space-time picture S ˆr0, 8q from px, sq to py, tq, written px, sq Ñ py, tq, if there exists a sequence of times s " s 0 ă s 1 ă ... ă s n´1 ă s n " t and spatial locations x " x 0 , x 1 , ..., x n " y such that i. for i " 1, ..., n, there is an arrow from x i´1 to x i at time s i .

ii. for i " 0, ..., n ´1, the vertical segments tx i u ˆps i , s i`1 q contain no death symbol.

In words, an active path is a connected oriented path that moves forward in time without crossing a death symbol and along the directions of the arrows. For instance, in Figure 1.1, there is an active path from p0, 0q to p1, tq. The contact process with initial configuration A Ă S is obtained by setting A A t :" ty P S : Dx P A such that px, 0q Ñ py, tqu Therefore, in our previous example, A t0u t " t1u.

The graphical construction provides a joint coupling of contact models with different transition rates : let λ

Chapter 1. Introduction

Thus, one has a non-decreasing growth with respect to λ 1 . On the other hand, it also provides a monotone coupling :

A Ă B ñ A A t Ă A B t ,
Therefore, the contact process is attractive and it also follows from the graphical construction that the contact process is additive (see D. Griffeath [START_REF] Griffeath | Additive and cancellative interacting particle systems[END_REF]) :

A AYB t " A A t Y A B t .

Upper invariant measure and duality

Since the partial order on Ω defined in (1.1.3) induces one on the set of probability measures on Ω, there will be a lowest and largest element on I with respect to this partial order.

If 0 denotes the configuration identically equal to 0, since 0 is an absorbing state then δ 0 is called the lower invariant measure for the contact process. The upper invariant measure can be constructed using attractiveness : choose the initial configuration as the biggest possible one, i.e. starting from Ξ 0 " S, and let µ t be the distribution of ξ t , so that µ 0 " δ 1 . Then µ t ď µ 0 . By attractiveness and applying the Markov property, we have µ t`s ď µ t for all s ą 0. Therefore, t Þ Ñ µ t is decreasing and in particular, for every increasing function f on Ω, the map t Þ Ñ ş Ω f dµ t is decreasing as well. Since PpΩq is compact for the weak topology, the limiting distribution µ :" lim tÑ8 µ t exists and is the upper invariant measure of the process. It is invariant as a limiting measure for the Markov process by Theorem 1.1.4. In particular, the measure µ has positive correlations.

Correlation inequalities will be crucial property in Section 2.6 where we will work in arbitrary large but finite spaces. A probability measure µ on Ω has positive correlations if ż

Ω f gdµ ě ż Ω f dµ ż Ω gdµ,
for all increasing functions f, g on Ω. A sufficient condition for a measure to have positive correlations is given by the following result.

Theorem 1.2.1 (C. Fortuin, P. Kasteleyn and J. Ginibre [START_REF] Fortuin | Correlation inequalities on some partially ordered sets[END_REF]). Suppose S is finite. Let µ be a probability measure on Ω such that for all ζ, ζ 1 P X µ 1 pmaxpζ, ζ 1 qqµ 2 pminpζ, ζ 1 qq ě µ 1 pζqµ 2 pζ 1 q Then µ has positive correlations.

A short story of the contact process

One essential property satisfied by the contact process is that it is self-dual [34, Proposition 6.5], that is, the dual process is again a contact process. For A, B Ă S, P λ 1 pΞ A t X B ‰ Hq " P λ 1 pΞ B t X A ‰ Hq (1.2.3) This property allows us to link an equality relation between survival probability and density of 1's under the upper invariant measure. Indeed, since tΞ t0u t`1 X S ‰ Hu Ă tΞ t0u t X S ‰ Hu for all t ě 0, t Þ Ñ tΞ t0u t X S ‰ Hu is non-increasing, lim tÑ8 P λ 1 pΞ t0u t X S ‰ Hq " P λ 1 p@t ě 0, Ξ t0u t

‰ Hq

By self-duality, applying (1.2.3) with A " t0u and B " S, one obtains P λ 1 pΞ t0u t X S ‰ Hq " P λ 1 pΞ S t X t0u ‰ Hq. The right-hand side is P λ 1 pΞ S t p0q " 1q, and by weak convergence of µ 0 to µ, one has lim tÑ8 P λ 1 pΞ S t p0q " 1q " μtξ : ξp0q " 1u

where µ stands for the upper invariant measure of pξ t q tě0 . By translation invariance of µ, lim tÑ8 P λ 1 pΞ t0u t X S ‰ Hq " lim tÑ8 P λ 1 pξ S t p0q " 1q " μtξ : ξpxq " 1u (1.2.4)

Survival and extinction

A key feature of the contact process lies in the fact its growth does not evolve spontaneously but depends on some neighbourhood. In words, the configuration 0 is a trap and a natural question is whether the individuals survive, that is, if there is infinitely often a site in state 1. The main feature of the contact process is that it exhibits a phase transition in the following way.

Define the survival event of the process by t@t ě 0, Ξ t ‰ Hu with the initial configuration ξ 0 " 1 t0u . The contact process is said to die out if P λ 1 p@t ě 0, Ξ t ‰ Hq " 0 and to survive strongly if P λ 1 p lim tÑ8 ξ t p0q " 1q ą 0.

The process is said to survive weakly if it survives but not strongly, that is, P λ 1 p@t ě 0, Ξ t ‰ Hq ą 0.

Using these definitions and monotonicity, we are now ready to define the two following critical values : λ c " inftλ 1 : P λ 1 p@t ě 0 Ξ t ‰ Hq ą 0u (1.2. if the process survives weakly then it survives strongly thus λ c ď λ s .

On the d´dimensional integer lattice Z d , one of the most important results about the contact process is the existence and uniqueness of a critical value λ c " λ s . Theorem 1.2.2 (T.E. Harris [START_REF] Harris | Contact interactions on a lattice[END_REF]). There exists a critical value λ c P p0, 8q such that the contact process survives if λ 1 ą λ c and dies out if λ 1 ă λ c , i.e. P λ 1 p@t ě 0, Ξ t ‰ Hq " 0 if λ 1 ă λ c , P λ 1 p@t ě 0, Ξ t ‰ Hq ą 0 if λ 1 ą λ c .

After having been an open question during about fifteen years, the critical behaviour has been given by Theorem 1.2.3 (C. Bezuidenhout and G.R. Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF]). The critical contact process dies out, that is, P λc p@t ě 0, Ξ t ‰ Hq " 0.

R. Holley and T.M. Liggett [START_REF] Holley | The survival of contact processes[END_REF] proved λ c ď 2 in the one-dimensional case. An improved upper bound 1.942 was given by T.M. Liggett [START_REF] Liggett | Improved upper bounds for the contact process critical value[END_REF]. More generally, one has for the general case d ě 1, p2d ´1q ´1 ď λ c ď 2d ´1, see N. Konno [START_REF] Konno | Phase transitions of interacting particle systems[END_REF] for further information on bounds of the contact process.

Hydrodynamic limits

Hydrodynamic limits are a device that arose in statistical physics to derive deterministic macroscopic evolution laws assuming the underlying microscopic dynamics are stochastic.

By way of illustration, consider the evolution of a system constituted of a large number of components (such as a fluid), one can examine and characterize the equilibrium states of the system through macroscopic quantities (such as temperature or pressure). Now, investigating the fluid in a volume which is small macroscopically but 1.3. Hydrodynamic limits large microscopically, the system is close to an equilibrium state and characterized by some spatial parameter. As the local equilibrium picture should evolve in a smooth way, at some time t the system is close to a new equilibrium state now characterized by a parameter depending on space and time. This space-time parameter evolves smoothly in time according to a partial differential equation, the hydrodynamic equation.

To take the limit from the microscopic to the macroscopic system, we need to introduce a suitable space-time scaling. Consider a microscopic space S N embedded in a corresponding macroscopic space S (e.g. S N " pZ{N Zq d and S " pR{Zq d ) so each microscopic vertex x P S N is associated to a macroscopic vertex x{N P S. Therefore, distance between particles converges to zero. Besides, we renormalize the time by linking a microscopic time t to a macroscopic time tθpN q (e.g. θpN q " N 2 ), since more time is needed in the macroscopic scale to observe movements of particles.

To investigate the hydrodynamic behaviour of interacting particle systems we shall prove that starting from a sequence of measures associated to some initial density profile ρ 0 , in the following sense lim

N Ñ8 µ N ˜ˇˇ1 N d ÿ xPS N Gpx{N qηpxq ´żS Gpuqρ 0 puqdu ˇˇą δ ¸" 0 (1.3.1)
for any δ ą 0 and continuous function G : S Ñ R, then at some renormalized time tθpN q, we obtain a state S tθpN q µ N associated to a new density profile ρ t p¨q that is a weak solution of a partial differential equation. That is, lim

N Ñ8 µ N ˜ˇˇ1 N d ÿ xPS N
Gpx{N qη tθpN q pxq ´żS Gpuqρ t puqdu ˇˇą δ ¸" 0.

(1.3.2)

In other words, the sequence of measures µ N integrates the density ρ t at the macroscopic point u P S in the same way than an equilibrium measure of density γpuq does. Since we shall work in a fixed space as N increases, we will examine the timeevolution of the empirical measures associated to the interacting particle system : for a configuration η P Ω, define the empirical measure π N pηq on S associated to η by

π N pηq " N ´d ÿ xPS N ηpxqδ x{N , (1.3.3)
where δ x represents the Dirac measure concentrated on x. This way, we can express (1.3.2) in terms of the empirical density, by integrating G with respect to π N . Since there is a one-to-one correspondence between a configuration η and empirical measure π N pηq, the measure π N t inherits the Markov property. The goal to derive the hydrodynamic limits is to prove the empirical measure π N t converges in probability to an absolutely continuous measure ρpt, uqdu where ρ t puq is the solution of a partial differential equation with initial condition ρ 0 .

Monographs dealing with hydrodynamic limits include A. De Masi and E. Presutti [START_REF] Masi | Mathematical methods for hydrodynamic limits[END_REF], H. Spohn [START_REF] Spohn | Large scale dynamics of interacting particle systems[END_REF] C. Kipnis and C. Landim [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF].

Chapter 1. Introduction

From life and nature

During the last decades, a better understanding of biological phenomena has arisen the need to study stochastic spatial processes. Authors such as R. Durrett, R. Schinazi, or J. Schweinsberg have deemed the relation of interacting particle systems to biological, ecological and medical frameworks. A quick interesting overview may be found in joint papers of R. Durrett with the biologist S. Levin [START_REF] Durrett | The importance of being discrete (and spatial)[END_REF][START_REF] Durrett | Stochastic spatial models : a user's guide to ecological applications[END_REF], and [START_REF] Durrett | Special invited paper : coexistence in stochastic spatial models[END_REF].

In this document, the biological phenomenon we are concerned is the so-called Sterile insect technique (SIT). Due to entomologists R.C. Bushland and E.F. Knipling's works [START_REF] Knipling | Possibilities of insect control or eradication through the use of sexually sterile males[END_REF] in the fifties, it is a pest control method whereby sterile individuals of the population to either regulate or eradicate are released. While sterile males compete with wild males, they eventually mate with (wild) females preventing the apparition of progenies. By repeated releases, we should be able to cause a variety of outcomes ranging from reduction to extinction.

The sterile insect technique

In the thirties and forties, the idea of designing a gene that actively spreads through a pest population without conveying some fitness advantage had arisen independently by A. S. Serebrovskii (Moscow State University), F. L. Vanderplank (Bristol Zoo and Tanzania Research Department) and E. F. Knipling (United States Department of Agriculture). Serebrovskii and Vanderplank both sought to achieve pest control through partial sterility that occurs when different species or genetic strains were hybridized (using chromosomal translocations or crossing) : competition between two interbreeding strains doesn't favour the fitter group, involving the genetic property called under-dominance which can actually cause the strain with greater fitness to die out.

Discovery and first success story. Discovery of induced mutagenesis by 1946 Nobel Prize H.J. Muller conducted Bushland and Knipling to use ionizing radiation in the sterilization process to get rid of the new world screw-worm fly (Cochliomyia hominivorax).

After successful eradication programs carried out in Curaçao and Florida in the late fifties, the technique was applied during the next decades to eradicate the screw-worm from the USA, Mexico, and Central America to Panama, until it has been declared a fly-free area.

The big picture. Food safety, quality and biodiversity have required demands at national and international levels for the development and introduction of area-wide (and biological approaches) for integrated management of pest control.

Fruit flies are a major interference in the marketing of fruit and vegetable commodities, preventing therefore important economic developments. The Mediterranean fruit 1.4. From life and nature fly (medfly) is a notorious insect pest threatening multi-million commodities export trade throughout the world.

In the seventies, a first large-scale program stopped the invasion of the medfly from Central America. Eradication from Mexico and maintaining the country free of this pest at an annual cost of US$ 8 million, has protected fruit and vegetable export markets of close to US$ 1 billion a year.

In Japan, the SIT was employed in the eighties and nineties to eradicate the melon fly in Okinawa and south-western islands, permitting access for fruits and vegetables produced in these islands to the main markets in the mainland. A program with Peru operates in Argentina, northern Chile and southern Peru. Chilean fruits have entered the US market for exports estimated to up to US$ 500 million per year.

More recently, the SIT is increasingly applied with eradication programs of fruit flies ongoing in Middle-East (Israel, Jordan, Palestine), South Africa, and Thailand ; in preparation in Brazil, Portugal, Spain, and Tunisia.

Economic benefits have been confirmed so that for medflies and other fruit flies, the current worldwide production capacity of sterile individuals has reached several billion a week. Future trends. Lauded for its attributes in terms of economics, environment and safety, the technique has successfully been able to get rid of populations threatening livestocks, fruits, vegetables, and crops. But besides economic reasons to involve SIT, public health issues have induced governments to request supports from International Atomic Energy Agency (IAEA) and Food and Agriculture Organization of the United Nations (FAO) for SIT initiatives to stem vector-borne diseases.

Time to unleash the mozzies ?

Thinking about the deadliest animal in the world, mosquitoes would not hit our minds. But one estimates about 1 million people per year die from mosquito-borne diseases, such as malaria, dengue fever, etc ... [Source : World health organization].

Urbanisation, globalisation and climate change have accelerated the spread and increased the number of outbreaks of new mosquito-borne diseases, such as the dengue.

Considered as the fastest growing disease, dengue fever is currently not cured by any vaccine or effective antiviral drug, meaning that mosquito control is the only viable option to control the disease at short notice. The SIT has the potential to reduce the targeted mosquitoes population to a level below which the disease is not transmitted. A first trial using sterile mosquitoes was conducted in El Salvador in the seventies, where 4.4 million sterile individuals were released in a 15 square km area over 22 weeks, eradicating successfully the targeted population. Going on a much larger area, total suppress of the population failed due to an immigration of local mosquitoes into the trial area. Being the highest endemic country of dengue, the brazilian government is highly concerned by the expansion of the dengue fever. According to pilot-scale releases in the state of Bahia started in june 2013, releases of genetically modified mosquitoes resulted in a 96% reduction of the wild population in the target area after 6 months-level maintained for a further 7 months using continued releases, at reduced rates, to avoid re-infestation.

The National Technical Commission for Biosecurity (CTNBio) in Brazil recently approved (april 2014) the commercial release of genetically modified mosquitoes in a bid to curb outbreaks of dengue fever. As of july 2014, the research program in the state of Bahia is waiting for an approval granted by the Brazilian Health Surveillance Agency (ANVISA) to ensue a scaling-up of the program. [Source : Comissão Técnica Nacional de Biossegurança (CTNBio), Agência Nacional de Vigilância Sanitária (ANVISA).]

Past mathematical models

Even if models of population dynamics are typically posed as difference or differential equations, such as predator-prey systems (whose Nicholson-Bailey and Lotka-Volterra models are the work horses), stochastic models give additional information on the expected variability of the resulting control. Some of them were developed by Kojima (1971), Bogyo (1975), Costello and Taylor (1975), Taylor (1976) and Kimanani and Odhiambo (1993), and they confirmed the former results of [START_REF] Knipling | Possibilities of insect control or eradication through the use of sexually sterile males[END_REF] [START_REF] Knipling | Possibilities of insect control or eradication through the use of sexually sterile males[END_REF] and others that used deterministic models.

As a former model, [START_REF] Knipling | Possibilities of insect control or eradication through the use of sexually sterile males[END_REF]Knipling ( , 1959) ) derived a simple numerical model foresha-1.5. The generalized contact process dowing most future modelling developments. The key feature of Knipling's models, and found in most of all subsequent models, is the ratio of fertile males to all males in the population. Simply modifying a geometric growth model,

F t`1 " λpW t {pS `Wt qqF t
where F t and W t is the population size of females and wild males at time t, λ is the growth rate per generation, R is the release rate of sterile individuals each generation. This yields an unstable positive equilibrium for F when R " R˚, where R˚" F pλ ´1q denotes the critical release rate, so that if R ą R˚then the population collapses while if R ă R˚then the population will increase indefinitely.

The question of the competitive ability of males was modelled amongst others by Berryman (1967), Bogyo et al. (1971), Berryman et al. (1973), Ito (1977), and Barclay (1982) all showing that the critical release rate increases as the competitive ability of sterilized individuals decreases.

For a general overview of the technique, we refer the reader to [START_REF] Dyck | Sterile insect technique : Principles and Practice in area-wide integrated pest management[END_REF].

The generalized contact process

In the further chapters, one constructs a contact process in random environment to lead a better understanding of this ecological phenomenon. Fix growth parameters λ 1 , λ 2 and release rate r.

One introduces the contact process in dynamic random environment (CP-DRE) on the graph S with parameters set pλ 1 , λ 2 , rq as an interacting particle system pξ t , ω t q tě0 P pt0, 1u ˆt0, 1uq S that evolves through the following dynamics. The environment part pω t q tě0 evolves independently according to 0 Ñ 1 at rate r, 1 Ñ 0 at rate 1, (1.5.1) while the contact process part evolves at x P S according to 0 Ñ 1 at rate ř

y:}y´x}"1 ´λ1 ξpyqp1 ´ωpyqq `λ2 ξpyqωpyq ¯, 1 Ñ 0 at rate 1. (1.5.2)
As we shall see, the most interesting case corresponds to λ 2 ď λ c ă λ 1 , where λ c denotes the critical value of the (basic) contact process. In words, the CP-DRE depicts a basic contact process whose growth rate is either subcritical or supercritical according to a time-evolving random environment which is parametrized by a rate r.

In our framework, one understands the environment as the space-time evolution of the sterile population released at rate r while the contact process stands for the wild population. When mixed up on a site, a competition between the two species occurs, slowing down the growth of the wild individuals to a subcritical rate λ 2 , if not, the wild individuals perform a supercritical contact process. Each individual dies spontaneously at rate 1.

In a traditional overview, the contact process part describes the spread of an infection, so that the environment is thought of as being an immune response, attempting to slow down the expansion of the infection.

We also make use of a different but equivalent outlook of this process, that is, one constructs a (single) multitype contact process pη t q tě0 on t0, 1, 2, 3u S , where each of these values corresponds to a possible combination of values taken by the process pξ t , ω t q tě0 . This way, a site x of S is empty if in state 0, occupied by type-1 individuals if in state 1, by type-2 individuals if in state 2 and occupied by both types simultaneously if in state 3.

It is important to underline that a site is occupied by a type of individuals and not as usual, by the number of individuals present standing on. We shall therefore rather think of a multicolour system.

Biologically speaking, one interprets the type-1 individuals as being the wild individuals and the type-2 as being the sterile individuals. Sites in state 3 containing both types represent sites where competition occurs. We say that sites in state 1 or 3 constitute the wild population.

Furthermore, in the multitype outlook we consider two kinds of action for the type-2 individuals that are reducing the growth rate in sites in state 3. In a so-called asymmetric case, type-2 individuals prevent births from occurring in sites they are standing on. Call it symmetric otherwise. Common transition rates for both cases at site x are given by 0 Ñ 1 at rate λ 1 n 1 px, ηq `λ2 n 3 px, ηq 1 Ñ 0 at rate 1 0 Ñ 2 at rate r 2 Ñ 0 at rate 1 1 Ñ 3 at rate r 3 Ñ 1 at rate 1 3 Ñ 2 at rate 1 (1.5.3) in which one adds the following transition in the symmetric case 2 Ñ 3 at rate λ 1 n 1 px, ηq `λ2 n 3 px, ηq.

(1.5.4)

As competition occurs in sites in state 3, growth rate λ 2 has to be lower than growth rate λ 1 of sites in state 1 where only type-1 individuals live. One thus makes the hypothesis :

λ 2 ă λ 1 . (1.5.5)
Here, since the presence of type-2 individuals dictate the growth rate of type-1 individuals, to even inhibit births in the asymmetric case, the type-2 individuals shape a dynamic random environment for the type-1 individuals.

The generalized contact process

Both outlooks of the process are linked by the following relations :

ηpxq " 0 Ø p1 ´ξpxqqp1 ´ωpxqq " 1 ηpxq " 1 Ø ξpxqp1 ´ωpxqq " 1 ηpxq " 2 Ø p1 ´ξpxqqωpxq " 1 ηpxq " 3 Ø ξpxqωpxq " 1
In a microscopic scale, we examine survival and extinction conditions for the population, after what, taking the hydrodynamic limit, we study the behaviour of the densities of each type of population at a macroscopic scale.

Phase transition in dynamic random environment

Set S as the d-dimensional integer lattice Z d , d ě 1. In Chapter 2, one investigates how the release rate affects the behaviour of the process.

First, we point out general properties of the system, such as necessary and sufficient conditions for the process to be monotone, then, only sufficient conditions to be in line with the construction of the process. The tricky part to prove these conditions lies in the definition of an order on the state space t0, 1, 2, 3u Z d , since a value on a given site does not correspond to the number of particles but a type. This is the interest of the next result.

Proposition. The symmetric multitype process is monotone, in the sense that, one can construct on a same probability space two symmetric multitype processes pη p1q t q tě0 and pη p2q t q tě0 with respective parameters pλ if and only if both parameters sets satisfy

1. λ p1q 2 ď λ p1q 1 , 2. λ p2q 2 ď λ p2q 1 , 3. λ p1q 1 ď λ p2q 1 , 4. λ p1q 2 ď λ p2q 2 , 5. r p1q ě r p2q 6. λ p1q 1 ď 1, 7. λ p1q 2 ď 1, 8. r p1q ě 1.
Essentials of SIT concern the control of the population by releasing sterile individuals, the question we address now is for which values of r does the wild population survive or die out ? For this, we prove the existence and uniqueness of a phase transition with respect to the release rate r for fixed growth rates λ 1 and λ 2 . The most interesting cases are discussed in the following results :

Theorem. Suppose λ 2 ď λ c ă λ 1 fixed. Consider the symmetric multitype process. There exists a unique critical value r c P p0, 8q such that the wild population survives if r ă r c and dies out if r ě r c .

Theorem. Suppose λ c ă λ 1 fixed. Consider the asymmetric multitype process. There exists a unique critical value s c P p0, 8q such that the wild population survives if r ă s c and dies out if r ě s c .

Chapter 1. Introduction

This actually confirms the former conclusions done by [START_REF] Knipling | Possibilities of insect control or eradication through the use of sexually sterile males[END_REF] in a deterministic model mentioned in Section 1.4.

Proofs strongly rely on the use of graphical representations and comparison with percolation processes that introduced M. Bramson and R. Durrett [START_REF] Bramson | A simple proof of the stability criterion of Gray and Griffeath[END_REF]. Using dynamic renormalization techniques from G. Grimmett et al. [START_REF] Barsky | Percolation in half-spaces : equality of critical densities and continuity of the percolation probability[END_REF][START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF], we are in particular able to describe the behaviour of the critical process. As a consequence, this allows us to discuss the competitive ability of the sterile individuals which was biologically exhibited (as mentioned in Section 1.4) : one shows the critical value increases as the competitiveness of the sterilized population decreases or as the fitness of the wild population increases.

We end up this chapter by considering the associated mean-field equations. This shows us a dynamical system featuring the densities of each type of individuals. There, we can explicit equilibria and mainly explicit numerical bounds on the transitional phase. We shall derive a rigorous proof of the convergence of the empirical densities to these macroscopic equations in Chapters 4 and 5.

Survival and extinction in quenched environment

In the previous chapter, we were unable to get a hand on bounds for the critical rate. Most of the arguments made use of theory of percolation, misfit to explicit criteria for the survival and extinction events. A way to come to this end is to consider the process pξ t , ωq tě0 by restricting the random environment to be initially fixed and setting S " Z.

Using former results obtained by T.M. Liggett [START_REF] Liggett | Spatially inhomogeneous contact processes[END_REF][START_REF] Liggett | The survival of one-dimensional contact processes in random environments[END_REF], one obtains in Chapter 3 several survival and extinction conditions for the process. In that way, we consider two kinds of growth rates in Z : one where the rates depend on the edges and one where the rates depend on the vertices. This yields numerical bounds on the transitional phase for the process to survive or die out.

After having investigated the behaviour of each type of individuals in a microscopic scale, we now turn into the study of the system in a macroscopic scale. When the microscopic evolution is more intricate, by a suitable scaling in time and space, we investigate the convergence of the empirical densities of each type of population.

Hydrodynamic limit in a bounded domain

In Chapter 4, set S " T d the d-dimensional torus, and assume the microscopic dynamics is driven by the asymmetric multitype process pη t q tě0 along with a diffusion process, modelling the migrations of the individuals. The diffusion process we consider here is a stirring process that exchanges two neighbouring occupation variables. Resulting with a reaction-diffusion process, we prove the convergence of the time-evolution of the empirical densities to the weak solution of a reaction-diffusion system.

The generalized contact process

Hydrodynamic limits with stochastic reservoirs or in infinite volume

One of the recurring reasons why the SIT fails, comes from an unexpected immigration in the system that prevents to maintain the pest population at a low level after regular releases. Such migrations with the external of the targeted area suggests the microscopic system is likely to be in non-equilibrium states.

In Chapter 5, one considers the microscopic time-evolution to be driven by the CP-DRE along with a rapid-stirring process. We consider a bounded cylinder connected to stochastic reservoirs at its boundaries with different densities in a stationary regime, creating and annihilating individuals. Such reservoirs create a flow through the system that put it in a nonequilibrium state, as dynamics within the bulk is no more reversible. Jointly with M. Mourragui and E. Saada, we establish the limiting equations given by a non-linear reaction-diffusion system with Dirichlet boundary conditions and a law of large numbers for the empirical currents. In a second step, we derive the hydrodynamic limit of the CP-DRE with rapid-stirring in infinite volume Z d .

Introduction

The Sterile insect technique concerns the control of a population by releasing sterile individuals of the same species, leading to a competition with the wild individuals to the reproduction. When a match with sterile individuals occurs, offsprings reach neither the adult phase nor sexual maturity, reducing the next generation.

This chapter is an attempt to understand the behaviour of the wild population with respect to the release of the competitive sterile individuals in this model. Following issues corresponding to biology and ecology, a wide class of multi-type contact processes has emerged. Relevant questions are to identify the mechanisms involving survival, existence or coexistence of species ; such questions have been topics of works such as the grassbushes-tree model by R. Durrett and G. Swindle [START_REF] Durrett | Are there bushes in a forest ? Stochastic Process[END_REF], a 2-type contact process by C. Neuhauser [START_REF] Neuhauser | Ergodic theorems for the multitype contact process[END_REF], a 3-type model by R. Durrett and C. Neuhauser [START_REF] Durrett | Coexistence results for some competition models[END_REF] for the spread of a plant disease.

The populations we consider are composed of wild males whereby sterile males are released at rate r to curb their development. We investigate the survival of the wild ones whose growth rate is time-evolving and randomly determined depending on the dynamics of the sterile individuals.

In Section 2.2, we describe the model and introduce some preliminary results about stochastic order and percolation. Then, we build graphically the particle system through Harris' graphical representation in Section 2.3. After exhibiting necessary and sufficient conditions for monotonicity properties in Section 2.4, we prove the existence and uniqueness of a phase transition with respect to the release rate in Sections 2.5 and 2.6.

Settings and results

The model

On the state space Ω " F S , where F " t0, 1, 2, 3u and S " Z d , the multitype contact process is an interacting particle system pη t q tě0 whose configuration at time t is η t P Ω, that is, for all x P Z d , η t pxq P F represents the state of site x at time t. Two sites x and y are nearest neighbours on Z d if }x ´y} " 1, also written x " y, and n i px, η t q stands for the number of nearest neighbours of x in state i, i " 1, 3.

One understands the model as follows : at time t, a site x in Z d is empty if in state 0, occupied by type-1 individuals if in state 1, by type-2 individuals if in state 2 and by both type-1 and type-2 individuals if in state 3.

Note that we only consider the type of individuals standing on each site and not their number. Moreover, we assume no limit on the number of female individuals, which is biologically a reasonable assumption (see Chapter 1).

Type-2 individuals act in two possible ways, they will reduce the growth rate of the type-1 individuals within sites in state 3. There, a competition occurs, so that the growth rate λ 2 shall be lower than the regular growth rate λ 1 in type-1 population where stand only wild individuals. Our basic assumption is thus,

λ 2 ă λ 1 . (2.2.1)
Furthermore, in a so-called asymmetric case, type-2 individuals will stem births on sites they occupy. Since we deal with the evolution of a population modelled by a particle system, we will often mingle the terms "individuals" and "particles".

Settings and results

The multitype contact process. Common transitions to both cases are the following : individuals on a site in state 1 (resp. 3) gives birth to type-1 individuals at rate λ 1 (resp. λ 2 ) on one of its 2d nearest neighbour sites, if empty. A type-2 individual is dropped independently and spontaneously at rate r on any site in Z d . Each type dies at rate 1, deaths are mutually independent. In the so-called symmetric case, births occurs on sites in state 2 as well.

Transition rates in x for a current configuration η that are common to both cases are :

0 Ñ 1 at rate λ 1 n 1 px, ηq `λ2 n 3 px, ηq 1 Ñ 0 at rate 1 0 Ñ 2 at rate r 2 Ñ 0 at rate 1 1 Ñ 3 at rate r 3 Ñ 1 at rate 1 3 Ñ 2 at rate 1 (2.2.2)
to which one adds the following transition in the symmetric case

2 Ñ 3 at rate λ 1 n 1 px, ηq `λ2 n 3 px, ηq. (2.2.3)
Therefore, the evolution of type-2 individuals occurs whatever the evolution of type-1 individuals is. Since type-2 individuals dictate the growth rate and even inhibit births in the asymmetric case, the type-2 individuals shape a dynamic random environment for the type-1 individuals.

In both cases, if η P Ω and x P Z d , denote by η i x P Ω, i P t0, 1, 2, 3u, the configuration obtained from η after a flip of x to state i :

η ÝÑ η i
x at rate cpx, η, iq, where @u P Z d , η i x puq "

" ηpuq if u ‰ x i if u " x (2.2.4)
Let L be the infinitesimal generator of pη t q tě0 , then for any cylinder function f on Ω :

Lf pηq " ÿ xPZ d 3 ÿ i"0 cpx, η, iq `f pη i x q ´f pηq ˘(2.2.5)
with infinitesimal transition rates, common to both cases,

cpx, η, 0q " 1 if ηpxq P t1, 2u cpx, η, 1q " " λ 1 n 1 px, ηq `λ2 n 3 px, ηq if ηpxq " 0 1 if ηpxq " 3 cpx, η, 2q " " r if ηpxq " 0 1 if ηpxq " 3 cpx, η, 3q " r if ηpxq " 1 (2.2.6)
Chapter 2. Phase transition on Z d and add the following rate in the symmetric case :

cpx, η, 3q " λ 1 n 1 px, ηq `λ2 n 3 px, ηq if ηpxq " 2.
Notice that all rates satisfy for all i P F , cpx, η, iq ě 0, sup Under these mild conditions, by Theorem 1.1.3 there exists a unique Markov process associated to the generator (2.2.5). Denote by pη A t q tě0 the process starting from A, i.e. such that η 0 " 1 A , in other words η 0 corresponds to the configuration containing sites in state 1 in A and empty otherwise. We care about the evolution of the wild population, i.e. individuals contained in sites in state 1 and 3. Define

H A t " tx P Z d : η A t pxq P t1, 3uu, (2.2.9)
as the set of sites containing the wild population at time t ě 0. Note that since η 0 " t0u,

H t0u 0 " tx P Z d : η t0u 0 pxq " 1u.
Denote by P λ 1 ,λ 2 ,r the distribution of pη t0u t q tě0 with parameters pλ 1 , λ 2 , rq. For fixed λ 1 and λ 2 , simplify by P r . Definition 2.2.1. The process pη t q tě0 with initial configuration η 0 " 1 t0u , survives if

P λ 1 ,λ 2 ,r p@t ě 0, H t0u t ‰ Hq ą 0 (2.2.10)
and dies out if

P λ 1 ,λ 2 ,r pDt ě 0, H t0u t " Hq " 1. (2.2.11)
Define the critical value according to the parameter r by r c " r c pλ 1 , λ 2 q :" inftr ą 0 : P r pDt ě 0, H t0u t " Hq " 1u (2.2.12) Indeed, the class t0, 2u is a trap : as soon as H t " H, the wild population is extinct while sterile individuals are constantly dropped along the time.

Recall λ c stands for the critical value of the basic contact process. The purpose of this chapter is to settle the following results.

We begin by a first set of conditions for the process to survive or die out, when λ 2 ă λ 1 are both smaller or larger than λ c : Proposition 2.2.1. Suppose λ 2 ă λ 1 ď λ c . For all r ě 0, both symmetric and asymmetric multitype processes with parameters pλ 1 , λ 2 , rq die out.

Settings and results

Proposition 2.2.2. Suppose λ c ă λ 2 ă λ 1 . For all r ě 0, the symmetric multitype process with parameters pλ 1 , λ 2 , rq survives.

The most interesting cases are given by Theorem 2.2.1. Suppose λ 2 ă λ c ă λ 1 . Consider the symmetric multitype process. There exists a unique critical value r c P p0, 8q such that if r ă r c , then the process survives and if r ą r c , then the process dies out. Theorem 2.2.2. Suppose λ c ă λ 1 and λ 2 ă λ 1 . Consider the asymmetric multitype process. There exists a unique critical value s c P p0, 8q such that if r ă s c , then the process survives and if r ą s c , then the process dies out.

In both cases, one has Theorem 2.2.3. The critical multitype process dies out.

The next two subsections are setting preliminaries to prove these results.

Necessary and sufficient conditions for attractiveness

We saw in Chapter 1 the stochastic order between two processes is related to the total order defined on the set of values taken by both processes, here on F " t0, 1, 2, 3u. In a biological context, setting an order between types of individuals does not make any sense, but mathematically it allows us to construct a monotone model and to compare different dynamics as well. This is the purpose of Section 2.4, using Theorem 2.2.4 below. Elements of F can be understood as species of respective types A, B, C and D. A process can be made attractive by reordering its space of values. Subsequently, denote by A the state 2, by B the state 0, by C the state 3 and by D the state 1, ordered by

A ă B " A `1 ă C " B `1 ă D " C `1.
(2.2.13)

Extending conditions obtained by T. Gobron and E. Saada [START_REF] Gobron | Couplings, attractiveness and hydrodynamics for conservative particle systems[END_REF] for conservative particle systems, D. Borrello [START_REF] Borrello | Stochastic order and attractiveness for particle systems with multiple births, deaths and jumps[END_REF] has settled necessary and sufficient conditions to non conservative dynamics to determine stochastic order between two processes. Particularly, [10, section 2.2.2] deals with multitype contact processes corresponding to our framework. We will see that this order is actually the only possible one that preserves the stochastic order.

Let x, y P Z d be two neighbouring sites and α, β P F , rewrite the transition rates of pη t q tě0 with notations of [START_REF] Borrello | Stochastic order and attractiveness for particle systems with multiple births, deaths and jumps[END_REF], for k P t1, 2u, as [START_REF] Borrello | Stochastic order and attractiveness for particle systems with multiple births, deaths and jumps[END_REF]Theorem 2.4] For all pα, βq P F 2 , pγ, δq P F 2 such that pα, βq ď pγ, δq (coordinate-wise, in the sense that α ď γ and β ď δ), h 1 ě 0, j 1 ě 0, an interacting particle systems pA t q tě0 with transition rates pR 0,k α,β , P `k β , P ´k α q is stochastically larger than an interacting particle system pB t q tě0 with transition rates p r R 0,k α,β , r P `k β , r P ´k α q if and only if

' R 0,k α,
iq ÿ kąδ´β`j 1 r Π 0,k α,β ď ÿ ląj 1 Π 0,l γ,δ and iiq ÿ kąh 1 r Π ´k,0 α,β ě ÿ ląγ´α`h 1 Π ´l,0 γ,δ (2.2.15)
One has for the asymmetric multitype process pη t q tě0 , with the order (2.2.13), the following rates.

R 0,2 D,B " λ 1 , R 0,2 C,B " λ 2 , P 1 A " P 1 C " 1, P ´1 B " P ´1 D " r, P ´2 C " P ´2 D " 1, (2.2.16)
to which, one adds the following rates if we consider the symmetric multitype process.

R 0,2 D,A " λ 1 , R 0,2 C,A " λ 2 . ( 2

Oriented percolation

In the following, we give a brief description presented by R. Durrett [START_REF] Durrett | Ten lectures on particle systems[END_REF] about oriented percolation and the comparison theorem, and their correspondence with interacting particle systems. The first application of this technique was done by M. Bramson and R. Durrett [START_REF] Bramson | A simple proof of the stability criterion of Gray and Griffeath[END_REF] for spin systems.

Construction.

Here is a description of an oriented (site) percolation process with parameter p. Consider the bi-dimensional even lattice L " tpx, nq P Z 2 : x `n is even, n ě 0u.

From L, construct an oriented graph by drawing successively an oriented bond from px, nq to px `1, n `1q and one from px, nq to px ´1, n `1q. Let tωpx, nq, px, nq P Lu be random variables taking their values in t0, 1u that indicate whether a site of L is open [START_REF] Andjel | Survival of multidimensional contact process in random environments[END_REF] or closed (0). We define their distribution in what follows.

There is an (oriented) open path from px, nq to py, mq, denoted by px, mq Ñ py, nq, if there exists a sequence of points x " x n , ..., x m " y such that px k , kq P L, |x k ´xk`1 | " 1 for n ď k ď m ´1 and ωpx k , kq " 1 for n ď k ď m. Since in our further setup, our constructions will set dependencies between the ωpx, nq's, we say that the ωpx, nq's are M -dependent with density at least 1´γ, for positive M and γ, if whenever px k , n k q 1ďkďI is a finite sequence such that }px i , n i q ´px j , n j q} 8 ą M for i ‰ j then

P pωpx i , n i q " 0 for 1 ď i ď |I|q ď γ I .
Oriented percolation is understood as a mimic of the crossing of fluids through some porous materials along a given direction, as a flow of water in a porous rock. Therefore, open sites are understood as air spaces the fluid can reach and turning them into wet sites if reached. Varying the microscopic porosity of the spaces (given by the distribution of ω), percolation processes exhibit a macroscopic phase transition from a permeable percolating regime to an impermeable non-percolating regime.

Given an initial condition W 0 Ă 2Z " tx P Z : px, 0q P Lu, we introduce the process of wet sites at time n ě 0 by W n :" ty : px, 0q Ñ py, nq for some x P W 0 u Let W 0 n be the process starting from W 0 0 " t0u and define C 0 :" tpy, nq : p0, 0q Ñ py, nqu as the set of points reached by the origin p0, 0q through an oriented open path. It is also called the connected open component or cluster from the origin. When the latter is infinite, that is, t|C 0 | " 8u, we say that percolation occurs.

A natural question is whether percolation occurs or not. The following result shows that if the density of open sites is high enough then percolation occurs with positive probability : Theorem 2.2.5 (R. Durrett [START_REF] Durrett | Lecture notes on particle systems and percolation[END_REF]). If γ ď 6 ´4p2M `1q 2 , then

P p|C 0 | ă 8q ď 1{20
Percolation processes that will arise are M -dependent but since most of the literature concerns percolation with independent random variables, next theorem tells us how a Mdependent process stochastically dominates the measure of a 0-dependent percolation. Let π p be the product measure of an independent percolation process with density p, i.e. with cylinder probabilities π p pω : ωpx, nq " 1 @px, nq P G; ωpx, nq " 0 @px, nq P Hq " p |G| p1 ´pq |H| .

where G, H are finite subsets of L. We have in our setup, Chapter 2. Phase transition on Z d Theorem 2.2.6 (Liggett,Schonmann and Stacey [59]). Let µ be a 1-dependent Bernoulli distribution. If µpωpx, nq " 1q ě 1 ´p1 ´?pq 2 a.s. for all px, nq P L with p ě 1{4, then µ ě π p .

So far, the link between an interacting particle system and a percolation process is still missing, this is the point of what follows.

Comparison theorem. The next result gives general conditions guaranteeing a process to dominate an oriented percolation.

(H) Comparison Assumptions. Let be pξ t q tě0 a translation invariant finite range process such that ξ t P F Z d , constructed from a graphical representation. Given positive integers L, T , k 0 and j 0 , define for pm, nq P L, space-time regions

R m,n " p2mLe 1 , nT q ``r´k 0 L, k 0 Ls d ˆr0, j 0 T s ˘(2.2.20)
where pe 1 , ..., e d q stands for the canonical basis in R d . Let M :" maxpk 0 , j 0 q, the regions R m,n and R m 1 ,n 1 are disjoint if }pm, nq ´pm 1 , n 1 q} 8 ą M .

Let H be collection of configurations determined by the values of ξ in r´L, Ls d . We declare pm, nq P L to be wet if ξ nT P τ 2mLe 1 H, where τ Le 1 stands for the translation by L in the direction e 1 .

Suppose, for all pm, nq P L, there exists a good event G m,n depending only on the graphical representation of the particle system in R m,n such that P pG m,n q ě 1 ´θ (θ ą 0) and so that if pm, nq is wet, then on G m,n , pm `1, n `1q and pm ´1, n `1q do as well, that is, ξ pn`1qT P τ 2pm´1qLe 1 H and ξ pn`1qT P τ 2pm`1qLe 1 H.

Let X n " tm : pm, nq P L, ξ nT P τ 2mLe 1 Hu be the set of wet sites at time t. Then, Theorem 2.2.7. [START_REF] Durrett | Ten lectures on particle systems[END_REF]Theorem 4.3] If the comparison assumptions (H) hold, then one can define random variables ωpx, nq so that for all n ě 0, X n dominates an M -dependent oriented percolation with initial configuration W 0 " X 0 and density at least 1 ´γ, that is, W n Ă X n for all n.

Graphical construction

In parallel to the analytical construction provided by the Hille-Yosida theorem 1.1.2, the multitype contact process can be constructed from a collection of independent Poisson processes [START_REF] Harris | Nearest-neighbor Markov interaction processes on multidimensional lattices[END_REF] : n ě 1u with rate 1 and for any y such that y " x, tT x,y n : n ě 1u with rate λ 1 . Let tU x n : n ě 1u be independent uniform random variables on p0, 1q, independent of the Poisson processes.

At space-time point px, A x n q, put a " " to indicate, if x is occupied by type-1 individuals (resp. empty), that it turns into state 3 (resp. state 2) which corresponds to transitions 0 Ñ 2 and 1 Ñ 3. At px, D 1,x n q (resp. at px, D 2,x n q), put an "X" (resp." ") to indicate at x, that a death of type-1 occurs corresponding to transitions 3 Ñ 2 and 1 Ñ 0 (resp. of type-2, corresponding to transitions 3 Ñ 1 and 2 Ñ 0). At times T x,y n , draw an arrow from x to y and two kinds of actions occur following the occupation at x : if x is occupied by type-1 individuals, the arrow indicates a birth in y of a type-1 individual if y is empty or in state 2, corresponding to transitions 0 Ñ 1, and 2 Ñ 3 for the symmetric case ; if x is occupied by type-3 individuals giving birth at rate λ 2 ă λ 1 , check at px, T x,y n q if U x n ă λ 2 {λ 1 to indicate, if success, that the arrow is effective so that a birth in y of a type-1 individual occurs if y is empty, or in state 2 for the symmetric case. In the asymmetric case, births occur only if y is not in state 2.

See Figure 2.1 for an example of the time-evolution of both processes starting from an identical initial configuration.

For s ď t, there exists an active path from px, sq to py, tq in Z d ˆR`i s there exists a sequence of times s " s 0 ă s 1 ă ... ă s n´1 ă s n " t and a sequence of corresponding spatial locations x " x 0 , x 1 , ..., x n " y such that : i. for i " 1, ..., n ´1, vertical segments tx i u ˆps i , s i`1 q do not contain any X's. ii. for i " 1, ..., n, there is an arrow from x i´1 to x i at times s i and if x i´1 ˆsi is lastly preceded by a " " this arrow exists only if U x i´1 s i ă λ 2 {λ 1 . and in the asymmetric case, substitute ii. by ii'. for i " 1, ..., n, there is an arrow from x i´1 to x i at times s i while tx i u ˆsi is not lastly preceded by a " ", while if x i´1 ˆsi is lastly preceded by a " " this arrow is effective if

U x i´1 s i ă λ 2 {λ 1 .
Consider the process pA A t q tě0 , the set of sites at time t reached by active paths starting from an initial configuration A 0 " A, containing sites in state 1 in A and 0 otherwise :

A A t " ty P Z : Dx P A such that px, 0q Ñ py, tqu Then A A t " H A t , with H A t defined in (2.2.9) so that A A t represents the wild population at time t starting from an initial configuration A of type-1 individuals.

From the graphical representation, the particle system pA A t q tě0 is additive [32, Chapter II] : for any initial configuration B such that A Ă B, then

A A t Ă A B t . On Figure 2.1, A t0u t
" t´1, 0, 1u for the asymmetric case. This graphical representation allows us to couple multitype contact processes starting from different initial Starting from η 0 " 1 t0u , following the arrows, if U 0 1 ă λ 2 λ 1 and U 0 2 ă λ 2 λ 1 , the wild population occupies at time t the set H t " t´1, 0, 1u in the asymmetric case and the set H t " t´2, ´1, 0, 1u in the symmetric case.

Attractiveness and stochastic order

configurations by imposing the evolution to obey to the same Poisson processes. Other kinds of couplings would be possible through the analytical construction of the process as we will see later. By way of illustration, A t1u t " H and A t0,1u t " t´1, 0, 1u in the asymmetric case, A t1u t " t2u and A t0,1u t " t´2, ´1, 0, 1, 2u in the symmetric case. More generally, graphical representations allow to couple processes with different dynamics as well, we investigate this question furthermore thereafter.

Attractiveness and stochastic order

Recall pη t q tě0 denotes the multitype contact process with parameters (λ 1 , λ 2 , r) and pξ t q tě0 denotes the basic contact process with growth rate λ 1 . Most of the proofs below rely on the construction of a markovian coupled process.

We defined a partial order on F Z d between two configurations η p1q and η p2q by (1.1.3) and (2.2.13). Here we shall settle necessary and sufficient conditions, then only sufficient, to obtain several properties of stochastic order with which we will work. We begin with the symmetric multitype contact process since it contains the transitions of the asymmetric one and of the basic contact process. Proposition 2.4.1. The symmetric multitype process is monotone, in the sense that, one can construct on a same probability space two symmetric multitype processes pη p1q t q tě0 and pη p2q t q tě0 with respective parameters pλ if and only if all parameters satisfy

1. λ p1q 2 ď λ p1q 1 , 2. λ p2q 2 ď λ p2q 1 , 3. λ p1q 1 ď λ p2q 1 , 4. λ p1q 2 ď λ p2q 2 , 5. r p1q ě r p2q 6. λ p1q 1 ď 1, 7. λ p1q 2 ď 1, 8. r p1q ě 1.
Remark conditions 1. and 2. are the assumptions made from the construction of the process, see (2.2.1).

Proof of Proposition 2.4.1. Let pη p1q t q tě0 and pη p2q t q tě0 be two symmetric processes with parameters pλ p1q 1 , λ p1q 2 , r p1q q and pλ p2q 1 , λ p2q 2 , r p2q q respectively. Apply Theorem 2.2.4 with j 1 , h 1 P t0, 1u (one can check they are the only non trivial possible values). Necessary and sufficient conditions on the rates for pη p2q t q tě0 to be stochastically larger than pη p1q t q tě0 are given by relations (2.2.15) with pα, βq ď pγ, δq, that is,

ÿ kąδ´β`j 1 Π 0,k,p1q α,β,p1q ď ÿ ląj 1 Π 0,l,
`1tα " CuP ´2,p1q C ě 1th 1 " 0u1tl " 2u1tγ ´α " 1u ´1tγ " D, α " CuP ´2,p2q D `1tγ " C, α " BuP ´2,p2q C 1th 1 " 0u1tl " 2u1tγ ´α " 0u ´1tγ " α " DuP ´2,p2q D `1tγ " α " CuP ´2,p2q C 1th 1 " 0u1tl " 1u1tγ ´α " 0u ´1tγ " α " DuP ´1,p2q D `1tγ " α " CuP ´1,p2q C 1th 1 " 1u1tl " 2u1tγ ´α " 0u ´1tγ " α " DuP ´2,p2q D `1tα " γ " CuP ´2,p2q C ¯. (2.4.3)
These inequalities can also be explicitly rewritten as

1tj 1 " 0u1tk " 2u `1tβ " A, δ " Bu `1tβ " B, δ " Cu ˘`λ p1q 1 1tα " Du `λp1q 2 1tα " Cu 1tj 1 " 0u1tk " 2u `1tβ " δ " Bu `1tβ " δ " Au ˘`λ p1q 1 1tα " Du `λp1q 2 1tα " Cu 1tj 1 " 0u1tk " 1u ´1tβ " δ " Au `1tβ " δ " Cu 1tj 1 " 1u1tk " 2u `1tβ " δ " Bu `1tβ " δ " Au ˘`λ p1q 1 1tα " Du `λp1q 2 1tα " Cu ď 1tj 1 " 0u ´1tl " 2u `1tδ " Bu `1tδ " Au ˘`λ p2q 1 1tγ " Du `λp2q 2 1tγ " Cu ˘1tj 1 " 0u1tl " 1u `1tδ " Au `1tδ " Cu ˘1tj 1 " 1u1tl " 2u `1tδ " Bu `1tδ " Au ˘`λ p2q 1 1tγ " Du `λp2q 2 1tγ " Cu ˘(2.4.4) 32
2.4. Attractiveness and stochastic order and 1th 1 " 0u `1tk " 1ur p1q p1tα " Bu `1tα " Du 1th (IV) h 1 " 0,α " B, γ " 1 `α " C in (2.4.5) give r p1q ě 1 stated by condition 8. while in other scenarios, one retrieves redundantly the above conditions or tautological inequalities such as "1 ě 0". Finally, one obtained the necessary conditions stated from 1. to 8. Now, we construct a coupled process pη p1q t , η p2q t q tě0 on Ω ˆΩ such that η p1q 0 ď η p2q 0 . According to the given order (2.2.13) on F , as η p1q 0 ď η p2q 0 :

1 " 0u1tk " 2u `1tα " Cu `1tα " Du 1th 1 " 1u1tk " 2up1tα " Cu `1tα " Duq ě 1th 1 " 0u1tl " 2u1tγ " 1 `αu ´1tγ " Cu `1tγ " Du 1th 1 " 0u1tl " 2u1tγ " αu `1tγ " Cu `1tγ " Du 1th 1 " 0u1tl " 1u1tγ " αur p2q p1tγ " Bu `1tγ " Duq `1th 1 " 1u1tγ " αu1tl " 2up1tγ " Cu `1tγ " Duq.
n 1 px, η p1q 0 q `n3 px, η p1q 0 q ď n 1 px, η p2q 0 q `n3 px, η p2q 0 q.
We saw that it is possible to construct the coupled process either through generators or through a graphical representation, via a collection of independent Poisson processes whose rates are given by the parameters of the processes. The coupling of two processes on a graphical construction is provided by coupling the Poisson processes related to births and releases.

In what follows, we construct the coupling through generators. The three following tables depict the infinitesimal transitions of the coupled process. 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q λ p2q 1 n 1 px, η p2q q ´λp1q 1 n 1 px, η p1q q `λp2q 2 n 3 px, η p2q q ´λp1q 2 n 3 px, η p1q q r p2q r p1q ´rp2q p1, 1q ÝÑ

$ & % p0, 0q p3, 3q p3, 1q 1 r p2q r p1q ´rp2q p2, 2q ÝÑ $ & % p3, 3q p2, 3q p0, 0q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q λ p2q 1 n 1 px, η p2q q ´λp1q 1 n 1 px, η p1q q `λp2q 2 n 3 px, η p2q q ´λp1q 2 n 3 px, η p1q q 1 p3, 3q ÝÑ " p1, 1q p2, 2q 1 1 p2, 0q ÝÑ $ ' ' & ' ' % p3, 1q p2, 1q p0, 0q p2, 2q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q λ p2q 1 n 1 px, η p2q q ´λp1q 1 n 1 px, η p1q q `λp2q 2 n 3 px, η p2q q ´λp1q 2 n 3 px, η p1q q 1 r p2q p2, 3q ÝÑ $ & % p3, 3q p0, 1q p2, 2q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q 1 1 p2, 1q ÝÑ $ ' ' & ' ' % p3, 1q p2, 0q p0, 1q p2, 3q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q 1 1 r p2q p3, 1q ÝÑ $ & % p2, 0q p1, 1q p3, 3q 1 1 r p2q p0, 1q ÝÑ $ ' ' & ' ' % p2, 3q p1, 1q p2, 1q p0, 0q r p2q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q r p1q ´rp2q 1 Table 2.1 transition rate p0, 3q ÝÑ $ ' ' & ' ' % p1, 1q p0, 1q p2, 2q p2, 3q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q p1 ´λp1q 1 qn 1 px, η p1q q `p1 ´λp1q 2 qn 3 px, η p1q q 1 r p1q ´1
λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q λ p2q 1 n 1 px, η p2q q ´λp1q 1 n 1 px, η p1q q `λp2q 2 n 3 px, η p2q q ´λp1q 2 n 3 px, η p1q q 1 r p1q p1, 2q ÝÑ $ ' ' & ' ' % p1, 3q p3, 2q p1, 0q p0, 2q λ p2q 1 n 1 px, η p2q q `λp2q 2 n 3 px, η p2q q r p1q 1 1 p1, 3q ÝÑ $ & % p0, 2q p3, 3q p1, 1q 1 r p1q 1 p3, 0q ÝÑ $ ' ' & ' ' % p1, 0q p2, 0q p3, 1q p3, 2q 1 1 λ p2q 1 n 1 px, η p2q q `λp2q 2 n 3 px, η p2q q r p2q p3, 2q ÝÑ $ & % p3, 3q p1, 0q p2, 2q λ p2q 1 n 1 px, η p2q q `λp2q 2 n 3 px, η p2q q 1 1 Table 2.3
To verify all the rates above are well defined, one decomposes n 1 px, η piq q and n 3 px, η piq q, pi " 1, 2q, as follows n 1 px, η p2q q " |ty " x : η p2q pyq " η p1q pyq " 1u| `|ty " x : η p2q pyq " 1, η p1q pyq " 3u| `|ty " x : η p2q pyq " 1, η p1q pyq P t0, 2uu|, n 3 px, η p2q q " |ty " x : η p2q pyq " η p1q pyq " 3u| `|ty " x : η p2q pyq " 3, η p1q pyq P t0, 2uu|, n 1 px, η p1q q " |ty " x : η p2q pyq " η p1q pyq " 1u| n 3 px, η p1q q " |ty " x : η p2q pyq " η p1q pyq " 3u| `|ty " x : η p2q pyq " 1, η p1q pyq " 3u|, in which case, we decompose the rate

λ p2q 1 n 1 px, η p2q q ´λp1q 1 n 1 px, η p1q q `λp2q 2 n 3 px, η p2q q ´λp1q 2 n 3 px, η p1q q 35
Chapter 2. Phase transition on Z d " pλ p2q 1 ´λp1q 1 q|ty " x : η p2q pyq " η p1q pyq " 1u| `pλ p2q 1 ´λp1q

2 q|ty " x : η p2q pyq " 1, η p1q pyq " 3u| `pλ p2q 2 ´λp1q

2 q|ty " x : η p2q pyq " η p1q pyq " 3u| `λp2q

1 |ty " x : η p2q pyq " 1, η p1q pyq P t0, 2uu| `λp2q

2 |ty " x : η p2q pyq " 3, η p1q pyq P t0, 2uu|

(2.4.6)

which is non-negative under conditions 1. to 4. coming from (I) and (III) in inequalities (2.4.4)-(2.4.5).

Rates of Table 2.2 are non-negative thanks to conditions 6. to 8., given by inequalities (II)-(i)(ii) with β " B, δ " C. Condition 5. is used by Tables 2.1 and 2.3 that correspond to a basic coupling while Table 2.2 uses a different coupling. Table 2.3 is listing transitions of the coupled process starting from configurations that do not preserve the defined partial order, nevertheless, starting from an initial configuration where it does, dynamics of the coupling given by Tables 2.1 and 2.2 do not reach states of Table 2.3.

For a coupled process pη p1q t , η p2q t q tě0 starting from an initial configuration such that η p1q 0 ď η p2q 0 , since transitions of the two first Tables preserve the order on F , the markovian coupling we just constructed is increasing :

r P pη p1q 0 ,η p2q 0 q pη p1q t ď η p2q t q " 1 for all t ą 0 (2.4.7)
where r P pη p1q 0 ,η p2q 0 q stands for the distribution of pη p1q t , η p2q t q tě0 starting from the initial configuration pη p1q 0 , η p2q 0 q. We can wonder if there exists an other order than (2.2.13) for which this statement (and the following ones as well) holds. By trying out other orders in inequalities (2.2.15) of Theorem 2.2.4 , we deduce that the one defined by (2.2.13) is the only order possible here to preserve the stochastic order.

After having obtained necessary and sufficient conditions, we investigate sufficient conditions only, with which we shall work subsequently. Proposition 2.4.2. The symmetric process pη t q tě0 is monotone, in the sense that, one can construct on a same probability space two symmetric processes pη p1q t q tě0 and pη p2q t q tě0 with respective parameters pλ

p1q 1 , λ p1q 2 , r p1q q andpλ p2q 1 , λ p2q 2 , r p2q q satisfying η p1q 0 , η p2q 0 P t0, 1u Z d , such that η p1q 0 ď η p2q 0 ùñ η p1q t ď η p2q t for all t ě 0 a.s. (2.4.8)
if all parameters satisfy 2.4. Attractiveness and stochastic order

1. λ p1q 2 ď λ p1q 1 , 2. λ p2q 2 ď λ p2q 1 , 3. λ p1q 1 ď λ p2q 1 , 4. λ p1q 2 ď λ p2q 2 ,

r p1q ě r p2q

Proof. Given our initial conditions, possible states for the coupled process keep laying in Table 2.1 of Proposition 2.4.1 and the coupled process does not reach any state of Tables 2.2 and 2.3. One can therefore omit conditions 4. to 6. of the previous Proposition 2. 4.1 and transition rates from the couple p0, 3q can be defined through a basic coupling even if it does not preserve the order : Since Tables 2.1, 2.4 and 2.3 correspond to a basic coupling, to construct such processes on a same probability space via the graphical representation, one define from Section 1.2 mutually independent Poisson processes : tT x,y n , n ě 1u with rate λ p2q 1 , tD 1,x n , n ě 1u with rate 1, tD (iv) Decrease w.r.t. r : if pη p1q t q tě0 and pη p2q t q tě0 have respective parameters pλ 1 , λ 2 , r p1q q and pλ 1 , λ 2 , r p2q q such that r p1q ě r p2q with λ 2 ă λ 1 , then

transition rate p0, 3q ÝÑ $ ' ' & ' ' % p1, 3q p0, 1q p0, 2q p2, 3q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q 1 1 r p1q
η p1q 0 ď η p2q 0 ùñ η p1q t ď η p2q t a.s., for all t ě 0.
A consequence related to Corollary (2.4.1)-(iv) is the non-increase of the survival probability with respect to the release rate r for fixed λ 1 , λ 2 : Corollary 2.4.2. Suppose λ 2 and λ 1 fixed. If pη t q tě0 has initial configuration η 0 " 1 t0u , the mapping r Þ ÝÑ P r p@t ě 0, H t ‰ Hq is a non-increasing function.

Proof. Indeed if pη p1q t q tě0 and pη p2q t q tě0 are two processes such that η p1q 0 , η p2q 0 P t0, 1u Z d and with respective parameters pλ 1 , λ 2 , r p1q q and pλ 1 , λ 2 , r p2q q such that r p1q ď r p2q , then according to Corollary 2.4.1,

H p2q 0 Ă H p1q 0 ùñ H p2q t Ă H p1q t ,
for all t ě 0.

One defined the asymmetric process as a particular case of the symmetric process where the transition from state 2 to state 3 does not occur. One can thus, in a similar way to Propositions 2.4.1 and 2.4.2, obtain necessary and sufficient conditions, then, only sufficient conditions, for the monotonicity of the asymmetric process.

Proposition 2.4.3. The asymmetric process pη t q tě0 is monotone in the sense that, conditions

1. λ p1q 2 ď λ p1q 1 , 2. λ p2q 2 ď λ p2q 1 , 3. λ p1q 1 ď λ p2q 1 , 4. λ p1q 2 ď λ p2q 2 ,
5. r p1q ě r p2q , are sufficient to construct on a same probability space two asymmetric process pη p1q t q tě0 and pη p2q t q tě0 with respective parameters pλ p1q 1 , λ p1q 2 , r p1q q and pλ p2q 1 , λ p2q 2 , r p2q q and with initial condition η p1q 0 , η

p2q 0 P t0, 1u Z d , such that η p1q 0 ď η p2q 0 ùñ η p1q t ď η p2q t
a.s., for all t ě 0.

(2.4.9)

Attractiveness and stochastic order

Proof. As in the proof of Proposition 2.4.2, one applies Theorem 2.2.4 with j 1 , h 1 P t0, 1u to two asymmetric processes pη p1q t q tě0 and pη p2q t q tě0 with respective parameters pλ p1q 1 , λ p1q 2 , r p1q q and pλ p2q 1 , λ p2q 2 , r p2q q. Using relations (2.4.2)-(2.4.3) with the corresponding rates of both processes given by (2.2.16), with pα, βq ď pγ, δq, one has

1tj 1 " 0u ˜1tk " 2u ´1tβ " δ " Bu `λp1q 1 1tα " Du `λp1q 2 1tα " Cu ˘1tk " 1u ´1tβ " δ " Au `1tβ " δ " Cu ¯1tj 1 " 0u1tk " 2u ´1tβ " B, δ " Cu `λp1q 1 1tα " Du `λp1q 2 1tα " Cu ˘1tj 1 " 1u1tk " 2u ´1tβ " δ " Bu `λp1q 1 1tα " Du `λp1q 2 1tα " Cu ˘ď 1tj 1 " 0u ´1tl " 2u1tδ " Bu `λp2q 1 1tγ " Du `λp2q 2 1tγ " Cu 1tl " 1u `1tδ " Au `1tδ " Cu ˘1tj 1 " 1u ´1tl " 2u1tδ " Bu `λp2q 1 1tγ " Du `λp2q 2 1tγ " Cu ˘(2.4.10)
while the second relation leaves (2.4.5) unchanged. One deduces the following necessary conditions : (I)

j 1 P t0, 1u, δ " β " B in (2.4.10) give (i) α " γ " C, β " B, δ " C : λ p1q 2 ď λ
(II) j 1 " 0, β " B, δ " 1 `β " C in (2.4.10) give (i) α " D : λ p1q 1 ď 1. (ii) α " C : λ p1q 2 ď 1. The relation (2.4.5) staying unchanged, one has (III) h 1 " 0, γ " α P tB, Du in (2.4.5) give r p1q ě r p2q stated by condition 5. (IV) h 1 " 0,α " B, γ " 1 `α " C in (2.4.5) give r p1q ě 1. The obtained necessary conditions are 1. λ p1q 2 ď λ p1q 1 , 2. λ p2q 2 ď λ p2q 1 , 3. λ p1q 1 ď λ p2q 1 , 4. λ p1q 2 ď λ p2q 2 , 5. r p1q ě r p2q 6. λ p1q 1 ď 1, 7. λ p1q 2 ď 1, 8. r p1q ě 1.
As for Proposition 2.4.1, these conditions allow us to construct an increasing markovian coupling. As in Proposition 2.4.2, given our initial configurations, state p0, 3q is not possible for the coupled process. One can thus dispense conditions 6 to 8. and sufficient conditions to settle an increasing markovian coupling as in Proposition 2.4.2 are given by Chapter 2. Phase transition on

Z d 1. λ p1q 2 ď λ p1q 1 , 2. λ p2q 2 ď λ p2q 1 , 3. λ p1q 1 ď λ p2q 1 , 4. λ p1q 2 ď λ p2q 2 , 5. r p1q ě r p2q
Details of the dynamics of the coupled process pη p1q t , η p2q t q tě0 are summarized in the following tables. P t0, 1u Z d , this markovian coupling is increasing since the transitions of the coupled process belong to Table 2.5 and r P pη p1q 0 ,η p2q 0 q pη p1q t ď η p2q t q " 1 for all t ą 0 (2.4.11)

transition rate p0, 0q ÝÑ $ ' ' & ' ' % p1, 1q p0, 1q p2, 2q p2, 0q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q λ p2q 1 n 1 px, η p2q q ´λp1q 1 n 1 px, η p1q q `λp2q 2 n 3 px, η p2q q ´λp1q 2 n 3 px, η p1q q r p2q r p1q ´rp2q p1, 1q ÝÑ $ & % p0, 0q p3, 3q p3, 1q 1 r p2q r p1q ´rp2q p2, 2q ÝÑ p0, 0q 1
p3, 3q ÝÑ " p1, 1q p2, 2q 1 1 p2, 0q ÝÑ $ & % p2, 1q p0, 0q p2, 2q λ p2q 1 n 1 px, η p2q q `λp2q 2 n 3 px, η p2q q 1 r p2q p2, 3q ÝÑ " p0, 1q p2, 2q 1 1 p2, 1q ÝÑ $ & % p2, 0q p0, 1q p2, 3q 1 1 r p2q p3, 1q ÝÑ $ & % p2, 0q p1, 1q p3, 3q 1 1 r p2q p0, 1q ÝÑ $ ' ' & ' ' % p2, 3q p1, 1q p2, 1q p0, 0q r p2q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q r p1q ´rp2q 1 Table 2.5 transition rate p0, 3q ÝÑ $ ' ' & ' ' % p1, 3q p0, 1q p0, 2q p2, 3q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q 1 1 r p1q
' ' & ' ' % p1, 1q p3, 2q p3, 0q p0, 0q λ p2q 1 n 1 px, η p2q q `λp2q 2 n 3 px, η p2q q r p2q r p1q ´rp2q 1 p0, 2q ÝÑ $ & % p1, 2q p0, 0q p2, 2q λ p1q 1 n 1 px, η p1q q `λp1q 2 n 3 px, η p1q q 1 r p1q p1, 2q ÝÑ $ & % p3, 2q p1, 0q p0, 2q r p1q 1 1 p1, 3q ÝÑ $ & % p0, 2q p3, 3q p1, 1q 1 r p1q 1 p3, 0q ÝÑ $ ' ' & ' ' % p1, 0q p2, 0q p3, 1q p3, 2q 1 1 λ p2q 1 n 1 px, η p2q q `λp2q 2 n 3 px, η p2q q r p2q p3, 2q ÝÑ " p1, 0q p2, 2q 1
where r P pη p1q 0 ,η p2q 0 q denotes the distribution of pη p1q t , η p2q t q tě0 starting from the initial configuration pη p1q 0 , η p2q 0 q. One can compare the symmetric with the asymmetric process as well. Proposition 2.4.4. Let pη t q tě0 be an asymmetric process and pχ t q tě0 be a symmetric process, both with parameters pλ 1 , λ 2 , rq and η 0 , χ 0 P t0, 1u Z d such that λ 2 ă λ 1 , then for all t ě 0, η 0 ď χ 0 ñ η t ď χ t a.s. for all t ě 0 Proof. Apply Theorem 2.2.4 with an asymmetric process pη t q tě0 and a symmetric process pχ t q tě0 with parameters pλ 1 , λ 2 , rq. Necessary and sufficient conditions on the parameters to obtain a stochastic order are given by (2.4.2)-(2.4.3) that become

1tj 1 " 0u1tk " 2u1tβ " δ " Bu `λ1 1tα " Du `λ2 1tα " Cu 1tj 1 " 0u1tk " 2u1tβ " B, δ " Cu `λ1 1tα " Du `λ2 1tα " Cu 1tj 1 " 0u1tk " 1u ´1tβ " δ " Au `1tβ " δ " Cu 1tj 1 " 1u1tk " 2u1tβ " δ " Bu `λ1 1tα " Du `λ2 1tα " Cu ˘41 Chapter 2. Phase transition on Z d ď 1tj 1 " 0u ´1tl " 2u `1tδ " Bu `1tδ " Au ˘`λ 1 1tγ " Du `λ2 1tγ " Cu 1tl " 1u `1tδ " Au `1tδ " Cu ˘1tj 1 " 1u1tl " 2u `1tδ " Bu `1tδ " Au ˘`λ 1 1tγ " Du `λ2 1tγ " Cu ˘(2.4.12)
and

1th 1 " 0u ´1tk " 1urp1tα " Bu `1tα " Duq `1tk " 2up1tα " Cu `1tα " Duq ¯`1th 1 " 1u ´1tk " 2up1tα " Cu `1tα " Duq ě 1th 1 " 0u1tγ " αu ´1tl " 1urp1tγ " Bu `1tγ " Duq `1tl " 2up1tγ " Cu `1tγ " Duq 1th 1 " 0u1tγ " 1 `αu ´1tl " 2up1tγ " Cu `1tγ " Duq 1th 1 " 1u1tγ " αu ´1tl " 2up1tγ " Cu `1tγ " Duq ¯(2.4.13)
These equations exhibit the following necessary conditions :

(I) j 1 P t0, 1u, δ " β " B,α " C, γ " D in (2.4.12) give : λ 2 ď λ 1 (II) h 1 " 0, α " B, γ " 1 `α " C in (2.4.13) give r ě 1
As previously, condition r ě 1 is necessary to construct an increasing markovian coupled process in a general framework, but if one restricts the initial conditions to satisfy η 0 ď χ 0 and η 0 , χ 0 P t0, 1u Z d , condition λ 2 ď λ 1 is sufficient and the coupled process can be constructed through the following transitions :

Attractiveness and stochastic order

transition rate

p0, 0q ÝÑ $ & % p1, 1q p0, 1q p2, 2q λ 1 n 1 px, ηq `λ2 n 3 px, ηq λ 1 pn 1 px, χq ´n1 px, ηqq `λ2 n 3 px, χq ´n3 px, ηqq r p1, 1q ÝÑ " p0, 0q p3, 3q 1 r p2, 2q ÝÑ " p0, 0q p2, 3q 1 λ 1 n 1 px, χq `λ2 n 3 px, χq p3, 3q ÝÑ " p1, 1q p2, 2q 1 1 p2, 0q ÝÑ $ & % p2, 1q p0, 0q p2, 2q λ 1 n 1 px, χq `λ2 n 3 px, χq 1 r p2, 3q ÝÑ " p0, 1q p2, 2q 1 1 p2, 1q ÝÑ $ & % p2, 0q p0, 1q p2, 3q 1 1 r p3, 1q ÝÑ $ & % p2, 0q p1, 1q p3, 3q 1 1 r p0, 1q ÝÑ $ & % p2, 3q p1, 1q p0, 0q r λ 1 n 1 px, ηq `λ2 n 3 px, ηq 1 Table 2.8 transition rate p0, 3q ÝÑ $ ' ' & ' ' % p2, 3q p1, 3q p0, 1q p0, 2q r λ 1 n 1 px, ηq `λ2 n 3 px, ηq 1 1 Table 2.9
Chapter 2. Phase transition on Z d transition rate

p1, 0q ÝÑ $ & % p1, 1q p3, 2q p0, 0q λ 1 n 1 px, χq `λ2 n 3 px, χq r 1 p0, 2q ÝÑ $ ' ' & ' ' % p1, 3q p0, 3q p0, 0q p2, 2q λ 1 n 1 px, ηq `λ2 n 3 px, ηq λ 1 n 1 px, χq `λ2 n 3 px, χq 1 r p1, 2q ÝÑ $ & % p3, 2q p0, 0q p1, 3q r 1 λ 1 n 1 px, χq `λ2 n 3 px, χq p1, 3q ÝÑ " p0, 2q p3, 3q 1 r p3, 0q ÝÑ $ ' ' & ' ' % p1, 0q p2, 0q p3, 1q p3, 2q 1 1 λ 1 n 1 px, χq `λ2 n 3 px, χq r Table 2.10
As in (2.4.6), the second rate in Table 2.8 is positive. Starting from an initial configuration such that η 0 ď χ 0 and η 0 , χ 0 P t0, 1u Z d , the coupled process does not reach any configuration of Tables 2.9 and 2.10. Condition λ 2 ď λ 1 is sufficient to obtain η t ď χ t a.s., for all t ě 0.

Finally, one settles two comparisons between a basic contact process and a multitype process.

Proposition 2.4.5. Let pξ t q tě0 be a basic contact process on t0, 1u Z d with growth rate λ 1 and let pχ t q tě0 be a symmetric multitype process with parameters pλ 1 , λ 2 , rq such that λ 2 ă λ 1 . Then, χ 0 ď ξ 0 ñ χ t ď ξ t a.s. for all t ě 0 Proof. Consider the basic contact process pξ t q tě0 viewed as a symmetric multitype process with parameters pλ

p2q 1 , λ p2q 2 , r p2q q with λ p2q 1 " λ 1 , λ p2q 2 " 0, r p2q " 0. Values A and C do not exist for the process ξ t , retrieving the proof of Proposition 2.4.1, relations (2.4.2)-(2.4.3) become 1tj 1 " 0u1tk " 2u1tβ " δ " Bu `λ1 1tα " Du `λ2 1tα " Cu ˘1tj 1 " 0u1tk " 2u ´1tβ " A, δ " Bu `λ1 1tα " Du `λ2 1tα " Cu 1tj 1 " 1u1tk " 2u1tβ " δ " Bu `λ1 1tα " Du `λ2 1tα " Cu ď `1tj 1 " 0u `1tj 1 " 1u ˘1tl " 2u1tδ " Buλ 1 1tγ " Du (2.4.14) 44
2.4. Attractiveness and stochastic order and

1th 1 " 0u ´1tk " 1urp1tα " Bu `1tα " Duq `1tk " 2up1tα " Cu `1tα " Duq ¯`1th 1 " 1u1tk " 2up1tα " Cu `1tα " Duq ě 1th 1 " 0u1tγ " αu1tl " 2u1tγ " Du `1th 1 " 0u1tγ " 1 `αu1tl " 2u1tγ " Du `1th 1 " 1u1tγ " αu1tl " 2u1tγ " Du (2.4.15)
that exhibit the following necessary condition : Therefore, under condition condition λ 2 ď λ 1 λ 1 pn 1 px, ξq ´n1 px, χqq ´λ2 n 3 px, χq " pλ 1 ´λ2 qn 3 px, χq `λ1 |ty " x : χpyq P t0, 2u, ξpyq " 1u| is non-negative, and

j 1 P t0, 1u, β " δ " B, α " C, γ " D in (2.4.
p0, 0q ÝÑ $ & % p2, 0q p1, 1q p0, 1q r λ 1 n 1 px, χq `λ2 n 3 px, χq λ 1 pn 1 px, ξq ´n1 px, χqq ´λ2 n 3 px, χq p1, 1q ÝÑ " p0, 0q p3, 1q 1 r p2, 0q ÝÑ $ & % p3, 1q p2, 1q p0, 0q λ 1 n 1 px, χq `λ2 n 3 px, χq λ 1 pn 1 px, ξq ´n1 px, χqq ´λ2 n 3 px, χq 1 p2, 1q ÝÑ $ & % p2, 0q p0, 1q p3, 1q 1 1 λ 1 n 1 px, χq `λ2 n 3 px, χq p3, 1q ÝÑ " p2, 0q p1, 1q 1 1 p0, 1q ÝÑ $ & % p1, 1q p2, 1q p0, 0q λ 1 n 1 px, χq `λ2 n 3 px, χq r 1 p1, 0q ÝÑ $ & % p1, 1q p3, 0q p0, 0q λ 1 n 1 px, χq `λ2 n 3 px, χq r 1 p3, 0q ÝÑ $ & % p2, 0q p1, 0q p3, 1q 1 1 λ 1 n 1 px, ξq
χ 0 ď ξ 0 ñ χ t ď ξ t a.s.,
for all t ě 0.

For next proposition, p r ξ t q tě0 is a basic contact process on t2, 3u Z d whose dynamics is given by the following transitions in

x P Z d 2 Ñ 3 at rate λ 2 n 3 px, r ξq, 3 Ñ 2 at rate 1 (2.4.19)
Proposition 2.4.6. Let pη t q tě0 be a symmetric multitype process with parameters pλ 1 , λ 2 , rq such that λ 2 ď λ 1 . Then

r ξ 0 ď η 0 ñ r ξ t ď η t a.s.
, for all t ě 0.

Proof. Use once again Theorem 2.2.4 to obtain necessary and sufficient conditions for a stochastic order. For the process p r ξ t q tě0 , values B and D are not reached and rates are given by ( 2

`1tj 1 " 0u `1tj 1 " 1u ˘1tk " 2u1tβ " δ " Auλ 2 1tα " Cu `1tj 1 " 0u1tk " 2u1tδ " B, β " Auλ 2 1tα " Cu ď 1tj 1 " 0u ´1tl " 2u `1tδ " Bu `1tδ " Au ˘`λ 1 1tγ " Du `λ2 1tγ " Cu 1tl " 1u1tδ " Au 1tj 1 " 1u1tl " 2u `1tδ " Bu `1tδ " Au ˘`λ 1 1tγ " Du `λ2 1tγ " Cu ˘(2.4.20)
and

1th 1 " 0u1tk " 2u1tα " Cu `1th 1 " 1u1tk " 2u1tα " Cu ě 1th 1 " 0u1tγ " αu1tl " 2u1tγ " Cu `1th 1 " 0u1tγ " 1 `αu1tl " 2u1tγ " Du `1th 1 " 1u1tγ " αu1tl " 2u1tγ " Cu (2.4.21)
exhibiting the following conditions :

j 1 P t0, 1u, β " δ " A, α " C, γ " D in (2.4.20)
give λ 2 ď λ 1 . Inequality (2.4.21) gives no condition on the rates and condition λ 2 ď λ 1 is sufficient to construct the coupled process p r ξ t , η t q tě0 via the following dynamics :

2.5. Phase transition transition rate p3, 1q ÝÑ " p2, 0q p3, 3q 1 r p2, 0q ÝÑ $ & % p2, 2q p3, 1q p2, 1q r λ 2 n 3 px, r ξq λ 1 n 1 px, ηq `λ2 pn 3 px, ηq ´n3 px, r ξqq p2, 2q ÝÑ $ & % p3, 3q p2, 3q p2, 0q λ 2 n 3 px, r ξq λ 1 n 1 px, ηq `λ2 pn 3 px, ηq ´n3 px, r ξqq 1 p3, 3q ÝÑ " p2, 2q p3, 1q 1 1 p2, 3q ÝÑ $ & % p2, 2q p2, 1q p3, 3q 1 1 λ 2 n 3 px, r ξq p2, 1q ÝÑ $ & % p3, 1q p2, 3q p2, 0q λ 2 n 3 px, r ξq r 1 p3, 2q ÝÑ " p2, 0q p3, 3q 1 λ 1 n 1 px, ηq `λ2 n 3 px, ηq p3, 0q ÝÑ $ & % p3, 1q p3, 2q p2, 0q λ 1 n 1 px, ηq `λ2 n 3 px, ηq r 1
whose rate λ 1 n 1 px, ηq `λ2 pn 3 px, ηq ´n3 px, r ξqq is well defined : since n 3 px, r ξq " |ty " x : r ξpyq " 3, ηpyq " 1u| `|ty " x : r ξpyq " 3, ηpyq " 3u| n 1 px, ηq " |ty " x : r ξpyq " 3, ηpyq " 1u| `|ty " x : r ξpyq " 2, ηpyq " 1u| n 3 px, ηq " |ty " x : r ξpyq " 3, ηpyq " 3u| `|ty " x : r ξpyq " 2, ηpyq " 3u| along with λ 2 ď λ 1 , gives

λ 1 n 1 px, ηq `λ2 pn 3 px, ηq ´n3 px, r
ξqq " pλ 1 ´λ2 q|ty " x : r ξpyq " 3, ηpyq " 1u| `λ2 p|ty " x : r ξpyq " 2, ηpyq " 3u| `λ1 |ty " x : r ξpyq " 2, ηpyq " 1u|q

Phase transition

In this section, we take advantage of all the stochastic order relations between processes established in Section 2.4 to derive results for a phase transition of the multitype process pη t q tě0 , in both symmetric and asymmetric cases. According to Definition Chapter 2. Phase transition on Z d 2.2.1, we assume the multitype process to have initial configuration η 0 " 1 t0u and note

η t " η t0u t .
As announced in Section 2.2, we first deal with the cases where λ 2 ă λ 1 are both smaller or larger than λ c .

Proof of Proposition 2.2.1. Let pξ t q tě0 be a basic contact process with growth rate λ 1 and let pη t q tě0 be a symmetric multitype process with parameters pλ 1 , λ 2 , rq such that η 0 ď ξ 0 . By Proposition 2.4.5, pξ t q tě0 is stochastically larger than pη t q tě0 . Since λ 1 ď λ c , pξ t q tě0 is subcritical, thus, the symmetric multitype process dies out.

The extinction of the asymmetric multitype process is a consequence of the extinction of the symmetric process and Proposition 2.4.4.

Proof of Proposition 2.2.2. Let p r

ξ t q tě0 be a basic contact process with growth rate λ 2 on t2, 3u Z d and let pη t q tě0 be a symmetric multitype process with parameters pλ 1 , λ 2 , rq. By Proposition 2.4.6, pξ t q tě0 is stochastically lower than pη t q tě0 . Since λ 2 ą λ c , the process p r ξ t q tě0 is supercritical and therefore, the symmetric multitype process survives.

We now turn to Theorems 2.2.1 and 2.2.2, for which we shall prove : Theorem 2.5.1. Assume λ 2 ă λ c ă λ 1 fixed. Let pη t q tě0 be the symmetric multitype process. Then, (i) there exists r 0 P p0, 8q such that if r ă r 0 then the process pη t q tě0 survives.

(ii) there exists r 1 P p0, 8q such that if r ą r 1 then the process pη t q tě0 dies out. Theorem 2.5.2. Assume λ c ă λ 1 and λ 2 ă λ 1 fixed. Let pη t q tě0 be the asymmetric multitype process. Then, (i) there exists s 0 P p0, 8q such that if r ă s 0 then the process pη t q tě0 survives.

(ii) there exists s 1 P p0, 8q such that if r ą s 1 then the process pη t q tě0 dies out.

These results imply Theorems 2.2.1 and 2.2.2, that is, the existence of a unique phase transition with a critical value r c (resp. s c ) defined in (2.2.12). Indeed, relying on Theorems 2.5.1 and 2.5.2, by monotonicity given by Corollary 2.4.2 one has r 0 " r 1 (resp. s 0 " s 1 ).

We shall prove both theorems in Subsections 2.5.2 and 2.5.3. One concludes for the critical case by proving Theorem 2.2.3 in Subsection 2.6.

Before that, subsection 2.5.1 deals with consequences of Theorems 2.5.1 and 2.5.2 along with monotonicity results of Section 2.4.

Behaviour of the critical value with varying growth rates

Suppose the existence of the critical value r c guaranteed in virtue of Theorems 2.5.1 and 2.5.2, one investigates the behaviour of r c when growth rates λ 1 and λ 2 are moving. One manages to prove monotonicity between growth rates and the release rate, in the sense that, the more competitive the species is (i.e. the higher the parameter λ 2 is) or the fittest the species is (i.e. the higher the parameter λ 1 is), the higher the release rate is (i.e. the higher the critical value r c is) : Proposition 2.5.1. For j " 1, 2, the function λ j Þ ÝÑ r c pλ j q is non-increasing.

Proof. We consider j " 2 as the case j " 1 is similar. Let pη t q tě0 and pη 1 t q tě0 be two multitype processes with respective parameters pλ 1 , λ 2 , rq and pλ 1 , λ 1 2 , rq. By Theorems 2.5.1 and 2.5.2, existence and uniqueness of the critical values r c and r 1 c associated to those processes are guaranteed. We now show that if λ 2 ă λ 1 2 , then r c ď r 1 c . By contradiction, suppose r c ą r 1 c . Let r be fixed be such that r c ą r ą r 1 c . Since

λ 2 ă λ 1 2 , by Corollary 2.4.1-(iii), if η 0 " η 1 0 then η t ď
P r p@t ě 0, H 1 t ‰ Hq ď P r 1 c p@t ě 0, H 1 t ‰ Hq " 0
But since r ă r c , the process pη t q tě0 survives : P r p@t ě 0, H t ‰ Hq ą 0. This contradicts η t ď η 1 t a.s., hence r c ď r 1 c .

Subcritical case

The following proof relies on a comparison of the multitype process with an oriented percolation process on the even grid L. Then we show that for the associated open sites, percolation occurs thanks to results we presented in Section 2.2.3.

We follow arguments used by N. Konno, R. Schinazi and H. Tanemura [START_REF] Konno | Coexistence results for a spatial stochastic epidemic model. Markov Process[END_REF] in the case of a spatial epidemic model.

Proof of Theorem 2.5.1 (i).

To simplify notations, choose d " 1 but the proof remains the same for any d ě 2. Introduce the following space-time regions, B " p´4L, 4Lq ˆr0, T s, B m,n " p2mLe 1 , nT q `B I " r´L, Ls, I m " 2mLe 1 `I for positive integers L, T to be chosen later, where pe 1 , ..., e d q denotes the canonical basis of R d . Notice they correspond to the boxes introduced in (2.2.20) with j 0 " 1, k 0 " 4.

Consider the process pη m,n t q tě0 restricted to the region B m,n , that is, constructed from the graphical representation where only arrival times of the Poisson processes occurring within B m,n are taken into account. Therefore, a birth on a site x P B m,n from some site y only occurs if y P B m,n . By Proposition 2.4.2 and Remark 2.4.1, one has

η m,n t ď η t ˇˇBm,n , (2.5.1) for all t ą 0 if η m,n 0 " η 0 ˇˇBm,n . Let k " t ?
Lu and define C " r´k, ks. One declares pm, nq P L to be wet if for any configuration at time nT such that there is a translate of C full with 1's in I m with I m containing only 0's and 1's, the process restricted to B m,n is such that at time pn `1qT there are a translate of C in I m´1 and a translate of C in I m`1 , both full of 1's, with I m´1 and I m`1 containing only 0's and 1's.

Let us show that the probability of a site pm, nq P L to be wet can be made arbitrarily close to 1 for L and T chosen sufficiently large. By translation invariance, it is enough to deal with the case pm, nq " p0, 0q.

Suppose I contains only 0's and 1's as well as the translate of C full of 1's and set r " 0 in B, that is, no more type-2 individuals arrive in the box B after time 0.

If type-2 individuals are present on the base p´4L, ´LqYpL, 4Lqˆt0u, the probability of the event E they all die by time T {2 is at least

´1 ´expp´T {2q ¯6L
which is larger than 1 ´ǫ for T and L chosen large enough. On E, the process restricted to the box B is now from time T {2 a supercritical contact process pξ m,n t q těT {2 with distribution r Ppξ m,n t P ¨q. But we have to make sure that at time T {2, there are still enough 1's for ξ m,n T {2 , for this we use the following result. Define τ pℓq " inftt ą 0 : Ξ r´ℓ,ℓs t ˇˇr ´ℓ,ℓs " Hu, the hitting time of the trap state 0 of the contact process starting from r´ℓ, ℓs and restricted to r´ℓ, ℓs ˆr0, T {2s. T. Mountford [START_REF] Mountford | A metastable result for the finite multidimensional contact process[END_REF] proved that r Ppτ pℓq ď exppℓqq ď expp´ℓq for ℓ large enough (2.5.2)

Partition C into M " t ? ku boxes, each of them being a translate of r0, M s. From each of these M boxes, say box j ď M run a supercritical contact process denoted by pζ j t q tě0 which coincides with the restriction of ξ m,n t to this box. Therefore for each x in this box J, as in 2.5.1, ζ j T {2 pxq ď ξ m,n T {2 pxq for all x P B m,n . Then for the union of these j boxes (j ď M ), the probability there is at least M 1's within C by time T {2 is after (2.5.2), with T such that exppM q ď T {2, at least r Ppτ pM q ě T {2q M ě r Ppτ pM q ě exppM qq M ě p1 ´expp´M qq M (2.5.3) which can be made larger than 1 ´ǫ, for M , i.e. L, large enough.

A result of R. Durrett and R. Schinazi [START_REF] Durrett | Asymptotic critical value for a competition model[END_REF] shows that for a contact process pξ t q tě0 , for any A Ă Z, except for a set with exponentially small probability, either Ξ A t " H, or ξ A t " ξ Z t on a linearly time growing set r´αt, αts : there exists α ą 0 such that for all A Ă Z, there exist positive constants C, γ such that

r PpΞ A t ‰ H, ξ A t pxq ‰ ξ Z t pxqq ď C expp´γtq (2.5.4)
where x P A `αt.

We applied this result with A Ă C which correspond to the numbers of 1's in the box. We just proved that |A| ą |M |. Moreover according to Proposition 2.6.1, one can 2.5. Phase transition choose k, and so L, large enough so that this supercritical contact process ξ A t starting from at least M 1's survives at time T {2 with probability close to 1, hence ξ A t ‰ H and (2.5.4) is valid. In this situation, taking T {2 " 9L{p2αq with L large enough, the process pξ m,n t q tě0 starting from at least M 1's in r´L, Ls at time T {2 will be coupled with a process ξ Z t on r´3L, 3Ls with probability at least 1 ´ǫ at time T . Hence, since 3L ą αT {2 ą 2L, by time T the contact process ξ A t started inside r´L, Ls has not reached the boundary of r´4L, 4Ls with probability close to 1. Then, the process ξ m,n t and the contact process ξ A t are the same with probability 1 ´ǫ in r´4L, 4Ls ; this way, the coupling of pξ m,n t q tě0 with pξ Z t q tě0 works so far with probability 1 ´ǫ if L is large enough.

Since the distribution of pξ Z t q tě0 is stochastically larger than the upper invariant measure ν (see Chapter 1 Section 1.1.2) of the contact process, on the survival event, ν loads a positive density ρ of 1's. Since ν is ergodic (see Chapter 1 Section 1.1.2), lim

LÑ8 1 2L `1 ´L ÿ x"´3L 1tηpxq " 1u " ρ ν -a.e.
In other words, as soon as L is large enough, under ν there are at least k 1's in any interval of length 2L`1 with ν-probability at least 1´ǫ. Since we obtained that pξ m,n t q tě0 is coupled to pξ Z t q tě0 by time T with probability at least 1 ´2ǫ, for L large enough, there are at least k 1's in r´3L, ´Ls at time T with probability at least 1 ´2ǫ and similarly, at least k 1's at time t in rL, 3Ls with probability at least 1 ´2ǫ as well for pξ m,n t q tě0 . Consequently, r Ppp0, 0q wetq ą 1 ´4ǫ, if r " 0.

(2.5.5)

Since B is a finite space-time region, for fixed L, T , one can pick r 0 ą 0 small enough so that the arrival times of a rate r Poisson process, such that r ă r 0 , in B occurs with probability at most ǫ. Let A L,T prq be the first arrival time of a rate r Poisson process in r´2L, 2Ls ˆr0, T s. P r pp0, 0q wetq ě P r pp0, 0q wet, A L,T prq ą T qP r pA L,T prq ą T q ě p1 ´4ǫqe ´rp4L`1qT ě 1 ´6ǫ

as soon as the exponent of the exponential is close to 0, i.e. by choosing r small enough. See Figure 2.2 for an illustration. Now construct a percolation process by defining the good event G m,n " tpm, nq wetu. Notice that G m,n depends only on the process constructed in B m,n , and for pa, bq P L, events G m,n and G a,b are independent if pm, nq and pa, bq are not neighbours. The events tG m,n , pm, nq P Lu are thus 1-dependent. By the comparison theorem 2.2.7, the process pη m,n t q tě0 restricted to regions B m,n is stochastically larger than a 1-dependent percolation process with probability 1 ´ǫ. By Lemma 2.2.5, one can choose ǫ small enough so that percolation occurs in the 1-dependent percolation process with density 1 ´ǫ.

I I 1 I ´1 I 2 I ´2 L ´L 4L ´4L ´6L 6L B B ´1,1 B 1,1 Figure 2.2:
There exist L, T such that p0, 0q is wet with P r -probability close to 1.

Supercritical case

In the following, one compares our particle system with a percolation process on Z 2 ˆZ`a nd uses arguments from Van Den Berg et al. [START_REF] Van Den Berg | Dependent random graphs and spatial epidemics[END_REF].

Proof of Theorem 2.2.1 (ii). Assume d " 2, the proof can similarly be extended to higher dimensions. For all pk, m, nq P Z 2 ˆZ`. Introduce the following space-time regions, for positive L and T to be chosen later.

A " r´2L, 2Ls 2 ˆr0, 2T s A k,m,n " A `pkL, mL, nT q B " r´L, Ls 2 ˆrT, 2T s B k,m,n " B `pkL, mL, nT q C " C bottom Ť C side C k,m,n " C `pkL, mL, nT q
where C bottom " tpm, n, tq P A : t " 0u C side " tpm, n, tq P A : |m| " 2L or |n| " 2Lu Consider a restriction of the process pη t q tě0 to A k,m,n , that is, the process pη k,m,n t q tě0 constructed from its graphical representation within A k,m,n .

One declares a site pk, m, nq P Z 2 ˆZ`t o be wet if the process pη k,m,n t q tě0 contains no wild individual in B k,m,n starting from any configuration in C k,m,n . Therefore it will be the same for η t ˇˇA k,m,n . Sites that are not wet are called dry.

For any ǫ ą 0, we show that for some chosen L and T any site pk, m, nq is wet with probability close to 1 when r is large enough. By translation invariance, it is enough to consider pk, m, nq " p0, 0, 0q. Set r " 8 in A. Then, the process pη k,m,n t q tě0 contains only sites in state 2 or 3 : sites in state 0 or 1 flip instantaneously into state 2 and 3 respectively. That is, pη k,m,n t q tě0 is in fact a contact process p r ξ k,m,n t q tě0 on t2, 3u r´2L,2Ls The contact process p r ξ t q tě0 on t2, 3u Z 2 with growth rate λ 2 ă λ c is subcritical.

If there is some wild individual in B, it should have come from a succession of births started somewhere in C. Starting from a site in C side , a path to B should last at least L 52 2.5. Phase transition sites ; according to C. Bezuidenhout and G. Grimmett [START_REF] Bezuidenhout | Exponential decay for subcritical contact and percolation processes[END_REF] there exists such a path with probability at most C expp´γLq, for some positive constants C, γ. Hence, PpDpx, tq P C side ˆr0, 2T s : px, tq Ñ Bq ď 4 `2T ˆp4L `1q ˘C expp´γLq Similarly, starting from the base C bottom , there exists a path lasting at least T sites with probability PpDpx, tq P C bottom : px, tq Ñ Bq ď p4L `1q 2 C expp´γT q

Consequently if r " 8, Ppp0, 0, 0q wet q ě 1 ´4`2 L ˆp4L `1q ˘Ce ´γL ´p4L `1q 2 Ce ´γT ě 1 ´ǫ{2,
for L and T large enough.

Since A is a finite space-time region, one can pick r large enough so that with probability at least 1 ´ǫ{2, an exponential clock with parameters r rings before any other so that there are no type-1 individuals in A with probability close to 1 : P r pk, m, nq wetq ě 1 ´ǫ for r large enough.

To construct a percolation process on Z 2 ˆZ`, one puts an oriented arrow from pk, m, nq to px, y, zq if n ď z and if A k,m,n XA x,y,z ‰ H. The event G k,m,n " tpk, m, nq wetu depends only on the graphical construction of the process within A k,m,n , furthermore, for all pk, m, nq P Z 2 ˆZ`, there is a finite number of sites px, y, zq P Z 2 ˆZ`s uch that A k,m,n X A x,y,z ‰ H. The percolation process is dependent but of finite range. The existence of a path of wild individuals for the particle system corresponds to a path of dry sites for the percolation and we proved that dry paths are unlikely.

Let us show that for all sites, there exists a finite random time after which there is no more wild individuals remaining. Follow the construction given by van den Berg et al. [START_REF] Van Den Berg | Dependent random graphs and spatial epidemics[END_REF].

Since the percolation is of finite range, there exists some positive constant Cpdq such that if the distance between two sites is at least Cpdq then they are mutually independent. For any x P Z 2 , define T x " suptt : η t pxq P t1, 3uu the last time where x is occupied by a wild individual. By translation invariance, deal with the case x " p0, 0q.

Let K ą 0, suppose T 0 ą T K, there exists some m P Z `such that p0, 0, mq is the end of a dry path starting from px, y, 0q with px, yq P Z 2 . The number of paths of length ℓ is at most δ ℓ . Moreover, a self-avoiding path of length ℓ contains at most νℓ mutually independent sites (i.e. whose in-between distance is at least Cpdq), ν ą 0. Hence,

PpT 0 ą T Kq ď ÿ měK´1 ÿ ℓěm δ ℓ `1 ´Pppk, m, nq wet q ˘νℓ (2.5.6)
For r large enough,the right-hand side tends to 0 when K goes to infinity. That is, T 0 is almost surely finite and the region A ˆrT x , 8q is wild individuals-free. For the percolation process, this means there is an infinite path of wet sites, hence the process pη t q tě0 dies out.

To sum up, we just showed there exist r 0 and r 1 such that r 0 ď r c ď r 1 , for r ď r 0 the process survives and for r ě r 1 the process dies out. This proves the existence of a phase transition for the symmetric multitype process. The proof of Theorem 2.5.1-(i) only uses that contact process with growth rate λ 1 is supercritical, this is also true to show the existence of s 0 in Theorem 2.5.2-(i). By Proposition 2.4.4, the asymmetric multitype process dies out as soon as the symmetric one does, existence of s 1 in Theorem 2.5.2-(ii) is then immediately guaranteed by Theorem 2.5.1-(ii). Though, one can remark that conditions of Theorem 2.5.2 are milder : one can actually show the existence of s 1 in a neater way. Indeed, retrieving briefly the proof of the supercritical case : assume λ 2 ą λ c : with the lack of the transition "2 Ñ 3" in the asymmetric case and choosing first r " 8, one notices for the subcritical contact process on t2, 3u Z 2 , there are no possible paths of wild individuals created by the 3's from the boundary C k,m,n up to extinction, but this occurs exponentially fast (see C. Bezuidenhout and G. Grimmett [START_REF] Bezuidenhout | Exponential decay for subcritical contact and percolation processes[END_REF]).

The critical process dies out

In this section, we prove Theorem 2.2.3 : the critical multi-type contact process dies out. i.e. P rc pH t ‰ H @t ě 0q " 0. Recall pη t q tě0 stands for the multitype process, starting from the initial configuration η 0 " 1 t0u .

One follows closely the arguments used by C. Bezuidenhout and G. Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF], well-exposed by T.M. Liggett [START_REF] Liggett | Stochastic interacting systems : contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]Chapter I.2]. We shall use both presentations.

The dynamic renormalization construction sees the time-evolution of the process in a suitable chosen scaling : space-time is divided into finite space-time regions. So far, this looks heavily like the comparison with oriented percolation we defined in Section 2.2. But here, instead of fixing every region initially, the idea is rather to determine their positions according to the past random position in the construction, along with the evolution of the process.

Let us sketch the contents of the proof.

Outline. The first step consists to observe that if the process survives in an arbitrary large box, then it reaches its boundaries densely. We shall estimate these densities at each side of a space-time region. This way, one can repeat this step by running the process in an other adjacent box starting from the boundary of the previous one and so on, conditionally on the fact that the starting configuration is dense enough. This is the second step. In connection with the proofs of Theorems 2.5.2 and 2.5.1 where we looked after having translations of occupied finite intervals at a given fixed time, here we look after having translations of the densities in some space-time slab. Now, knowing that at each stage, one can construct overall a path of adjacent boxes wherein the process survives and reaches the boundaries densely, it remains to compare the process with an oriented percolation process to extend the arguments to infinite space and time. As before, compare a space-time box to a vertex in the even lattice of an oriented percolation so that one declare a vertex to be wet if some good event associated to the box is a success. Conclude thanks to results about percolation theory, recalled in Section 2.2.

Local characterization of the survival event

We saw under specific hypothesis on r, the multi-type contact process survives with positive probability. Supposing it survives, one exhibits here several properties of growth satisfied by the process restricted to an arbitrary large box. Such results have been proved for the basic contact process by C. Bezuidenhout and G. Grimmett [START_REF] Bezuidenhout | The critical contact process dies out[END_REF], thanks to techniques of dynamic renormalization introduced by G. Grimmett et al. [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF][START_REF] Barsky | Percolation in half-spaces : equality of critical densities and continuity of the percolation probability[END_REF].

First note the arguments developed by [START_REF] Bezuidenhout | The critical contact process dies out[END_REF] rely on elementary properties of the contact process making them robust. They are also valid for the multitype process because the latter satisfies the following properties we have exhibited in previous sections : (A) contact process-like dynamics : one retrieves the growth rate λ 1 or λ 2 of a basic contact process, even if it is determined randomly. We will make use of the more suitable one depending on the situation. (B) attractiveness, by Section 2.4. (C) correlation inequalities : using correlation inequalities such as FKG inequality 1.2.1. Note that the use of (C) is possible because we shall work in finite space-time regions in the following. Such techniques have been several times exploited to study critical processes, including works by O. Garet and R. Marchand [START_REF] Garet | The critical branching random walk in a random environment dies out[END_REF] for a branching random walk, J. Steif and M. Warfheimer [START_REF] Steif | The critical contact process in a randomly evolving environment dies out[END_REF] for a randomly evolving contact process.

Covering of an arbitrary large box

Proposition 2.6.1. Suppose pη t q tě0 survives, then

lim nÑ8 P r pH r´n,ns d t ‰ H @tq " 1.
(2.6.1)

Proof. By attractiveness of the process pη t q tě0 (see Corollary 2.4.1), if A, B Ă Z d are such that A Ă B then P r pH B t ‰ H @t ě 0q ě P r pH A t ‰ H @t ě 0q ą 0. Since we assumed pη t q tě0 survives, lim AÒZ d P r pH A t ‰ H @tq " 1.

Chapter 2. Phase transition on Z d

Consider, for L ě 1 and A Ă Z d , the truncated process p L η A t q tě0 defined as the process pη t q tě0 starting from the initial configuration η 0 " 1 A constructed from the graphical representation in p´L, Lq d ˆr0, ts. Denote by p L H A t q tě0 the associated set of sites occupied by wild individuals at time t.

The next two results show that sites occupied by wild individuals are dense in some orthant of the top of a box of size p´L, Lq d ˆr0, T s. Following estimates are analogous to the ones we did previously in the proof of Theorem 2.5. ‰ H @t ą 0q.

On the other hand, for an initial configuration constituted of p2n `1q d wild individuals, the probability that these p2n `1q d wild individuals die before any birth is at least the probability the maximum of p2n `1q d independent exponential clocks with parameter 1 is smaller than the minimum of 2dp2n `1q d independent exponential clocks with parameter λ 2 , since λ 2 ă λ 1 . That is, Using FKG inequality (C), one shows that the truncated process contains a large number of occupied sites in some orthant of R d . For this, define the 2 d orthants of R d : for u " pu 1 , ..., u d q P t´, `ud ,

P r pH r´n,ns d t " H for some t|F s q ě " 1 1 `2dλ
O u :" tx " px 1 , ..., x d q P R d : sgnpx i q " u i , 1 ď i ď du.
By symmetry and reflexion with respect to the time axis, without loss of generality one can only consider the positive orthant i.e. when sgnpx i q " `for any 1 ď i ď d that we denote by O `:" tx " px 1 , ..., x d q P R d : sgnpx i q " `, 1 ď i ď du.

Proposition 2.6.3. Fix n ě 1, N ě 1 and L ě n,

P r p| L H r´n,ns d t X O `| ď N q 2 d ď P r p| L H r´n,ns d t | ď 2 d N q (2.6.5)
Proof. Along this proof, make us of (A) since we retrieve a basic contact process, so that one use the suitable growth rate depending on the ongoing estimate. First, remark that

| L H r´n,ns d t | " ÿ uPt´,`u d | L H r´n,ns d t X O u |.
All t| L H r´n,ns d t XO u |, u P t´, `ud u are independent, identically distributed and positively correlated by monotonicity : increasing with respect to growth rate Poisson process and decreasing with respect to death and release rates Poisson processes, using (B) by Corollary 2.4.1. So that by FKG inequality, for all u P t´, `ud :

˜Pr ˆ|L H r´n,ns d t X O `| ď N ˙¸2 d " ź uPt´,`u d P r ˆ|L H r´n,ns d t X O u | ď N ď P r ˆč uPt´,`u d ´|L H r´n,ns d t X O u | ď N ¯ď P r ˆÿ uPt´,`u d | L H t r´n,ns d X O u | ď 2 d N ď P r ˆ|L H r´n,ns d t | ď 2 d N ˙.
By Propositions 2.6.2 and 2.6.

A Ă Z d , let N A
S pL, T q be the cardinal of the set tpx, tq P SpL, T q X L H A : px 1 , s 1 q, px 2 , s 2 q P SpL, T q X L H A such that |s 1 ´s2 | ě 1u.

Proposition 2.6.4. Let pL j q jě1 and pT j q jě1 be two increasing sequences of integers.

For any integers M, N, n,

lim jÑ8 P r pN r´n,ns d pL j , T j q ď M qP r p| L j H r´n,ns d T j ď N q ď P r pH r´n,ns d t
" H for some tq (2.6.6)

Proof. Let F L,T be the sigma-algebra generated by the Poisson processes of the graphical representation of the process pη t q tě0 in p´L, Lq d ˆr0, T s. For each site of L H r´n,ns d T , there is a probability at least p1 `2dλ 1 q ´1 that a site does not give birth (exponential clock with parameter 1 associated to a death ringing before an exponential clock associated to a birth). By independance of the Poisson processes, the probability that none of x P L H T r´n,ns d contributes to the survival of the process is at least

´p1 `2dλ 1 q ´1¯| L H r´n,ns d T | .
For the lateral parts of p´L, Lq d ˆr0, T s, consider now a segment txu ˆr0, T s, where |x| 8 " L, and define px, t 1 q, ..., px, t j q a maximal set of 1-sparse time-wise points of the segment in SpL, T q X r´n, ns d i.e. such that for any points px, t i q and px, t j q in this set, then |t i ´tj | ě 1. Fix j ě 1, the segment

I " j ď k"1 txu ˆpt k ´1, t k `1q.
is of Lebesgue-measure at least 2j. There is no arrow in the graphical representation starting from a site of I with probability at least ´e´2jλ 1 ¯2d .

For each interval of length y in txu ˆr0, T qzI, the event no arrow occurs or an arrow occuring is preceded by a death or a slowdown symbol, occurs with probability at least

1 1 `2dλ 1 .
Consequently, no points of txu ˆr0, T s contributes to the survival of the process with probability at least

e ´4djλ 1 ´1 2dλ 1 ¯j.
Counting the contribution of all such x, P r pH r´n,ns d t " H for some t|F L,T q ě e ´4dkλ Then, consider two increasing sequences pT j q jě0 , pL j q jě0 and integers M, N , define

H j :" tN r´n,ns d S pL j , T j q `|L H r´n,ns d T j | ď M `N u. If G " tH r´n,ns d t
" H for some tu, by the martingale convergence theorem, lim jÑ8 P r pG|F L j ,T j q " 1 G a.s.

From (2.6.7), for all j ě 0, on H j , P r pG|F L j ,T j q is bounded below by some positive form and thus, lim jÑ8 H j Ă G. That is, lim jÑ8 P r pH j q ď P r p lim jÑ8 H j q ď P r pGq.

Furthermore, by FKG inequality (C),

P r pN r´n,ns d pL, T q `|L H r´n,ns d T | ď M `N q ě P r pN r´n,ns d S pL, T q ď M, | L H r´n,ns d T | ď N q ě P r pN r´n,ns d S pL, T q ď M qP r p| L H r´n,ns d T | ď N q
this concludes the proof.

As for the top of the box (Proposition 2.6.3), one can control the number of occupied sites on the lateral parts of the box p´L, Lq d ˆr0, T s. For this, introduce for i " 1, ..., d2 d and u P t´, `ud , the 2 d sides of the box by S u i pL, T q :" tpx, tq P Z d ˆr0, T s, x i " u i L, sgnpx j q " u j pj ‰ iqu 59 Chapter 2. Phase transition on Z d and N A i pL, T q the cardinal of the set tpx, tq P S u i pL, T q X L H A : px 1 , s 1 q, px 2 , s 2 q P SpL, T q X L H A such that |s 1 ´s2 | ě 1u.

By symmetry and reflexion with respect to the time axis, it is enough to look only at the positive coordinates :

S `pL, T q :" S p`,...,`q 1 pL, T q " tpx, tq P Z d ˆr0, T s, x 1 " L, x j ě 0 pj ‰ iqu.

Proposition 2.6.5. For any integers M, L, T and n ă L,

P r pN r´n,ns d `pL, T q ď M q d2 d ď P r pN r´n,ns d S pL, T q ď M d2 d q
Proof. Remark that tN r´n,ns d i pL, T q, 1 ď i ď d2 d u are identically distributed and positively correlated. Moreover,

N r´n,ns d S pL, T q ď d2 d ÿ i"1 N r´n,ns d i pL, T q.
So, as for Proposition 2.6.4, one has by FKG inequality,

P r pN r´n,ns d `pL, T q ď M q d2 d " d2 d ź i"1 P r pN r´n,ns d i pL, T q ď M q ď P r p d2 d č i"1 N r´n,ns d i pL, T q ď M q ď P r pN r´n,ns d S pL, T q ď M d2 d q.
Space-time conditions Proposition 2.6.6. Suppose pη t q tě0 survives. For any ǫ 6 ą 0, there exist integers n, L, T ą 0 such that Let N be sufficiently large so that N points in Z d contain at least N 1 points which are p2n `1q-sparse in L 8 -distance. Choose now N 1 sufficiently large so that

P r ˆL`2n H r´n,ns d T `1 Ą x `
" 1 ´Pr p n`1 H t0u t Ą r´n, ns d q ı N 1 ď δ.
Likewise, choose M sufficiently large so that M points in Z d contain at least M 1 points which are p2n `1q-sparse. Choose now M 1 sufficiently large so that

" 1 ´Pr `n`1 H t0u 1 Ą r0, 2ns ˆr´n, ns d´1 ˘ıM 1 ď δ. Fix n, L, N , the map t Þ Ñ P r p| L H r´n,ns d t | ě 2 d N q is continuous and lim nÑ8 P r p| L H r´n,ns d t
| ą 2 d N q " 0, by Proposition 2.6.2, there exist two increasing sequences L j Ò 8 and T j Ò 8 such that for all j ě 1,

P r p| L j H r´n,ns d T j | ą 2 d N q " 1 ´δ.
Using Proposition 2.6.4, there exists some j 0 for which, P r pN r´n,ns d S pL j 0 , T j 0 q ą M d2 d q ą 1 ´δ.

Considering L " L j 0 and T " T j 0 , applying Propositions 2.6.3 and 2.6.5, one has

P r p| L H r´n,ns d T X r0, Lq d | ą 2 d N q ě 1 ´δ1{2 d and P r pN r´n,ns d S pL, T q ą M d2 d q ą 1 ´δ1{d2 d .
In other words, because the Poisson processes used in the graphical representation are independent in different space-time regions, The next result links the previous estimates with a percolation process.

P r ˆL`2n H r´n,ns d T `1 Ą x `

Block constructions

The following two constructions rely on the geometry of the boxes only, proofs are similar to the ones of [5, Lemma 18] and [5, Lemma 19] respectively. They allow us to position the successive boxes adjacently and well centred. Proposition 2.6.8. Suppose pη t q tě0 survives. For any ǫ 8 " ǫ 8 pǫ 7 q ą 0 and fix k P N, there exist integers n, a, b such that n ă a for which : for all px, sq P r´a, as d ˆr0, bs, with P r ´probability at least 1 ´ǫ8 , there exists a translate py, tq `r´n, ns d ˆt0u satisfying : i. py, tq P ra, 3as ˆr´a, as d´1 ˆr5b, 6bs.

ii. From px, sq `r´n, ns d ˆt0u, there exist active paths reaching any points of py, tq r´n, ns d ˆt0u lying within the region r´5a, 5as d ˆr0, 6bs.

The idea is to repeat sufficiently enough the Proposition 2.6.7 in order to translate the center px, sq of a box to the center py, tq of another box, so that if the first box is occupied, then the second one is as well and so on [see Figure 2.3].

Proof. Choose n, L, T as in Proposition 2.6.7 . Define a " 2L `n and b " 2T . One can thus construct boxes as following : noting one needs to recentre within the box py, tq P ra, 3as ˆr´a, as d´1 ˆr5b, 6bs :

(1) for 2 ď i ď d, for some current centre pz, rq such that z i ě 0 or z i ă 0, it suffices to move it in the opposite direction. Since a ě 2L, the ith coordinate does not leave out of r´a, as.

62 2.6. The critical process dies out

(2) Move the spatial coordinate to reach ra, 3as. Since it always moves by at least 2L `n and 2L `n ě 2a, it reaches ra, 3as in at most four steps.

(3) Move the time coordinate to reach 5b. As it moves between T and 2T , it reaches 5b after four to ten steps. As b " 2T , it does not overcross 6b by 10 steps.

As each step depend only of Poisson processes within the region r´5a, 5as d by disjoint time intervals, by Proposition 2.6.7, this construction succeeds with probability at least p1 ´ǫ7 q 10 ": 1 ´ǫ8 .

Iterating k times the previous result, one obtains (see Figure 2.3) :

Proposition 2.6.9. Suppose pη t q tě0 survives. For any ǫ 9 " ǫ 9 pǫ 8 q ą 0 and k P N fixed, there exist δ ą 0, and integers n, a, b such that n ă a for which : For all px, sq P r´a, as d r0, bs, with P r ´probability at least p1´ǫ 9 q k , there exists a translated py, tq`r´n, ns d ˆt0u such that : i. py, tq P p2ka `r´a, asq ˆr´a, as d´1 ˆp5kb `r0, bsq.

ii. From px, sq `r´n, ns d ˆt0u, there exist active paths reaching any point of py, tq r´n, ns d ˆt0u lying within the region R " k´1 ď j"0 p2ja `r´5a, 5asq ˆr´5a, 5as d´1 ˆp5jb `r0, 6bsq.

Comparison with oriented percolation For p, q P Z such that q ě 0 and p `q is even, define v p,q :" r´a, as d ˆr0, bs `pp2kae d ˆq5kbq and S " ď qě0 p`q even ´T `pp2kae d ˆq5kbq ¯,

where T " r´a, as d´1 ˆ!px d , tq P Z ˆR`:

0 ď t ď p5k `1qb, ´5a ˘at{b ď x d ď 5a ˘at{b ) .
Here, S is a cross shaped nesting of successive boxes (as in Figure 2.3) using reflections and symmetries. Similarly to [5, Lemma 21], one has Theorem 2.6.1. If pη t q tě0 survives, there exist integers n, a such that P r pη r´n,ns d t survives in Z ˆr´5a, 5as d´1 ˆr0, 8qq ą 0 Proof. Adapting the proof of [5, Lemma 21], fix δ ą 0 and ǫ ą 0 such that 1 ´ǫ ą 1 ´δ. Choose n, a, b as in Proposition 2.6.9. Construct random variables tZ n piq " pI n piq, P n piqq : n ě 0, i ě 0u, where I n piq P t0, 1u and P n piq P Z d ˆr0, 8q such that P n piq is undefined if I n piq " 0. Fix Z 0 p0q " p1, 0q.

For defined random variables tZ n piq : n ď N, i ě 0u, construct recursively Z N `1piq " pI N `1piq, P N `1piqq as follows. 1. the random variable I N `1piq is 1 if : for j P ti, i ´1u, I N pjq " 1, P N pjq `r´n, ns d is joined to every point of some translate of r´n, ns d centred in v 2i´N ´1,N `1 within S. 2. if I N `1piq " 1 then P N `1piq is defined as the centre of some translate of r´n, ns d . With this construction, if for any n, ti ě 0 : I n piq " 1u ‰ H then the process pη r´n,ns d t q tě0 survives in Z ˆr´5a, 5as d´1 ˆr0, 8q. It remains then to show P r pti ě 0 : I n piq " 1u ‰ H @n ě 0q ą 0. Define F n " σpZ k piq, i ě 0, 0 ď k ď nq and by Proposition 2.6.9 : P r pI n`1 piq " 1|F n q ą 1 ´δ on tI n pi ´1q " 1 or X n piq " 1u But conditionally to F n , variables tI n piq, i ě 0u are 1-dependent. By Theorem 2.2.6, one can construct Bernoulli random variables whose product measure of density p is lower than the distribution of the 1-dependent variables. By taking 1 ´p close to 1, by Lemma 2.2.5, one has I n piq " 1 for an infinity of pairs pn, iq with positive probability.

Extinction of the critical case

Using the foregoing dynamic block construction, one concludes to the Theorem 2.2.3 : Proof of Theorem 2.2.3. Let r ą 0 be such that the process pη t q tě0 survives. In the block constructions established in Propositions 2.6.8 and 2.6.9, each event depends only on the graphical representation of the process in each box p2ja`r´5a, 5asqˆr´5a, 5as d´1 p5jb `r0, 6bsq, for j ě 0. Then, Propositions 2.6.8 and 2.6.9 are preserved with P r`δprobability for some δ ą 0. From Theorem 2.6.1, the process pηq tě0 survives in r `δ. But since r `δ ď r c , then r ă r c . That is, whenever the process survives, r stands below r c : the critical process dies out.

The mean-field model

Consider in this section the mean-field model associated to the multitype process, both symmetric and asymmetric. This is a deterministic and non-spatial process where all individuals are mixed up, leading to study the densities of each type of particles overall.

Mean-field models give rise to differential systems and are interesting to compare stochastic behaviours, as previously studied, with corresponding deterministic behaviours. We investigate here the equilibria of these differential systems, first in the asymmetric model, and in the symmetric model then. Since existence of such equilibria yields the existence of a critical value, we survey the mean-field equations in order to exhibit conditions on the parameter r to deduce bounds on the critical value r c .

Subsequently, let u i be the density of type-i individuals for i " 1, 2, 3. Overall, one has u 1 `u2 `u3 " 1´u 0 . Furthermore, in connection with the definition of wild and sterile individuals, we consider as well v 1 , resp. v 2 , the density of the wild individuals (type-1 and type-3 individuals), resp. the sterile individuals (type-2 and type-3 individuals), and the density of empty sites v 0 " u 0 . Relations between the u-system and the v-system are described by

$ & % u 1 " 1 ´v0 ´v2 u 2 " 1 ´v0 ´v1 u 3 " v 0 `v1 `v2 ´1 .
(2.7.1)

Since we consider densities, both systems satisfy u i P r0, 1s for i " 0, 1, 2, 3, v i P r0, 1s for i " 0, 1, 2.

(2.7.2)

Asymmetric multitype process

Assuming total mixing, the mean-field model associated to the asymmetric multitype process is given by :

$ & % u 1 1 " 2dpλ 1 u 1 `λ2 u 3 qu 0 `u3 ´pr `1qu 1 u 1
2 " ru 0 `u3 ´u2 u 1 3 " ru 1 ´2u 3 .

(2.7.3)

Chapter 2. Phase transition on Z d

This system admits two equilibria :

pu 1 , u 2 , u 3 q " ´0, r r `1, 0 ¯, ´1 r `1 ´r `2 4dλ 1 `2dλ 2 r , r r `1 ´r 2 ´1 r `1 ´r `2 4dλ 1 `2dλ 2 r ¯, r 2pr `1q ´rpr `2q 2p4dλ 1 `2dλ 2 rq ¯.
Note that the first equilibrium gives pu 1 , u 2 , u 3 q " ´0, r r`1 , 0 ¯which puts a positive density on the sterile individuals and none on the others, which corresponds to the extinction of the process.

$ ' ' & ' ' % v 1 0 " ´2d ´pλ 2 ´λ1 qv 0 `λ2 v 1 `pλ 2 ´λ1 qv 1 `λ1 ´λ2 ¯v0 ´pr `2qv 0 ´v1 ´v2 `2 v 1 1 " 2d ´pλ 2 ´λ1 qv 0 `λ2 v 1 `pλ 2 ´λ1 qv 2 `λ1 ´λ2 ¯v0 ´v1 v 1
2 " rp1 ´v2 q ´v2 (2.7.4) This system gives rise to an equilibrium :

pv 0 , v 1 , v 2 q " ´2 `r 4dλ 1 r `2dλ 2 r , r `2 2pr `1q ´pr `2q 2 2p4dλ 1 `2dλ 2 rq , r r `1¯.
In particular, by checking conditions (2.7.2), one highlights a condition : the density v 1 is non-negative as soon as 4dλ 1 `2dλ 2 r ą pr `1qpr `2q.

which gives the following condition

r ă 2dλ 2 ´3 `ap2dλ 2 ´3q 2 ´8p1 ´2dλ 1 q 2 (2.7.5)
This indicates a lower bound for the phase transition.

Symmetric multitype process

The mean-field equations associated to the symmetric multitype process are :

$ & % u 1 1 " 2dpλ 1 u 1 `λ2 u 3 qu 0 `u3 ´pr `1qu 1 u 1 2 " ru 0 `u3 ´u2 ´2dpλ 1 u 1 `λ2 u 3 qu 2 u 1 3 " ru 1 `2dpλ 1 u 1 `λ2 u 3 qu 2 ´2u 3 (2.7.6)
As previously, this system admits one trivial equilibrium :

pu 1 , u 2 , u 3 q " ´0, r r `1, 0 2.7.
The mean-field model retrieving once again a situation related to the extinction of the process, by a positive density of sterile individuals and none of the wild ones. We derive the non-trivial equilibrium thanks to the corresponding v-system :

$ ' ' & ' ' % v 1 0 " ´2d ´pλ 2 ´λ1 qv 0 `λ2 v 1 `pλ 2 ´λ1 qv 1 `λ1 ´λ2 ¯v0 ´pr `2qv 0 ´v1 v 2 `2 v 1 1 " 2d ´pλ 2 ´λ1 qv 0 `λ2 v 1 `pλ 2 ´λ1 qv 2 `λ1 ´λ2 ¯qp1 ´v1 q ´v1 v 1
2 " rp1 ´v2 q ´v2 (2.7.7) Let us determine the non-trivial equilibrium. Last line of (2.7.7) gives already v 2 " r r `1. Using relations of (2.7.1) in (2.7.7), according to v 1 1 " 0, an equilibrium pv 0 , v 1 , v 2 q satisfies in particular v 1 " 2dpλ 1 u 1 `λ2 u 3 qp1 ´v1 q (2.7.8)

checking v 1 cannot be equal to 1, one then has

v 1 1 ´v1 " 2dpλ 1 u 1 `λ2 u 3 q (2.7.9) and v 1 ‰ 1. (2.7.10)
On the other hand, from the u-system (2.7.6) with relations (2.7.1) and using condition (2.7.10),

u 1 1 " 0 ô v 1 v 0 1 ´v1 `p2 `rqv 0 `v1 ´r `2 r `1 " 0 u 1 2 " 0 ô pr `2qv 0 `v1 ´r `2 r `1 `v0 v 1 1 ´v1 u 1
3 " 0 ô p1 ´v1 q 2 `p1 ´v1 qp 1 r `1 ´pr `1qv 0 q ´v0 " 0 By solving the last line with respect to p1 ´v1 q, one has 1 ´v1 " pr `1qv 0 or 1 ´v1 " ´1 r `1.

To deduce the value of v 1 , we investigate both possibilities. Using (2.7.9)

1. if 1 ´v1 " ´1 r `1, But since this value is negative, necessarily 1 ´v1 ‰ ´1 r `1. 2. if 1 ´v1 " pr `1qv 0 , using (2.7.9) v 0 solves 2dpλ 1 `λ2 rqpr `1qv 2 0 ´p2dλ 1 `2dλ 2 r `r `1qv 0 `1 " 0. This implies

v 0 " 1 r `1 or v 0 " 1 2dλ 1 `2dλ 2 r .
Chapter 2. Phase transition on Z d (a) if v 0 " 1 r`1 , then by relations (2.7.1),

u 1 " 0, u 3 " 1, v 2 " 1 `u2 , which is a contradiction. (b) if v 0 " 1 2dλ 1 `2dλ 2 r
, then

v 1 " r `1 2dpλ 1 `λ2 rq , v 2 " r r `1.
Verifying this v-system to be a set of densities by condition 2.7.2, one case highlights a condition on r : v 1 ď 1 if and only if rp1 ´2dλ 2 q ď 2dλ 1 ´1. In the case where λ 2 ď 1{p2dq, then one has the condition

r ď 2dλ 1 ´1 1 ´2dλ 2 . (2.7.11)
Consequently, a non-trivial equilibrium of 2.7.7 is given by

pv 0 , v 1 , v 2 q " ´1 2dλ 1 `2dλ 2 r , r `1 2dλ 1 `2dλ 2 r , r r `1¯( 2.7.12)
To put in a nutshell, this survey of equilibria associated to both mean-field models, in symmetric and asymmetric case, gave us the bounds (2.7.5) and (2.7.11) for the phase transition.

We will turn into a rigorous proof of the convergence of the empirical densities to these reaction-diffusion systems. This is dealt with the hydrodynamic limits in Chapters 4 and 5.

Introduction

One considers here the unidimensional contact process on Z, either in an inhomogeneous (deterministic) environment, or in a quenched random environment where the growth parameter takes two possible values depending on the environment. Previously in Chapter 2, we investigated the contact process in a dynamic random environment, for which we exhibited a phase transition. Nevertheless, through the use of percolation theory, we were not able to explicit rigorous numerical bounds on the phase transition, but we are now.

Here, we are concerned by two kinds of quenched random environment on Z : in the first case, growth rates are randomly chosen according to each vertex ; in the second case, growth rates are chosen randomly on each oriented edges.

The contact process in random environment has already been studied in many ways to understand how a random rate affects the behaviour of the process. In an unidimensional case, M. Bramson, R. Durrett and R. Schonmann [START_REF] Bramson | The contact process in a random environment[END_REF] exhibited an intermediary phase where the process survives without growing linearly. In higher dimensions, N. Madras, R. Schinazi and R. Schonmann [START_REF] Madras | On the critical behavior of the contact process in deterministic inhomogeneous environments[END_REF] showed there exist choices of a random death rate for which the critical contact process survives. Several survival and extinction conditions have been given successively by T.M. Liggett [START_REF] Liggett | Spatially inhomogeneous contact processes[END_REF][START_REF] Liggett | The survival of one-dimensional contact processes in random environments[END_REF], C. Newman and S. Volchan [START_REF] Newman | Persistent survival of one-dimensional contact processes in random environments[END_REF] in dimension 1 and E. Andjel [START_REF] Andjel | Survival of multidimensional contact process in random environments[END_REF], A. Klein [START_REF] Klein | Extinction of contact and percolation processes in a random environment[END_REF] in higher dimensions.

We will rely on [START_REF] Liggett | Spatially inhomogeneous contact processes[END_REF][START_REF] Liggett | The survival of one-dimensional contact processes in random environments[END_REF] whose model and results are described in Section 3.2 before taking advantage of them by illustrating them in our framework. We expose our results when growth rates are depending on vertices in Section 3.3 and depending on edges in Section 3.4. To conclude the chapter, we obtain by the two previous sections a list of numerical bounds in Section 3.5

Settings and results

Preliminaries

The contact process in random environment introduced by T.M. Liggett [START_REF] Liggett | Spatially inhomogeneous contact processes[END_REF][START_REF] Liggett | The survival of one-dimensional contact processes in random environments[END_REF] is a Markov process pχ t q tě0 on t0, 1u Z whose transitions at each site x P Z are given by 0 Ñ 1 at rate ρpxqχpx `1q `λpxqχpx ´1q 1 Ñ 0 at rate δpxq (3.2.1)

where the family t `δpxq, ρpxq, λpxq ˘, x P Zu stands for the random environment which is an ergodic stationary process. See Figure 3.1. If t `δpxq, ρpxq, λpxq ˘, x P Zu is chosen deterministic, hence inhomogeneous, we will refer to it as the inhomogeneous contact process. Let ω be the random environment. For almost-every realization of ω, the process pχ t q tě0 survives if P ω p@t ě 0, X t ‰ Hq ą 0

k ´2 k ´1 k k `1 k `2 λpk `2q ρpk

Settings and results

and dies out if P ω p@t ě 0, X t ‰ Hq " 0.

T.M. Liggett [START_REF] Liggett | Spatially inhomogeneous contact processes[END_REF][START_REF] Liggett | The survival of one-dimensional contact processes in random environments[END_REF] settled survival and extinction conditions for such a process, among which : Theorem 3.2.1 (T.M. Liggett [START_REF] Liggett | Spatially inhomogeneous contact processes[END_REF]).

1. The inhomogeneous contact process dies out if for all n ě 0, Adaptating the results above of T.M. Liggett [START_REF] Liggett | Spatially inhomogeneous contact processes[END_REF][START_REF] Liggett | The survival of one-dimensional contact processes in random environments[END_REF], one is able to exhibit extinction and survival conditions leading us to explicit numerical bounds on the phase transition of the contact process in quenched random environment.

ÿ

The model

Our framework is the following. One describes the environment as a configuration over the sites of Z. Let p P p0, 1q, define a random environment ω P t0, 1u Z where each site x P Z is free (0) with probability 1 ´p or slowed-down (1) with probability p, independently of any other site.

The contact process in random environment we consider here is a contact process pχ t q tě0 with state space t0, 1u Z and quenched environment ω. Let λ 1 and λ 2 be growth parameters such that

λ 2 ď λ c ă λ 1 , (3.2.2)
where λ c stands for the critical growth rate of the basic contact process on t0, 1u Z .

Recall from previous chapter that some release rate r was curbing the expansion of a supercritical contact process with λ 1 ą λ c to a subcritical rate λ 2 ď λ c . Subsequently, for r P p0, 8q, p " r{pr `1q (3.2.3) stands for (in connection with the previous chapter) the probability a site is slowed down (corresponding to the minimum of two exponential clocks with parameters r and 1). Deaths occur at rate 1.

The process pχ t q tě0 is still monotone according to Chapter 2 Section 2.4.

Denote by P ω λ 1 ,λ 2 ,r the distribution of pχ t q tě0 with parameters pλ 1 , λ 2 , rq and environment ω. For fixed parameters λ 1 and λ 2 satisfying λ 2 ď λ c ă λ 1 , simplify by P ω r . For any A Ă Z, define X A t :" tx P Z : χ A t pxq " 1u, where χ A t denotes the process at time t started from the initial configuration χ 0 " 1 A . If A " t0u, simplify by X t " X t0u t . Consider subsequently two kinds of random environment : one depending of the vertices and one depending on the edges of the graph.

Random growth on vertices

Consider the dynamics where growth rates are affected to vertices. If λ v pkq is the growth rate from site k P Z : a birth at site k occurs at rate λ v pk ´1q if k ´1 is occupied plus at rate λ v pk `1q if k `1 is occupied, where

λ v pkq " λ 1 p1 ´ωpkqq `λ2 ωpkq (3.3.1) See Figure 3.2.
Based on the notations of Section 3.2, one has λpk `1q " ρpk ´1q " λ v pkq and λ v pkq " ρ v pkq for all k P Z. where for j P Z, λ v pjq is defined by (3.3.1).

Random growth on vertices

k ´2 k ´1 k k `1 k `2 λ v pk `1q λ v pk `2q λ v pkq λ v pk `1q λ v pk ´1q λ v pkq λ v pk ´2q λ v pk ´1q

Extinction conditions

Proof. Introduce a modified version pα t q tě0 of the process pχ t q tě0 where a death at site x P Z occurs uniquely if αpx ´1q " 0 or αpx `1q " 0, while births occur at the same rate than pχ t q tě0 : 0 Ñ 1 at rate ÿ where n 0 px, αq " ř y:|y´x|"1 1tαpyq " 0u stands for the number of neighbours of site x that are in state 0. This way, if initially the set tx P Z : α 0 pxq " 0u is a non-empty interval then for all t ą 0, tx P Z : α t pxq " 0u is still an interval of Z until it potentially disappears in case α t is identically equal to 1 on Z. In the non-empty case, considering times at which a flip occurs, each end of this interval moves respectively as a birth and death chain : the rightmost zero evolves according to

k Ñ k `1 at rate 1 and k Ñ k ´1 at rate λ v pk `1q
and the leftmost zero evolves according to k Ñ k `1 at rate λ v pk ´1q and k Ñ k ´1 at rate 1

For m, ℓ P Z such that m ă 0 ă ℓ, consider the initial condition

α 0 pxq " " 0 if m ď x ď ℓ, 1 otherwise.
Since both rightmost and leftmost zeros move as birth-death chains Z, it remains to study their hitting time of 0. Define pR n q ně0 and pL n q ně0 two corresponding Markov chains, whose respective transition matrices Q R and Q L are defined by

Q R pk, k `1q " 1 1 `λv pk `1q , Q R pk, k ´1q " λ v pk `1q 1 `λv pk `1q , Q L pk, k `1q " λ v pk ´1q 1 `λv pk ´1q , Q L pk, k ´1q " 1 1 `λv pk ´1q
.

For a P Z, note P R a and P L a their respective probability measures conditionally in R 0 " a and L 0 " a. Denote by pS n q ně1 the flipping times and consider pα n q ně1 , the skeleton-Markov chain corresponding to pα t q tě0 , such that α n " α Sn for all n ě 1. Then, P ω r pα n pxq " 0q " P R ℓ pT R 0 " 8qP L m pT L 0 " 8q, where T R 0 " infpn ě 0 : R n " 0q and T L 0 " infpn ě 0 : L n " 0q are the hitting times of zero for both birth and death chains. By a known result on birth-death processes (see §I.4 [START_REF] Schinazi | Classical and spatial stochastic processes[END_REF] for instance), and has for any site x P Z,

lim nÑ8 P ω r pα n pxq " 0q ě ℓ´1 ř k"0 k ś j"0 λ v pj `1q ř kě0 k ś j"0 λ v pj `1q 0 ř k"m`1 0 ś j"k λ v pj ´1q ř kď0 0 ś j"k λ v pj ´1q (3.3.5)
By (3.3.2), this limit tends to 1 when m goes to ´8 and ℓ goes to 8. With a death rate equal to 1, for all m ă 0 ă ℓ, there exists almost surely some time t where α t pxq " 0, for all x P Z. Coupling the processes pα t q tě0 and pχ t q tě0 starting from such times t, if χ 0 ď α 0 then the dynamics of the coupled process pχ t , α t q tě0 is given by the following transitions : transition rate Since we assumed λ 1 ą λ c , the right-hand side is positive and (3.3.8) is an upper bound on the transitional phase with respect to λ 1 and λ 2 for the extinction of the process.

Survival conditions

Applying Theorem 3.2.2, one gets

Theorem 3.3.2. Assume ÿ jě0 E ω r ˜1 λ v pjq j ź k"1 λ v pkq `λv pk ´1q `1 λ v pkqλ v pk ´1q ¸ă 8.
Then the process pχ t q tě0 in random environment survives.

The lack of independence in the product of the terms of this series disables us to obtain explicit conditions for survival of the process. Nevertheless, by defining the randomness on the edges rather than on the vertices, meaning that the growth rates emanating from a site k respectively to k `1 and to k ´1 are randomly chosen for each k P Z, we are able to explicit bounds on r with respect to λ 1 and λ 2 .

Random growth on oriented edges

Let tpρ e pkq, λ e pkqq, k P Zu be an ergodic, stationary and i.i.d. sequence. For the random growth on oriented edges, given a site k P Z, a birth from k to k `1 occurs at rate λ e pk `1q and independently of a birth from k to k ´1 occuring at rate ρ e pk ´1q. See Figure 3 Based on the notations provided in Section 3.2, one has λpkq " λ e pkq and ρpkq " ρ e pkq.

.3 k ´2 k ´1 k k `1 k `2 λ e pk `2q

Random growth on oriented edges

Extinction conditions

Theorem 3.2.1 permits to obtain the following criteria. Theorem 3.4.1. The process in random environment dies out if the two following assertions are satisfied.

i.

E ω r λ e pkq ă 1, ii.

1 ´Eω

r 1 λ e pkq ă E ω r 1 λ e pkq ´1 ´Eω r λ e pkq ¯.
that is, if λ 2 ă 1 and r ą λ 1 ´1 1 ´λ2 .

Proof. Computing the expectation of the growth rates, conditions on r for the process to die out are given by : (i) can be rewritten using (3.2.3)

λ 1 p1 ´pq `λ2 p ă 1 ô rp1 ´λ2 q ą λ 1 ´1, therefore, as λ 1 ´1 ą 0 since λ 1 ą λ c ą 1, one has again λ 2 ă 1 (3.4.1)
and the condition

r ą λ 1 ´1 1 ´λ2 (3.4.2)
On the other hand, piiq is

Aprq " 2r 2 1 ´λ2 λ 2 `r´2 ´λ2 λ 1 `2 ´λ1 λ 2 ´2¯`2 1 ´λ1 λ 1 ą 0.
The roots of the polynomial are real since its corresponding discriminant ∆ is nonnegative, ∆ " 1

λ 2 1 λ 2 2 pλ 1 ´λ2 q 2 ´pλ 1 `λ2 ´2q 2 `4λ 1 λ 2

Roots

are therefore given by δ ˘" pλ 1 `λ2 ´2qpλ 1 `λ2 q ˘pλ 1 ´λ2 q a pλ 1 `λ2 ´2q 2 `4λ 1 λ 2 4λ 1 p1 ´λ2 q Consequently, the process in random environment survives as soon as r satisfies pr ´δ`q pr ´δ´q ą 0 Since λ 2 ă 1 and λ 1 ą 1, one has δ `δ´" p1 ´λ1 qλ 2 p1 ´λ2 qλ 1 ă 0. Both roots δ ´and δ `are of opposite sign and Aprq ą 0 if

r ą δ `, (3.4.3)
(because δ ´ă 0). Notice condition 3.4.2 implies that r ą δ `.

Survival conditions

Applying Theorem 3.2.2 to our case where the sequence tρpkq, λpkq, k P Zu is i.i.d., we get Theorem 3.4.2. The process in random environment survives if for all j ě 0,

E ω r ´1 λ e pj `1q
¯´E ω r λ e pkq `ρe pk ´1q `1 λ e pkqρ e pk ´1q ¯j ă 1 that is, if λ 2 ă 1 `?2 ă λ 1 and r ă λ 2 ´λ1 ´?2 ´1λ

1 ´λ2 ´?2 ´1¯.
Proof. The (geometric) series converges as soon as

E ω r λ e p0q `ρe p0q `1 λ e p0qρ e p0q ă 1, that is, if 2E ω r 1 λ e pkq `Eω r 1 λ e pkqρ e pk ´1q " 2λ 1 `1 λ 2 1 p1 ´pq 2 `2λ 1 `λ2 `1 λ 1 λ 2 pp1 ´pq `2λ 2 `1 λ 2 2 p 2
smaller than 1 i.e. using (3.2.3) if,

Aprq :" r 2 " λ 2 1 p2λ 2 `1q ´λ2 1 λ 2 2 ı `rλ 1 λ 2 " 2pλ 1 `λ2 `1q ´2λ 1 λ 2 ı `"λ 2 2 p2λ 1 `1q ´λ2 1 λ 2 2 ı ă 0. (3.4.4)
The associated discriminant is ∆ " 8λ 2 1 λ 2 2 pλ 1 ´λ2 q 2 . And the roots of Aprq are δ A

´" ´λ2 ´λ1 `?2 ´1λ

1 ´λ2 `?2 ´1¯ă 0 and δ A `" ´λ2 ´λ1 ´?2 ´1λ

1 ´λ2 ´?2 ´1¯, which is positive if λ 1 ą 1 `?2, (3.4.5)
and λ 2 ă 1 `?2, this last condition is cleared by the assumption (3.2.2), as λ 2 ă λ c ď 1 `?2. In this case, the process survives if r is such that

r ă δ A `.
(3.4.6)

Numerical bounds on the transitional phase

Numerical bounds on the transitional phase

Back to the basic contact process

Assume r " 0, then for all x P Z, ωpxq " 0 a.s. and λ e pxq " ρ e pxq " λ 1 . We thus recover the one-dimensional basic contact process with growth rate λ 1 . In this case, our estimates lead to the following bound for λ c . Corollary 3.5.1. For the one-dimensional basic contact process,

λ c ď 1 `?2.
Proof. According to (3.4.4) in the proof of Theorem 3.4.2, the process survives if

λ 2 1 2λ 1 ´1 ą 0, that is, if λ 1 ą 1 `?2.
Recall on Z, λ c P r1.539, 1.942s. This bound is quite rough but its advantage is that we derived it simply. Consequently, one first deduces a bound on the critical value λ c of the one-dimensional basic contact process : λ c ď 1 `?2 » 2.41.

The phase transition

From results obtained in the previous section, one gets the following numerical bounds for a phase transition. By choosing parameters λ 1 and λ 2 satisfying (3. that the necessary condition λ 2 ă 1 disables us to conclude to an upper bound for values of λ 2 . In a similar way, condition (3.4.5) of Theorem 3.4.2 imposes λ 1 to be larger than 1 `?2, disabling us to find an explicit lower bound on the transitional phase in such cases.

Introduction

In this chapter, one derives the hydrodynamic limit on the d-dimensional torus of the asymmetric multitype contact process defined in Chapter 1.

The work here is is based on the entropy method due to M. Z. Guo, G. C. Papanicolaou and S. R. S. Varadhan [START_REF] Guo | Nonlinear diffusion limit for a system with nearest neighbor interactions[END_REF] to prove the hydrodynamic behaviour of a large class of interacting particle systems through the investigation of the time-evolution of the entropy and arguments by C. Kipnis, S. Olla and S.R.S Varadhan [START_REF] Kipnis | Hydrodynamics and large deviation for simple exclusion processes[END_REF], using martingales techniques.

This chapter is a preliminary to the next one, it introduces many involved quantities and we detail here classical computations that appear in both chapters. It is organized as follows. We begin by describing the model and the main result in Section 4.2, which is subsequently proved in Section 4.3, while classical proofs concerning the block estimates are proved in Section 4.4.

In the Appendix 4.A, we deal with a construction of an auxiliary process, a trick introduced by M. Mourragui [START_REF] Mourragui | Comportement hydrodynamique des processus de sauts, de naissances et de morts[END_REF], in case of unbounded rates. Whereafter, we expose some lengthy computations surrounding the reference measure (Appendix 4.B) and reminders on the Skorohod topology (Appendix 4.D).

Notations and Results

Let T d N " pZ{N Zq d be the d´dimensional discrete microscopic torus t0, ..., N ´1u d and T d " pR{Zq d be the corresponding macroscopic torus r0, 1q d .

The model

Define E N " t0, 1, 2, 3u T d N .
The model we investigate is a reaction-diffusion process pη t q tě0 given by the generator

L N :" L N,R,D " N 2 L D N `LR N , (4.2.1)
where N 2 L D N stands for the generator of a rapid-stirring process, defined for any function

f on E N by N 2 L D N f pηq " N 2 ÿ x,yPT d N }x´y}"1
´f pη x,y q ´f pηq ¯, (4.2.2) here, }x} " max 1ďjďd |x j | denotes the max norm for x P Z d , and η P E N , η x,y is the configuration obtained from η by exchanging the occupation variables ηpxq and ηpyq of two neighbouring sites x, y P T d N , that is,

η x,y pzq " $ & % ηpxq if z " y, ηpyq if z " x, ηpzq if z ‰ x, y,
and L R N is the generator of the asymmetric CP-DRE defined in Chapter 2, which is given for any cylinder function f on E N by L R N f pηq "

3 ÿ i"0 cpx, η, iq ´f pη i x q ´f pηq ¯(4.2.3)
with η P E N , η i x is the configuration obtained from η by flipping the state of site x into the state i P t0, 1, 2, 3u, that is,

η i x pzq " " i if z " x, ηpzq if z ‰ x.

Notations and Results

while the rate function c is defined by cpx, η, 0q " 1 if ηpxq P t1, 2u, cpx, η, 1q "

$ ' & ' % βpx, ηq :" λ 1 ř yPT d N }y´x}"1 η 1 pyq `λ2 ř yPT d N }y´x}"1 η 3 pyq if ηpxq " 0, 1 if ηpxq " 3, cpx, η, 2q " " r if ηpxq " 0, 1 if ηpxq " 3, cpx, η, 3q " r if ηpxq " 1. (4.2.4)
Since the conserved quantities for the generator L D N concern the total number of particles of each type i P t1, 2, 3u, one defines the product measure

ν N p ψ on E N by ν N p ψ pηq :" ź xPT d N 1 Z p ψ exp ˜3 ÿ i"0 rψ i 1tηpxq " ius ¸(4.2.5)
where Z p ψ "

3 ř i"0
exppψ i q is the normalization constant, for p ψ " pψ 0 , ψ 1 , ψ 2 , ψ 3 q such that ψ 0 , ψ 1 , ψ 2 , ψ 3 P R are parameters. Because of a high use of indicator functions, we shall simplify the notation by η i pxq " 1tηpxq " iu, for x P T d N and i " 1, 2, 3.

As usual, we parametrize the measure by the conserved quantities (see for instance R. Marra and M. Mourragui [START_REF] Marra | Phase segregation dynamics for the Blume-Capel model with Kac interaction[END_REF]). By a change of variables (see Appendix 4.B for details), given parameters ρ 1 , ρ 2 , ρ 3 such that ρ i ě 0 and ρ 1 `ρ2 `ρ3 ď 1, one defines the product measure for p ρ " pρ 1 , ρ 2 , ρ 3 q by ν N p ρ p¨q " ν N Ψpρ 1 ,ρ 2 ,ρ 3 q p¨q and ρ 0 " 1 ´ρ1 ´ρ2 ´ρ3 . To do changes of variables, it will be more convenient to write the measures as follows :

ν N p ρ pηq " exp ! 3 ÿ j"0 ÿ xPT d N ̺ j η j pxq ) (4.2.8)
with ̺ j " log ρ j (4.2.9)

Since conserved quantities are densities of three types of particles, we need to work with three dimensional vectors whose i-th component is associated to the type i. These vectors will be distinguished with a hat. For any configuration η, define the empirical measure of type i on E N by π N,i pηq " 1

N d ÿ xPT d N η i pxqδ x N , ( 4

Hydrodynamics for the reaction-diffusion process

Let M 1 `be the subset of M of all positive measures absolutely continuous with respect to the Lebesgue measure with positive density bounded by 1 :

M 1
`" π P M : πpduq " ρpuqdu and 0 ď ρpuq ď 1 a.e.

( .

Fix T ą 0. Let Dpr0, T s, pM 1 `q3 q be the set of right-continuous with left limits trajectories with values in pM 1 `q3 , endowed with the Skorohod topology and equipped with its Borel σ´algebra.

For any probability measure µ on E N , denote by P N µ N the probability measure on Dpr0, T s, E N q of the process pη t q tPr0,T s with generator L N and by E N µ N the corresponding expectation. Consider Q N µ " P N µ N ˝pp π N q ´1 the law of the process `p π N t pη t q ˘tPr0,Ts .

Notations and Results

Fix T ą 0. A sequence of probability measures is associated to a density profile p γ " pγ 1 , γ 2 , γ 3 q : T d Ñ r0, 1s 3 if for any δ ą 0 and any function p

G P C 1 pT d , R 3 q, lim N Ñ8 µ N !ˇˇˇx p π N pηq, p Gp¨qy ´xp γp¨q, p Gp¨qy ˇˇą δ ) " 0 , (4.2.13)
Denote by p ρ " pρ 1 , ρ 2 , ρ 3 q : r0, T s ˆTd Ñ r0, 1s 3 a typical macroscopic trajectory. We shall show that the macroscopic time-evolution of empirical density p π N is given by a reaction-diffusion system

# B t p ρ " ∆p ρ `p Rpp ρq in T d ˆp0, T q, p ρ 0 p¨q " p γp¨q in T d , (4.2.14) where p R " pR 1 , R 2 , R 3 q : r0, 1s 3 Ñ R 3 is defined by $ ' & ' % R 1 pρ 1 , ρ 2 , ρ 3 q " 2dpλ 1 ρ 1 `λ2 ρ 3 qρ 0 `ρ3 ´ρ1 pr `1q , R 2 pρ 1 , ρ 2 , ρ 3 q " rρ 0 `ρ3 ´ρ2 , R 3 pρ 1 , ρ 2 , ρ 3 q " rρ 1 ´2ρ 3 , ( 4
3 ř i"1 ż T d G i puqH i puqdu.
The rest of this chapter is devoted to prove the following result.

Theorem 4.2.1. Let p γ : T d Ñ r0, 1s 3 be an initial continuous profile and pµ N q N ě1 be a sequence of probability measures with µ N a probability measure on E N for each N associated to p γ. The sequence of random measures pp π N t q N ě1 converges weakly in probability as N goes to infinity to the absolutely continuous measure p π t pduq " p ρpt, uqdu whose density p ρpt, uq " pρ 1 , ρ 2 , ρ 2 qpt, uq is the unique weak solution of the reaction-diffusion system (4.2.14). That is, for any t P r0, T s, any δ ą 0 and any function p

G P CpT d , R 3 q lim N Ñ8 P N µ N
!ˇˇˇx p π N pη t q, p Gp¨qy ´xp ρ t p¨q, p Gp¨qy ˇˇą δ ) " 0 .

The hydrodynamic limit

For any function p G " pG 1 , G 2 , G 3 q P C 1,2 pr0, T s ˆTd ; R 3 q, by Dynkin's formula

M N,i t " xπ N,i t , G i,t y ´xπ N,i 0 , G i,0 y ´t ż 0 L N xπ N,i s , G i,s yds ´ż t 0 xπ N,i s , B s G i,s yds (4.3.1)
is a Q N µ N ´martingale with respect to the σ´algebra F t " σpη s , s ď tq. To derive the hydrodynamic behaviour of the reaction-diffusion process, one needs to prove that the above martingale vanishes as N goes to infinity. To this purpose, apply the generator L N to the function η Ñ η i pxq so that the integral part of M N,i t is depicted as follows.

N 2 L D N xπ N,i t , G i,t y " N 2 N d ÿ xPT d N d ÿ j"1
G i,t px{N q ´ηi,t px `ej q `ηi,t px ´ej q ´2η i,t pxq

" xπ N,i t , ∆ N G i,t p¨qy, where ∆ N G i,t px{N q " N 2 d ÿ j"1
pG i,t ppx `ej q{N q `Gi,t ppx ´ej q{N q ´2G

L R N xπ N,1 t , G 1,t y " 1 N d ÿ xPT d N G 1,t px{N q ˜λ1 ÿ y:}y´x}"1 η 1,t pyq `λ2 ÿ y:}y´x}"1 η 3,t pyq ¸η0,t pxq `xπ N,3 t , G 1,t y ´pr `1qxπ N,1 t , G 1,t y, L R N xπ N,2 t , G 2,t y " rxπ N,0 t , G 2,t y `xπ N,3 t , G 2,t y ´xπ N,2 t , G 2,t y ,L R N xπ N,3 t , G 3,t y " rxπ N,1 t , G 3,t y ´2xπ N,3 t , G 3,t y. (4.3.5)
Thus, to close the equations we need to replace the local function of η which is the term in L R N η 1 pxq by a functional of the empirical densities given by p π N t defined in (4.2.11). This is the purpose of the replacement lemma and the blocks estimates, exposed in Sections 4.3.2 and 4.4.

The hydrodynamic limit

Next, we need to characterize all the limit points of the sequence pQ N µ N q N ě1 : their existence comes from by the tightness of the sequence of measures, it is proved in Section 4.3.1, then, the identification and uniqueness of the limit points as weak solutions of (4.2.14) conclude the proof in Sections 4.3.3 and 4.3.4.

Tightness

Existence of limit points is guaranteed by the following lemma. Lemma 4.3.1 (Tightness). The sequence pQ N µ q N ě1 is tight and all its limit points Q

μ satisfy Q μ´p π : 0 ď p π t puq
ż t 0 ! L N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL N xπ N,i s , G i y ) ds. (4.3.11)
We postpone the detailed computations to Appendix 4.C. By Lemma 4.C.1, one has

N 2 ż t 0 # L D N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL D N xπ N,i s , G i y + ds ď CpGqtN ´d ż t 0 ! L R N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL R N xπ N,i s , G i y ) ds ď Cpλ 1 , λ 2 , rqt}G i } 2 2 N ´d
where Cpλ 1 , λ 2 , rq stands for the supremum of the bounded rates since all involved rates in factor in (4.C.6) are positive. Therefore, combining both estimates,

xM N,i y t ď `Cpλ 1 , λ 2 , rq}G i } 2 2 `CpGq ˘t N d , (4.3.12)
which converges to zero as N Ñ 8, one deduces (4.3.9) by using Doob's martingale inequality.

To prove (4.3.10), on one hand,

ˇˇN 2 L D N xπ N,i t , G i y ˇˇ" ˇˇxπ N,i t , ∆ N G i y ˇˇď }∆G i } 1 ,
where ∆G stands for the Laplace operator ∆G "

d ř j"1
B 2 e j G when B e j is the first derivative in the j-th direction. On the other hand, since all rates (4.2.4) are bounded, by ( 4 that is, any limit point is supported on trajectories such that p π t is absolutely continuous with respect to the Lebesgue measure for all t P r0, T s. 1

Replacement lemma

N d log P N µ N ¨1 N d ż ÿ xPT d N τ x V ǫN pη t qdt ě a '" ´8, where V k pηq :" ˇˇˇˇ1 p2k `1q d ř }y}ďk τ y φpηq ´r φpp η k p0qq ˇˇˇˇ.
Proof. For any γ ą 0, by Markov's inequality,

P N ν N p ρ ¨1 N d ż ÿ xPT d N τ x V ǫN pη t qdt ě a 'ď expp´γN d aq ¨EN ν N p ρ » -exp ¨γ T ż 0 ÿ xPT d N τ x V ǫN pη t qdt 'fi fl . Introduce in L 2 pν N p ρ q the operator A N,γ :" 1 2 pL N `pL N q ‹ q `γ ÿ xPT d N τ x V ǫN (4.3.14)
where pL N q ‹ is the adjoint of L N in L 2 pν N p ρ q. Fix T ą 0, by Feynman-Kac formula (see [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]Appendix 1.7]), for all t P r0, T s, the unique solution of the differential equation

$ & % B t upt, ηq " 1 2 pL N `pL N q ˚qupt, ηq `γ ř xPT d N τ x V ǫN upt, ηq up0, ηq " 1 (4.3.15)
is given by upt, ηq "

E N ν N p ρ » -up0, ηq exp ¨T ż 0 γ ÿ xPT d N τ x V ǫN pη s qds 'fi fl .
By the spectral decomposition of the auto-adjoint operator A N,γ ,

λ ǫN pγq " sup }u}"1 xA N,γ u, uy (4.3.16)
is the largest eigenvalue of the operator A N,γ , so that

E N ν N p ρ » -exp ¨T ż 0 γ ÿ xPT d N τ x V ǫN pη s qds 'fi fl ď exppT λ ǫN pγqq hence, 1 N d log P N ν N p ρ ¨1 N d ż T 0 ÿ xPT d N τ x V ǫN pη s qds ě a 'ď 1 2N d T λ ǫN pγq ´γa.
It thus remains to show for all γ ą 0,

lim ǫÑ0 lim N Ñ8 λ ǫN pγq N d " 0, (4.3.17)
in which case one would have for all γ ą 0,

lim ǫÑ0 lim N Ñ8
1

N d log P N ν N p ρ ¨1 N d ż T 0 ÿ xPT d N τ x V ǫN pη s qds ě a 'ď ´γa
and conclusion will follow by letting γ go to infinity. By Rayleigh-Ritz variational formula,

λ ǫN pγq " sup f N PL 2 pν N p ρ q }f N } L 2 "1 ¨ż γ ÿ xPT d N τ x V ǫN pηqpf N q 2 pηqdν N p ρ pηq `xL N f N , f N y ' " sup f N PL 2 pν N p ρ q }f N } L 2 "1 ˜ż γ ÿ xPT d N τ x V ǫN pηqpf N q 2 pηqdν N p ρ pηq `N 2 xL D N f N , f N y `xL R N f N , f N y
Ȩstimate the reaction part as follows.

xL R N f N , f N y " ÿ xPT d N 3 ÿ i"0 ż cpx, η, iqf N pηq ´f N pη i x q ´f N pηq ¯dν N p ρ pηq " ÿ xPT d N 3 ÿ i"0 ż cpx, η, iqf N pηqf N pη i x qdν N p ρ pηq ´ÿ xPT d N 3 ÿ i"0 ż cpx, η, iqf N pηq 2 dν N p ρ pηq ď ÿ xPT d N 3 ÿ i"0 ż cpx, η, iq ´f N pηq 2 `1 4 f N pη i x q 2 ¯dν N p ρ pηq ´ÿ xPT d N 3 ÿ i"0 ż cpx, η, iqf N pηq 2 dν N p ρ pηq
where we used the inequality AB ď 1 2a A 2 `a 2 B 2 for A, B, a ą 0 with a " 2. Use formulas of changes of variables given by Lemma 4.B.2 to bound the first sum by the L 2 -norm of f N and the fact that f N is a density with respect to ν N p ρ to bound the second integral :

xL R N f N , f N y ď Cpλ 1 , λ 2 , rq 4 ÿ xPT d N 3 ÿ i"0 ÿ j‰i ż f N pη i x q 2 η j pxqdν N p ρ pηq 90 4.3. The hydrodynamic limit " Cpλ 1 , λ 2 , rq 4 ÿ xPT d N 3 ÿ i"0 ÿ j‰i ż ρ j ρ i f N pηq 2 η i pxqdν N p ρ pηq ď Cpp ρq Cpλ 1 , λ 2 , rq 4 N d
Hence,

1

N d λ ǫN pγq " sup f N PL 2 pν N p ρ q }f N } L 2 "1 ˜1 N d ż γ ÿ xPT d N τ x V ǫN pf N q 2 pηqdν N p ρ pηq `N 2´d xL D N f N , f N y ¸`C 0
for some positive constant C 0 " Cpp ρqCpλ 1 , λ 2 , rq{4. By reversibility of the measure with respect to the generator

L D N , D D N p|f N |q ď D D N
pf N q and one can take the supremum over functions

f in L 2 pν N p ρ q such that }f } L 2 pν N p ρ " 1 to the supremum over non-negative functions f in L 2 pν N p ρ q such that } ? f } L 2 pν N p ρ " 1. Recall ν N p ρ is reversible with respect to the generator L D N but not L R N .
Going back to the upper bound of λ ǫN pγq,

1 N d λ ǫN pγq ď sup f N ě0,f N PL 2 pν N p ρ q } ? f N } L 2 "1 ˜1 N d ż γ ÿ xPT d N τ x V ǫN f N pηqdν N p ρ pηq `N 2´d xL D N a f N , a f N y ¸`C 0 ď sup f N ě0,f N PL 2 pν N p ρ q } ? f N } L 2 "1 ˜ż 1 N d γ ÿ xPT d N τ x V ǫN f N pηqdν N p ρ pηq ´N 2´d D D N pf N q ¸`C 0 , where D D N pf N q " ÿ x,yPT d N }x´y}"1 ż ´af N pη x,y q ´af N pηq ¯2 dν N p ρ pηq
is the Dirichlet form associated to the generator of stirring. Since φ is bounded, there exists some positive constant C such that ÿ

xPT d N V ǫN pηq ď CN d ,
one can thus restrict the supremum over functions f N satisfying

D D N pf N q ď CN d´2
To get (4.3.17), it remains to show for all positive C,

lim ǫÑ0 lim N Ñ8 sup f N PA N 1 N d ż ÿ xPT d N τ x V ǫN pηqf N pηqdν N p ρ pηq " 0, (4.3.18)
where

A N :" ! f N P L 2 pν N p ρ q : f N ě 0, } a f N } L 2 " 1, D D N pf N q ď CN d´2
) This limit will follow from the blocks estimates. On one hand, the one block estimate ensures the average of local functions in some large microscopic boxes can be replaced by their mean with respect to the grand-canonical measure parametrized by the particles density in these boxes. While the two blocks estimate ensures the particles density over large microscopic boxes and over small macroscopic boxes is very close. Let us first state the block estimates, we postpone their proofs to the next section. Let us prove that the limit (4.3.18) is a consequence of these two previous lemmas.

1

N d ż ÿ xPT d N τ x V ǫN pηqf N pηqdν N p ρ pηq " 1 N d ż ÿ xPT d N τ x ˇˇˇˇ1 p2ǫN `1q d ÿ }y}ďǫN τ y φpηq ´r φpp η ǫN p0qq ˇˇˇˇf N pηqdν N p ρ pηq ď 1 N d ż ÿ xPT d N τ x ˇˇˇˇ1 p2ǫN

Identification of limit points

Now we show that any limit point of the sequence pQ N µ N q N ě1 is concentrated on trajectories that are weak solutions of the reaction-diffusion system (4.2.14). For this, we come back to the martingale M N,i t defined in (4.3.1), which satisfies (4.3.9). We focus on the case i " 1 since it is the only one for which we need to use the replacement lemma. Define

B 1 ǫ pp π N t q " xπ N,1 t , G 1,t y ´xπ N,1 0 , G 1,0 y ´t ż 0 xπ N,1 s , B s G 1,s yds ´t ż 0 xπ N,1 s , ∆ N G 1,s yds ´t ż 0 xπ N,3 s , G 1,s yds `t ż 0 pr `1qxπ N,1 s , G 1,s yds ´t ż 0 1 N d ÿ xPT d N G 1,s px{N q2dλ 1 xπ N,1 s , ι ǫ p¨´x{N qyxπ N,0 s , ι ǫ p¨´x{N qyds ´1 N d t ż 0 ÿ xPT d N G 1,s px{N q2dλ 2 xπ N,3 s , ι ǫ p¨´x{N qyxπ
lim ǫÑ0 lim N Ñ8 Q N µ N ´sup 0ďtďT ˇˇB 1 ǫ pp π N t q ˇˇą a ¯" 0.
If Q μ is a limit point of the sequence pQ N µ N q N ě1 , the mapping p π Þ ÝÑ B 1 ǫ pp π T q is continuous in Skorohod topology, taking the limit as N goes to infinity,

lim ǫÑ0 Q μ˜ˇx π 1 T , G 1,T y ´xπ 1 0 , G 1,0 y ´T ż 0 xπ 1 s , B s G 1,s yds ´T ż 0 xπ 1 s , ∆G 1,s yds ´T ż 0 xπ 3 s , G 1,s yds `T ż 0 pr `1q ă π 1 s , G 1,s yds ´T ż 0 ż T d ! G 1,s puq2dλ 1 `π1 s ˚ιǫ ˘`π 0 s ˚ιǫ ˘)ds du ´T ż 0 ż T d ! G 1,s puq2dλ 2 `π3 s ˚ιǫ ˘`π 0 s ˚ιǫ ˘)ds du ˇˇˇˇą a ¸" 0.
In virtue of Lemma 4.3.1, all limit points are absolutely continuous with respect to the Lebesgue measure on T d , that is, if p π s " p ρps, uqdu, then for all i P t0, 1, 2, 3u, pπ i t ˚ιǫ qpuq converges in L 2 pT d q to ρ i pt, uq as ǫ goes to 0. Hence, du xR 1 pp π 1 s q, G 1,s puqy ˇˇą a ¯" 0 For i " 2, 3, the martingales M N,i t do not provide local functions of η so that one has immediately the following limits.

Q μ´ˇx π 1 T , G 1,T y ´xπ 1 0 , G 1,0 y ´T ż 0 xπ 1 s , B s G 1,s yds ´T ż 0 xπ 1 s , ∆G 1
Q μ´ˇx π i T , G i,T p¨qy ´xπ i 0 , G i,0 ¨qy ´T ż 0 B s xπ i s , G i,s p¨qyds ´T ż 0 xπ i s , ∆G i,s p¨qyds ´ż T 0 ds ż T d du xR i pp π i s q, G i,s puqy ˇˇą a ¯" 0
Finally, any limit point is concentrated on trajectories p π t pduq " p ρpt, uqdu which are weak solutions of (4.2.14). where pz.uq denotes the usual inner product in R d . The set tψ z : z P Z d u forms an orthonormal basis of L 2 pT d q so that any function f P L 2 pT d q can be rewritten as : f " ÿ zPZ d xψ z , f yψ z , with x., .y standing for the inner product of L 2 pT d q. For any f, g P L 2 pT d q, one has ż

Uniqueness of weak solutions

T d f puqgpuqdu " ÿ zPZ d xψ z , f yxψ z , gy.
Consider now two such solutions of (4.2.14) p ρ p1q and p ρ p2q starting from an initial profile p γ. Note p m the difference p ρ p1q ´p ρ p2q and introduce R i M : r0, T s Ñ R the function

R i M ptq " ÿ zPZ d M p1 `a|z| 2 qpM `a|z| α q
xψ z , m i pt, ¨qyxψ z , m i pt, ¨qy.. Since p ρ pjq , j " 1, 2, satisfies (S1), R i M ptq converges as M Ñ 8 and as a Ñ 0 to

R i ptq :" ÿ zPZ d xψ z , m i pt, ¨qyxψ z , m i pt, ¨qy.
which is equal to }m i pt, ¨q} 2 L 2 T d by (4.3.26). By an integration by parts, note that xψ z , B e j f y " ´2πiz j xψ z , f y, for any function

f P C 1 pT d q. Now, differentiate R i M ptq, B t R i M ptq " 2 ÿ zPZ d M p1 `a|z| 2 qpM `a|z| α q xψ z , m i pt, ¨qyxψ z , B t m i pt, ¨qy " ´8π 2 ÿ zPZ d M |z| 2 p1 `a|z| 2 qpM `a|z| α q xψ z , m i pt, ¨qyxψ z , m i pt, ¨qy `ÿ zPZ d M p1 `a|z| 2 qpM `a|z| α q xψ z , m i pt, ¨qyxψ z , R i pp ρ p1q q ´Ri pp ρ p2q qy ď ´8π 2 ÿ zPZ d M p1 `a|z| 2 qpM `a|z| α q xψ z , m i pt, ¨qyxψ z , m i pt, ¨qy `ÿ zPZ d M p1 `a|z| 2 qpM `a|z| α q xψ z , m i pt, ¨qyxψ z , R i pp ρ p1q q ´Ri pp ρ p2q qy
where we used that |z| 2 ě 1 for all z ‰ 0. Then, integrating along the time and taking the limit as M Ñ 8 and a Ñ 0,

R i ptq ď R i p0q ´8π 2 ż t 0 R i psqds `ż t 0
ˇˇxm i ps, ¨q, R i pp ρ p1q q ´Ri pp ρ p2q qy ˇˇds.

Then, notice that p R is Lipschitz, |R i pp ρ p1q q ´Ri pp ρ p2q q| ď Cpλ 1 , λ 2 , rq

3 ÿ i"1 |m i |, for all i " 1, 2, 3.
Therefore,

3 ÿ i"1 }m i pt, ¨q} 2 L 2 pT d q ď 3 ÿ i"1 }m i p0, ¨q} 2 L 2 pT d q `3`´8 π 2 `Cpλ 1 , λ 2 , rq ˘3 ÿ i"1 ż t 0 }m i ps, ¨q} 2 L 2 pT d q ds
and one concludes the proof by Gronwall's inequality.

Proof of the replacement lemma

One follows the well-reviewed proofs provided by C. Kipnis and C. Landim [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]Chap. 5], originally introduced by [START_REF] Guo | Nonlinear diffusion limit for a system with nearest neighbor interactions[END_REF].

One block estimate

Proof of Lemma 4.3.2. Note that V k pηq depends only on configurations η through the occupation variables tηpxq, |x| ď ku. Therefore, one can project any probability density

Proof of the replacement lemma

f N on a space of configurations independent of N . Let f N pηq " 1

N d ř xPT d N τ x f N pηq. By translation invariance of the measure ν N p ρ , 1 N d ż ÿ xPT d N τ x V k pηqf N pηqdν N p ρ pηq " ż V k pηqf N pηqdν N p ρ pηq
For Λ k :" tx P T d N , |x| ď ku, define E k :" F Λ k . Now, denote by ν k p ρ the product measure ν N p ρ restricted to E k and for any probability density f N , denote by f k the conditional expectation of f N with respect to the σ´algebra σpηpxq, x P Λ k q, i.e. for all

η P E k f k pηq " 1 ν k p ρ pηq ż 1tη 1 , η 1 pxq " ηpxq x P Λ k uf N pη 1 qdν N p ρ pη 1 q
it is thus enough to show :

lim kÑ8 lim N Ñ8 sup f N :f N ďD D N pf N qďCN d´2 ż V k pηqf k pηqdν k p ρ pηq " 0
By convexity of the Dirichlet forms, if D D k denotes the Dirichlet form, associated to the stirring process, defined over the set of densities

f k : Λ k Ñ R `, then D D k pf k q ď CpkqN ´dD D N pf N q ď CpkqN ´dD D N pf N q, (4.4.1) so that D D k pf k q ď C 1 pkqN ´2. Therefore, it remains to show lim kÑ8 lim N Ñ8 sup f k :D D k pf k qďC 1 pkqN ´2 ż V k pηqf k pηqdν k p ρ pηq " 0
By compactness of the level set of D D k and lower semi-continuity of Dirichlet forms, lim

N Ñ8 sup f k :D D k pf k qďC 1 pkqN ´2 ż V k pηqf k pηqdν k p ρ pηq ď sup D D k pf k q"0 ż V k pηqf k pηqdν k p ρ pηq. Now, it is about to show lim kÑ8 sup f k :D D k pf k q"0 ż V k pηqf k pηqdν k p ρ pηq " 0.
A probability density f k , whose associated Dirichlet form is null, is constant over each hyperplane with a fixed number of type-i particles for all i. The set of measures tf k ν k p ρ : D D k pf k q " 0u is convex, we can restrict ourselves to its extremal elements which are uniform over the configurations with a fixed number of particles of each type i pi " 1, 2, 3q. For any vector p ℓ " pℓ 1 , ℓ 2 , ℓ 3 q P pr0, p2k `1q d s X N d q 3 such that ℓ 1 `ℓ2 97 Now fix a positive integer p increasing to infinity after k and decompose the set Λ k in cubes of length p2p `1q. Consider the set A " tp2p `1qx, x P Z d u Ş Λ k´p and list its elements by A " tx 1 , ..., x q u such that }x ℓ } ď }x j } for ℓ ď j.

Let B ℓ " x ℓ `Λp if 1 ď ℓ ď q. Note that B ℓ Ş B j " H if ℓ ‰ j and q Ť ℓ"1 B ℓ Ă Λ k . Define B 0 " Λ k z q Ť ℓ"1
B ℓ so that |B 0 | ď Cpk d´1 by construction, for some positive constant C. This way, the integral (4.4.3) is bounded by As p goes to infinity, since ν p ρ is product, by the law of large numbers this integral converges uniformly to 0 on every compact subset of R `.

q ÿ i"1 |B ℓ | |Λ k | ż ˇˇ1 |B ℓ | ÿ yPB ℓ τ y φpηq ´r φ ´p ℓ p2k `1q d ¯ˇˇd m k p ℓ pηq But |B 0 | ď

Equivalence of ensembles

To prove the closeness between the grand-canonical and the canonical measures, we derive the so-called equivalence of ensembles. Lemma 4.4.1 (Equivalence of ensembles). For every bounded function f : t0, 1, 2, 3u

T d Ñ R, lim kÑ8 sup p ℓ ˇˇm k p ℓ pf q ´νk p ℓ{p2k`1q d pf q ˇˇ" 0
Proof of the equivalence of ensembles. For any m P N, let px 1 , ..., x m q P pΛ k q m and let

m i " m ř j"1
η i px j q. Denote by I i the set of sites that are in state i P t1, 2, 3u, i.e. I i " tx j , j " 1, ..., m : η i px j q " 1u, so that |I i | " m i .

Consider ℓ 0 " p2k `1q d ´ℓ1 ´ℓ2 ´ℓ3 and m 0 " m ´m1 ´m2 ´m3 . First, compute ν k p ρ ˜η1 px j q " 1, x j P I 1 ; η 2 px j q " 1, x j P I 2 ; η 3 px j q " 1,

x j P I 3 ; ÿ Λ k z 3 Ť i"1 I i η 1 pxq " ℓ 1 ´m1 ; ÿ Λ k z 3 Ť i"1 I i η 2 pxq " ℓ 2 ´m2 ; ÿ Λ k z 3 Ť i"1 I i η 3 pxq " ℓ 3 ´m3 " `p2k `1q d ´m˘!
pℓ 0 ´m0 q!pℓ 1 ´m1 q!pℓ 2 ´m2 q!pℓ 3 ´m3 q! p̺ 0 q ℓ 0 p̺ 1 q ℓ 1 p̺ 2 q ℓ 2 p̺ 3 q ℓ 3 by the expression of the measure ν N p ρ given in (4.2.8) and

ν k p ρ ˜ÿ xPΛ k η 1 pxq " ℓ 1 ; ÿ xPΛ k η 2 pxq " ℓ 2 ; ÿ xPΛ k η 3 pxq " ℓ 3 " p2k `1q d ℓ 0 !ℓ 1 !ℓ 2 !ℓ 3 ! p̺ 0 q ℓ 0 p̺ 1 q ℓ 1 p̺ 2 q ℓ 2 p̺ 3 q ℓ 3
Consequently, the canonical measure is given by m k p ℓ ´η1 px j q " 1, x j P I 1 ; η 2 px j q " 1, x j P I 2 ; η 3 px j q " 1,

x j P I 3 " `p2k `1q d ´m˘! p2k `1q d ! ℓ 0 ! pℓ 0 ´m0 q! ℓ 1 ! pℓ 1 ´m1 q! ℓ 2 ! pℓ 2 ´m2 q! ℓ 3 ! pℓ 3 ´m3 q!
while the grand-canonical measure is defined by ν k p ℓ{p2k`1q d pη 1 px j q " 1, x j P I 1 ; η 2 px j q " 1, x j P I 2 ; η 3 px j q " 1, x j P I 3 q " ˆℓ0 p2k `1q d Υ k p ℓ ´η1 px j q " 1, x j P I 1 ; η 2 px j q " 1, x j P I 2 ; η 3 px j q " 1,

x j P I 3 " ¨3 ś i"0 pℓ i q m i pp2k `1q d q m ‹ ‹ ' ¨¨3 ś i"0 ℓ i ℓ i ℓ i ´1 ℓ i ... ℓ i ´mi `1 ℓ i p2k`1q d p2k`1q d p2k`1q d ´1 p2k`1q d ... p2k`1q d ´m`1 p2k`1q d ‹ ‹ ' ´1‹ ‹ ' " ¨3 ś i"0 pℓ i q m i pp2k `1q d q m ‹ ‹ ' ¨¨3 ś i"0 ´1 ´1 ℓ i ¯... ´1 ´mi ´1 ℓ i 1 ´1 p2k`1q d ¯... ´1 ´m´1 p2k`1q d ¯‹ ‹ ' ´1‹ ‹ '
Taking now the maximum over p ℓ P p0, .., p2k `1q d q 3 , max p ℓ Υ k p ℓ ´η1 px j q " 1, x j P I 1 ; η 2 px j q " 1, x j P I 2 ; η 3 px j q " 1,

x j P I 3 ď ¨1 ´1 ´1 p2k`1q d ¯... ´1 ´m´1 p2k`1q d ¯' ´1
which tends to zero as k Ñ 8.

Two blocks estimate

Proof of Lemma 4.3.3. Begin by replacing the average over a small macroscopic box of size p2ǫN `1q d by the average over large microscopic boxes of size p2k `1q d , that is, for N large enough, one has

ˇˇη k i phq ´ηǫN i p0q ˇď ˇˇ1 p2k `1q d ÿ }y´h}ďk η i pyq ´1 p2ǫN `1q d ÿ }y}ďǫN 1 p2k `1q d ÿ }z´y}ďk η i pzq ˇ1 p2ǫN `1q d ÿ }y}ďǫN 1 p2k `1q d ÿ }z´y}ďk η i pzq ´1 p2ǫN `1q d ÿ }y}ďǫN η i pyq ˇď sup 2kă}h}ď2ǫN ˇˇη k i phq ´ηk i p0q ˇˇ`p 2k `1q d 2ǫN `1
It is thus enough to show : where η k i p0q and η k i phq depend only on configurations pηq over the set of occupation variables tηpxq, x P Λ h,k u, with Λ h,k :" t´k, ..., ku d Y ph `t´k, ..., ku d q.

lim kÑ8 lim ǫÑ0 lim N Ñ8 sup f N :D D N pf N qďCN D´2 sup 2k`1ď}h}ď2ǫN 1 N d ż ÿ xPT d N τ x ˇˇη k i p0q ´ηk i phq ˇˇf N pηqdν N
Denote by ν 2k p ρ the product measure ν N p ρ restricted to E k ˆEk and for any density f N , denote by f h,k the conditional expectation of f N with respect to the sigma-algebra σpηpxq, x P Λ h,k q. Let ζ and χ be two copies of η defined on E k , it is enough to prove

lim kÑ8 lim ǫÑ0 lim N Ñ8 sup f N :D D N pf N qďCN D´2 sup 2k`1ď}h}ď2ǫN ż ˇˇζ k i p0q ´χk i p0q ˇˇf h,k pζ, χqdν 2k p ρ pζ, χq " 0 (4.4.6)
Let g be a function on E k ˆEk , define the following Dirichlet forms corresponding to exchanges within two separate boxes and to exchanges between those two boxes, for two neighbouring sites x, y P Λ k D 1,k x,y pgq "

ż ´agpζ x,y , χq ´agpζ, χq ¯2dν 2k p ρ pζ, χq D 2,k
x,y pgq " ż ´agpζ, χ x,y q ´agpζ, χq ¯2dν 2k p ρ pζ, χq

∆ k pgq " ż ´ag k ppζ, χq 0 q ´ag k pζ, χq ¯2dν 2k p ρ pζ, χq
where pζ, χq 0 is obtained from pζ, χq by switching the values of ζp0q and χp0q. Define

D k pgq " D 1,k x,y pgq `D1,k x,y pgq `∆k pgq (4.4.7)
As for the one block estimate, one has the following upper bounds. For all x, y P Λ k such that }x ´y} " 1,

D 1,k x,y pf h,k q ď D D
x,y pf N q, and D 2,k x,y pf h,k q ď D D h`x,h`y pf N q As in (4.4.1), summing over each pair x, y P Λ k such that }x ´y} " 1 : ÿ

x,yPΛ k :}x´y}"1

D 1,k x,y pf h,k q `ÿ x,yPΛ k :}x´y}"1 D 1,k x,y pf h,k q ď 2CpkqN ´dD D N pf N q ď CpkqN ´2,
for any probability density whose Dirichlet form is bounded by CN d´2 . For the last one, To switch the occupations variables of ζp0q and χp0q, define a path from the origin to h by a sequence of sites x 0 , ..., x }h} 1 such that x 0 " 0, x }h} 1 " h and for each 0 ď k ď }h} 1 ´1, }x k`1 ´xk } 1 " 1, so that we have a telescopic summation b f N pη 0,h q ´bf N pηq "

∆ k pf h,k q ď ż ˆbf N pη 0,h q ´bf N pηq ˙2 dν N p ρ pηq. ( 4
}h} 1 ´1 ÿ k"0 ´bf N pη x k ,x k`1 q ´bf N pηq ¯.
By Cauchy-Schwarz inequality, from (4.4.8)

∆ k pf h,k q ď }h} 1 }h} 1 ´1 ÿ k"0 ż ´bf N pη x k ,x k`1 q ´bf N pηq ¯2dν N p ρ pηq which is equal to }h} 1 }h} 1 ´1 ř k"0 D D x k ,x k`1 pf N q. From (4.4.1), D D x k ,x k`1 pf N q ď N ´dD D N pf N q.
Moreover, }h} 1 ď 2ǫN , hence

∆ k pf h,k q ď }h} 2 1 N ´dD D N pf N q ď C 1 p2ǫq 2 .
To conclude the proof, it is thus enough to show that

lim kÑ8 lim ǫÑ0 sup D k pf qďCpkqǫ 2 ż ˇˇζ k p0q ´χk p0q ˇˇf pζ, χqdν 2k p ρ pζ, χq " 0 (4.4.9)
We conclude as for the 1-block estimate : we first let ǫ go to zero, then if f satisfies D k pf q " 0, it is constant on hyperplanes having a fixed total number of particles of each type i on Λ k Y `h `Λk ˘. The result is a consequence of the equivalence of ensembles.

4.A Construction of an auxiliary process

The reference measure ν N p ρ defined in (4.2.6) is only reversible with respect to the generator of stirring L D N . Assuming the occupation variables are unbounded, we would not be able to use the bound of the proof of the replacement lemma 4.3.1, a way to avoid this issue is to build an auxiliary reaction process whose generator is invariant (or reversible if the dynamics makes it possible, but this is not our case) with respect to the reference measure. We follow arguments presented by M. Mourragui [START_REF] Mourragui | Comportement hydrodynamique des processus de sauts, de naissances et de morts[END_REF], for births, deaths and jump processes.

.B.2], ż r L R N f pηqdν N p ρ pηq " ż ÿ xPT d N pαrf pη 1 x q ´f pηqs `r rpx, ηqrf pη 2 x q ´f pηqsq1tηpxq " 0udν N p ρ pηq `ż ÿ xPT d N pδ 1 rf pη 0 x q ´f pηqs `r rpx, ηqrf pη 3 x q ´f pηqsq1tηpxq " 1udν N p ρ pηq `ż ÿ xPT d N δ 2 rf pη 0 x q ´f pηqs1tηpxq " 2udν N p ρ pηq `ż ÿ xPT d N pδ 2 rf pη 1 x q ´f pηqs `δ1 rf pη 2 x q ´f pηqsq1tηpxq " 3udν N p ρ pηq " ż ÿ xPT d N f pηq " 1tηpxq " 0u ˆδ1 ρ 1 ρ 0 `δ2 ρ 2 ρ 0 ´α ´r0 1tηpxq " 1u ˆα ρ 0 ρ 1 `δ2 ρ 3 ρ 1 ´δ1 ´r1 ˙`1tηpxq " 2u ˆr0 ρ 0 ρ 2 `δ1 ρ 3 ρ 2 ´δ2 1tηpxq " 3u ˆr1 ρ 1 ρ 3 ´δ2 ´δ1 ˙ dν p ρ pηq.
A sufficient condition for this integral to be null is that each term between brackets vanishes. Therefore, posing δ 1 " δ 2 " 1, the measure ν N p ρ is invariant with respect to r L R N as soon as ´λ1 n 1 px, η s q λ2 n 3 px, η s q ¯1tη s pxq " 0uds are P N -martingales. Remark that, since n i px, ηq ď 2d and

r rpx, ηq " ρ 2 ´ρ3 ρ 0 1tηpxq " 0u `2ρ 3 ρ 1 1tηpxq " 1u (4.
λ 2 ă λ 1 , one has λ 1 n 1 px, ηq `λ2 n 3 px, ηq ď 2dλ 1 , for all x P T d N .
Rates r rpxq and α were defined in (4.A.2)-(4.A.3).

Starting from a common initial measure, one obtains the density via the Girsanov formula for jump processes [42, Proposition A1.2.6]. Since D x,i t have same jump rate, both are P N -and r P N -martingales, so that they vanish in the computation of the density while on the other hand,

dP N ν N p ρ d r P N ν N p ρ pη ¨q " exp # ÿ xPT d N ˜t ż 0 log r r 0 dI x,0 s ´t ż 0 `r ´r0 ˘1tη s pxq " 0udst ż 0 log r r 1 dI x,1 s ´t ż 0 `r ´r1 ˘1tη s pxq " 1uds `t ż 0 log ˆλ1 n 1 px, η s q `λ2 n 3 px, η s q α ˙dB x s ´ż t 0 ˆλ1 n 1 px, η s q `λ2 n 3 px, η s q ´α˙1 tη s pxq " 1uds ¸+, (4.A.8)
where the stochastic integral of a bounded continuous function f with respect to a jump process pI t q tě0 is defined by

ż t 0 f pη s qdI s " ÿ sďt f pη s´q pI s ´Is´q Proposition 4.A.1.
There exists a positive constant C such that

HpP N µ N | r P N ν N p ρ q ď CN d .
Proof. By definition of the entropy

HpP N µ N | r P N ν N p ρ q " ż log ¨dµ N dν N p ρ pη 0 q dP N ν N p ρ d r P N ν N p ρ pη ¨q' dP N µ N pη ¨q 105
Chapter 4. Hydrodynamic limit on the torus

" Hpµ N |ν N p ρ q `ż log ¨dP N ν N p ρ d r P N ν N p ρ pη ¨q' dP N µ N pη ¨q
Using (4.A.7), the result comes from (4.A.8) since the involved rates are bounded :

ż log ¨dP N ν N p ρ d r P N ν N p ρ pη ¨q' dP N µ N pη ¨q ď Cpλ 1 , λ 2 , r, r 0 , r 1 , αq ÿ xPT d N ´EN µ N ´Bx t ¯`E N µ N ´Ix,0 t ¯`E N µ N ´Ix,1 t ¯ď C 1 N d
First, prove this limit for the auxiliary process with infinitesimal generator r L N . Next, one concludes for the reaction-diffusion process using the entropy inequality given by (4.A.6). It is now about to prove the following. 1

Replacement lemma

N d log r P N ν N p ρ ¨1 N d ż T 0 ÿ xPT d N τ x V ǫN pη t qdt ě a '" ´8 (4.A.9)
Proof. The proof is very similar to the proof of Proposition 4.3.1 with the exception of estimating the term x r L R N a f N , a f N y. This is done as following.

x r L R N a f N , a f N y " ÿ xPT d N 3 ÿ i"0 ż r cpx, η, iq a f N pηq ´af N pη i x q ´af N pηq ¯dν N p ρ pηq " ÿ xPT d N 3 ÿ i"0 ż r cpx, η, iq ´af N pηq a f N pη i x q ´f N pηq ¯dν N p ρ pηq ď 1 4 ÿ xPT d N 3 ÿ i"0 ż r cpx, η, iqf N pηqdν N p ρ pηq `ÿ xPT d N ÿ i ż r cpx, η, iq ´f N pη i x q ´f N pηq ¯dν N p ρ pηq, using inequality AB ď 1 2a A 2 `a 2 B 2
for A, B, a ą 0 for the last bound. We deduce an estimate by Cauchy-Schwarz inequality to bound the first integral by the L 2 -norm of f N while the second integral is null since ν N p ρ is invariant with respect to the auxiliary generator L R N .

106 4.B. Properties of measures

4.B Properties of measures

Recall the measure we defined on T d N by ν N p ψ (4.2.6) for any vector p ψ " pψ 0 , ψ 1 , ψ 2 , ψ 3 q P R 4 :

ν N p ψ pηq :" ź xPT d N 1 Z p ψ exp ˜3 ÿ i"0 ψ i 1tηpxq " iu ¸(4.B.1)
where Z p ψ "

3 ř i"0
exppψ i q is the normalization constant. Using that 1tηpxq " 0u " 1 3

ř i"1 1tηpxq " iu, fix ψk " ψ k ´ψ0 for 1 ď k ď 3 so that ν N pψ 1 ,ψ 2 ,ψ 3 q pηq " ź xPT d N exp ˆ3 ř i"1 ψi 1tηpxq " iu 1 `3 ř i"1 exppψ i q .
To parametrize the invariant measure by the density of each type of particles, first deal with a change of variables as follows. Denote by Rp¨q the expectation of each occupation variable of a site x by type i under

ν N p ψ , R : ¨ψ1 ψ 2 ψ 3 'Þ ÝÑ 1 Z p ψ ¨exppψ 1 q exppψ 2 q exppψ 3 q '.
Let the vector of densities p ρ " pρ 1 , ρ 2 , ρ 3 q such that ρ i P r0, 1s and ρ 1 `ρ2 `ρ3 " 1 ´ρ0 . Then for all i " 0, 1, 2, 3, ρ i satisfies

ν N p ψ pηpxq " iq " 1 Z p ψ exppψ i q " ρ i . (4.B.2)
Proposition 4.B.1. The vector p ρ such that 1 ´ρ0 " ρ 1 `ρ2 `ρ3 is uniquely determined by the vector p ψ.

Proof. Since we parametrize the measures by p ρ, for all i :

1 1 `3 ř i"1 exppψ i q exppψ i q " ρ i .
And pψ 1 , ψ 2 , ψ 3 q solves the following system of equations $ & % exppψ 1 q " ρ 1 p1 `exppψ 1 q `exppψ 2 q `exppψ 3 qq exppψ 2 q " ρ 2 p1 `exppψ 1 q `exppψ 2 q `exppψ 3 qq exppψ 3 q " ρ 3 p1 `exppψ 1 q `exppψ 2 q `exppψ 3 qq which can be rewritten as $ ' ' ' ' & ' ' ' ' %

ψ 1 " logp ρ 1 ρ 0 q ψ 2 " logp ρ 2 ρ 0 q ψ 3 " logp ρ 3 ρ 0 q
One gets a triplet pρ 1 , ρ 2 , ρ 3 q such that 1 " ρ 0 `ρ1 `ρ2 `ρ3 , by the transformation Ψ :

Ψ : ¨ρ1 ρ 2 ρ 3 'Þ ÝÑ ¨log ˆρ1 1 ´ρ1 ´ρ2 ´ρ3 log ˆρ2 1 ´ρ1 ´ρ2 ´ρ3 log ˆρ3 1 ´ρ1 ´ρ2 ´ρ3 ˙‹ ‹ ‹ ‹ ‹ ‹ ' . (4.B.3)
where Ψ is the inverse function of R.

One can hence define uniquely a new product measure parametrize by the triplet p ρ " pρ 1 , ρ 2 , ρ 3 q by : ν N p ρ p.q :" νN Ψpρ 1 ,ρ 2 ,ρ 3 q p.q (4.B.4)

One gets a family of measures whose marginal is given by ν N p ρ pηpxq " kq " ρ k . In particular, ν N p ρ pηpxq " 0q " 1 ´ρ1 ´ρ2 ´ρ3 Lemma 4.B.1. The measure ν N p ρ is reversible with respect to the generator of rapidstirring process.

Proof. Let pζ t q tě0 be a stirring process with generator L D N on t0, 1, 2, 3u T d N . For any cylinder function f , by posing ξ " ζ x,y :

ż L D f pζqdν N p ρ pζq " ż ÿ x,yPT d N }x´y}"1 ´f pζ x,y q ´f pζq ¯dν N p ρ pζq " ż ÿ x,yPT d N }x´y}"1 f pζ x,y qdν N p ρ pζq ´ż f pζqdν N p ρ pζq " ż ÿ x,yPT d N }x´y}"1 f pξq ν N p ρ pξ y,x q ν N p ρ pξq dν N p ρ pξq ´ż f pζqdν N p ρ pζq,
and since (4.2.6) is product,

ν N p ρ pζ x,y q ν N p ρ pξ y,x q " 1.
A useful formula of change of variables : Let i, j P t0, 1, 2, 3u such that i ‰ j. For any cylinder functions f, g and α ą 0,

ż αf pη i x qgpηq1tηpxq " judν N p ρ pηq " ż α ρ j ρ i f pηqgpη j x q1tηpxq " iudν N p ρ pηq (4.B.5) Proof. Pose ξ " η i x , ż αf pη i x qgpηq1tηpxq " judν N p ρ pηq " ż αf pξqgpξ j x q1tξpxq " iu ν N p ρ pξ j x pxq " jq ν N p ρ pξpxq " iq dν N p ρ pξq " ż αf pξqgpξ j x q1tξpxq " iu ρ j ρ i dν N p ρ pξq
Define a generator L N by

L N " 3 ÿ i"0 cpx, η, iq ´f pη i x q ´f pηq ¯(4.B.6)
where for positive α, β, γ, κ,

α 1 , α 2 , β 1 , β 2 : cpx, η, 0q " " α 1 if ηpxq " 1 α 2 if ηpxq " 2 cpx, η, 1q " " α if ηpxq " 0 α 2 if ηpxq " 3 cpx, η, 2q " " r if ηpxq " 0 α 1 if ηpxq " 3 cpx, η, 3q " " r if ηpxq " 1 γ if ηpxq " 2 (4.B.7)
Lemma 4.B.3. Let pL N q ‹ be the adjoint of L N in L 2 pν N p ρ q, then pL N q ‹ is given for any cylinder function g on E N by :

pL N q ‹ gpηq " ÿ xPT d N # pα 1 ρ 1 ρ 0 gpη 1 x q ´αgpηq `α2 ρ 2 ρ 0 gpη 2 x q ´rgpηqq1 tηpxq"0u `pα ρ 0 ρ 1 gpη 0 x q ´α1 gpηq `α2 ρ 3 ρ 1 gpη 3 x q ´rgpηqq1 tηpxq"1u
`pr ρ 0 ρ 2 gpη 0 x q ´α2 gpηq `α1

ρ 3 ρ 2 gpη 3 x q ´γgpηqq1 tηpxq"2u `pr ρ 1 ρ 3 gpη 1 x q ´α2 gpηq `γ ρ 2 ρ 3 gpη 2 x q ´α1 gpηqq1 tηpxq"3u + ": ÿ xPT d N 3 ÿ i"0
c ‹ px, η, iqrgpη i q ´gpηqs 109 Chapter 4. Hydrodynamic limit on the torus Proof.

ż gpηqL N f pηqdν N p ρ pηq " ÿ xPT d N ż # ´α`f pη 1 x q ´f pηq ˘`r `f pη 2 x q ´f pηq ˘¯1 tηpxq"0u
`´α 1 `f pη 0 x q ´f pηq ˘`r `f pη 3 x q ´f pηq ˘¯1 tηpxq"1u `´α 2 `f pη 0 x q ´f pηq ˘`γ `f pη 3 x q ´f pηq ˘¯1 tηpxq"2u `´α 2 `f pη 1 x q ´f pηq ˘`α 1 `f pη 2 x q ´f pηq ˘¯1 tηpxq"3u

+ ¨gpηqdν N p ρ pηq " ÿ xPT d N ż # ´α1 ρ 1 ρ 0 gpη 1 x q ´αgpηq `α2 ρ 2 ρ 0 gpη 2 x q ´rgpηq ¯1tηpxq"0u `´α ρ 0 ρ 1 gpη 0 x q ´α1 gpηq `α2 ρ 3 ρ 1 gpη 3 x q ´rgpηq ¯1tηpxq"1u `´r ρ 0 ρ 2 gpη 0 x q ´α2 gpηq `α1 ρ 3 ρ 2 gpη 3 x q ´γgpηq ¯1tηpxq"2u `´r ρ 1 ρ 3 gpη 1 x q ´α2 gpηq `γ ρ 2 ρ 3 gpη 2 x q ´α1 gpηq ¯1tηpxq"3u + ¨f pηqdν N p ρ pηq " ż f pηqpL N q ‹ gpηqdν N p ρ pηq

4.C Quadratic variations computations

We prove in this section computations of the quadratic variation (4.3.11) of the martingale M N,i t defined in 4.3.1, for i " 1, 2, 3.

Lemma 4.C.1. xM N,i y t " N 2 2N 2d ż t 0 ÿ xPT d N ÿ z‰x,}z´x}"1
´Gi pz{N q ´Gi px{N q ¯2´η

i,s pzq ´ηi,s pxq ¯2ds

`1 N 2d ż t 0 ÿ xPT d N G 2 i px{N q ´1 ´2η i,s pxq ¯LR N η i,s pxqds (4.C.1)
Proof. The quadratic variation of M N,i t is given, for any function p

G P CpT d ; R 3 q, by xM N,i y t " ż t 0 # L N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL N xπ N,i s , G i y + ds
We shall prove the two following equalities :

N 2 ż t 0 # L D N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL D N xπ N,i s , G i y + ds 110 4.C. Quadratic variations computations " N 2 2N 2d ż t 0 ÿ xPT d N ÿ z‰x,}z´x}"1
´Gi pz{N q ´Gi px{N q ¯2´η

i,s pzq ´ηi,s pxq ¯2ds (4.C.2)

ż t 0 ! L R N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL R N xπ N,i s , G i y ) ds " 1 N 2d ż t 0 ÿ xPT d N G 2 i px{N q ´1 ´2η i,s pxq ¯LR N η i,s pxqds (4.C.3)
Let us prove first (4.C.2).

L D N xπ N,i s , G i y 2 " 1 N 2d ÿ xPT d N G 2 i px{N qL N η i,s pxq `1 N 2d ÿ x,yPT d N x‰y,}x´y}ą1 G i px{N qG i py{N q ´ηi,s pxqL D N η i,s pyq `ηi,s pyqL D N η i,s pxq 1 N 2d ÿ x,yPT d N x‰y,}x´y}"1 G i px{N qG i py{N q # ÿ zPT d N z‰y,|z´x|"1
´ηi,s pzqη i,s pyq ´ηi,s pxqη i,s pyq ÿ

uPT d N u‰y,}u´x}"1
´ηi,s puqη i,s pxq ´ηi,s pxqη i,s pyq ¯+ and

´2xπ N,i s , G i yL D N xπ N,i s , G i y " ´2 N 2d ÿ xPT d N G 2 i px{N qη i,s pxqL D N η i,s pxq ´1 N 2d ÿ x,yPT d N x‰y G i px{N qG i py{N q ¨# ÿ zPT d N z‰y,}z´x}"1
´ηi,s pzqη i,s pyq ´ηi,s pxqη i,s pyq ηi,s

pyq ´ηi,s pyqη i,s pxq `ÿ uPT d N u‰y,}u´x}"1
´ηi,s puqη i,s pxq ´ηi,s pxqη i,s pyq ¯`η i,s pxq ´ηi,s pyqη i,s pxq

+ ´1 N 2d ÿ x,yPT d N x‰y,}x´y}ą1 G i px{N qG i py{N q ´ηi,s pxqL D N η i,s pyq `ηi,s pyqL D N η i,s pxq so that, N 2 ż t 0 # L D N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL D N xπ N,i s , G i y + ds " N 2 N 2d ż t 0 ÿ xPT d N ÿ z:}z´x}"1
G 2 i px{N q ´ηi,s pzq ´ηi,s pxq ¯ds

´N 2 N 2d ż t 0 ÿ xPT d N ÿ z:}z´x}"1
G 2 i px{N q ´2η i,s pzqη i,s pxq ´ηi,s pxq ¯ds

´N 2 N 2d ż t 0 ÿ xPT d N ÿ z‰x,}z´x}"1
G i px{N qG i pz{N q ´ηi,s pzq ´2η i,s pzqη i,s pxq `ηi,s pxq ¯ds

" N 2 2N 2d ż t 0 ÿ xPT d N ÿ z‰x,}z´x}"1
´Gi pz{N q ´Gi px{N q ¯2´η

i,s pzq ´ηi,s pxq ¯2ds

On the other hand,

ż t 0 ! L R N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL R N xπ N,i s , G i y ) ds " 1 N 2d ż t 0 ÿ xPT d N G 2 i px{N qL R N η i,s pxqds (4.C.4) `1 N 2d ż t 0 ÿ y‰x G i px{N qG i py{N q ´ηi,s pxqL R N η i,s pyq `ηi,s pyqL R N η i,s pxq ¯ds ´2 N 2d ż t 0 ÿ xPT d N G 2 i px{N qη i,s pxqL R N η i,s pxqds ´1 N 2d ż t 0 ÿ y‰x G i px{N qG i py{N q ´ηi,s pxqL R N η i,s pyq `ηi,s pyqL R N η i,s pxq ¯ds " 1 N 2d ż t 0 ÿ xPT d N G 2 i px{N q ´1 ´2η i,s pxq ¯LR N η i,s pxqds (4.C.5)
Using 4.3.2, we have for each i " 1, 2, 3 :

´1 ´2η 1 pxq ¯LR N η 1 pxq " L R N η 1 pxq `2! pr `1qη 1 pxq `´λ 1 ÿ y:}y´x}"1 η 1 pyq `λ2 ÿ y:}y´x}"1 η 3 pyq ¯η1 pxq ) " ´λ1 ÿ y:}y´x}"1 η 1 pyq `λ2 ÿ y:}y´x}"1 η 3 pyq ¯p1 `η1 pxq ´η2 pxq ´η3 pxqq `η3 pxq `pr `1qη 1 pxq ´1 ´2η 2 pxq ¯LR N η 2 pxq " L R N η 2,s pxq `2η 2 pxq " rη 0 pxq `η3 pxq `η2 pxq ´1 ´2η 3 pxq ¯LR N η 3 pxq " L R N η 3 pxq `4η 3 pxq " rη 1 pxq `2η 3 pxq. 112

4.D. Topology of the Skorohod space

Gathering all these estimates, one has

ż t 0 ! L R N xπ N,i s , G i y 2 ´2xπ N,i s , G i yL R N xπ N,i s , G i y ) ds " ´λ1 ÿ y:}y´x}"1 η 1 pyq `λ2 ÿ y:}y´x}"1 η 3 pyq ¯pη 0 pxq `2η 1 pxqq
`rη 0 pxq `p2r `1qη 1 pxq `η2 pxq `4η 3 pxq (4.C.6)

4.D Topology of the Skorohod space

We summarize here some useful tips concerning the Skorohod space, see [7, Chapter 3] for further details.

Fix T ą 0. Recall Dpr0, T s, pM 1 `q3 q stands for the set of right-continuous with left limits trajectories with values in pM 1 `q3 , endowed with the Skorohod topology and equipped with its Borel σ´algebra.

Define a metric on M 1 `by introducing for every dense sequence of continuous functions tf k , k ě 1u on T d the distance δp¨, ¨q by δpµ, νq "

8 ÿ k"1 1 2 k |xµ, f k y ´xν, f k y| 1 `|xµ, f k y ´xν, f k y| (4.D.1)
The space M `is complete with respect to the endowed weak topology, and any set

A Ă M 1 `is relatively compact in M 1 `if and only if sup µPA xµ, 1y ă 8
Let E be a polish space equipped with the metric δp¨, ¨q and consider a sequence of probability measures pP N q N in Dpr0, T s, Eq. Let Λ be the set of increasing continuous functions on r0, T s. Define, for all λ P Λ, }λ} " sup The space Dpr0, T s, Eq equipped with the metric δp¨, ¨q is polish.

To extend Ascoli's theorem to the space Dpr0, T s, Eq, one introduces the modulus of continuity :

ω µ pγq " sup |t´s|ďγ δpµ s , µ t q (4.D.2)

A continuous function on r0, T s is uniformly continuous. To get something similar for functions in the Skorohod space, introduce Lemma 4.D.1. For all µ P Dpr0, T s, Eq and ǫ ą 0, there exists a sequence of times tt i u 0ďiďr such that 0 " t 0 ă t 1 ă ... ă t r " T and ω µ pt i ´ti´1 q ą ǫ, i " 1, ..., r.

For such a sequence tt i u 0ďiďr , one can define the modified modulus of continuity by P N pµ t P Kpt, ǫq c q ď ǫ.

ω
(2) For any ǫ ą 0, lim where T T stands for the set of stopping times bounded from above by T .

For the space M 1 `endowed with the weak topology, to prove the relative compactness for a sequence of measures pQ N µ N , N ě 1q defined in Dpr0, T s, M 1 `q, it is enough to check Prohorov's theorem 4.D.1 for real-valued processes by projecting the empirical measures with functions of a dense countable set of CpT d ; Rq : Proposition 4.D.4. Let tg k , k ě 1u be a dense countable set in CpT d q with g 1 " 1. A sequence of probability measures pQ N µ N q N ě1 is relatively compact in Dpr0, T s, M 1 `q if for any positive integer k, the sequence pQ N µ N g ´1 k q N ě1 in Dpr0, T s, Rq defined by

Q N µ N g ´1 k pAq " Q N µ N pπ N,i : xπ N,i , g k y P Aq is relatively compact.
We consider a generalized contact process represented by a two species process evolving either in a bounded domain in contact with particles reservoirs at different densities, or in Z d . In both cases we study the law of large numbers for current and densities.

Introduction

In this chapter, we consider a generalized contact process describing the evolution on a lattice of three types of populations labeled respectively by 1, 2 and 3. This process was introduced in [START_REF] Kuoch | Phase transition for a contact process with random slowdowns[END_REF] (in preparation), see Chapters 2 and 3, to model the sterile insect technique, developed by E. Knipling and R. Bushland (see for instance [START_REF] Knipling | Possibilities of insect control or eradication through the use of sexually sterile males[END_REF][START_REF] Dyck | Sterile insect technique : Principles and Practice in area-wide integrated pest management[END_REF]) in the fifties to control the New World screw worm, a serious threat to warm-blooded animals. This pest has been eradicated from the USA and Mexico only in recent decades. The technique works as follows : Screw worms are reared in captivity and exposed to Gamma rays. The male screw worms become sterile. If a sufficient number of sterile males are released in the wild then enough female screw worms are mated by sterile males so that the number of offspring will decrease generation after generation. This technique is well suited for screw worms, because female apparently mate only once in their lifetime, but is also being tried for a large variety of pests, including a current project to fight dengue in Brazil.

The particle system pη t q tě0 we look at has state space t0, 1, 2, 3u S , where S Ă Z d , typically d " 2. Each site of S is either empty (we say it is in state 0), occupied by wild screw worms only (state 1), by sterile screw worms only (state 2), or by wild and sterile screw worms together (state 3). We keep track only of the presence or not of the type of the male screw worms (and not of their number), and we assume that enough female are around as not to limit mating. A site gets sterile males at rate r independently of everything else (this corresponds to the artificial introduction of sterile males). The birth rate is 0 at sites in state 2, λ 1 at sites in state 1, and λ 2 at sites in state 3. We assume that λ 2 ă λ 1 to reflect the fact that at sites in state 3 the fertility is decreased. Deaths for each population occur at all sites at rate 1, being mutually independent.

If η denotes a current configuration, the transitional mechanism for the generalized contact dynamics at a site x can be summarized as follows :

0 Ñ 1 at rate λ 1 n 1 px, ηq `λ2 n 3 px, ηq 1 Ñ 0 at rate 1 0 Ñ 2 at rate r 2 Ñ 0 at rate 1 1 Ñ 3 at rate r 3 Ñ 1 at rate 1 2 Ñ 3 at rate λ 1 n 1 px, ηq `λ2 n 3 px, ηq 3 Ñ 2 at rate 1 (5.1.1)

where n i px, ηq is the number of nearest neighbors of x in state i for i " 1, 3. This dynamics has been studied in S " Z d in [START_REF] Kuoch | Phase transition for a contact process with random slowdowns[END_REF], see Chapter 2, where a phase transition

Introduction

in r was exhibited : Assuming that λ 2 ď λ c ă λ 1 , where λ c denotes the critical value of the d-dimensional basic contact process, there exists a critical value r c such that the populations in states 1 and 3 survive for r ă r c , and die out for r ě r c .

Our goal in the present chapter is to add to the previous contact dynamics displacements of populations within S infinite volume case, as well, in the finite volume case, as departures from S and immigrations to S. We are interested in the evolution of the empirical densities of the 3 types of populations, for which we establish hydrodynamic limits. The limiting equations are given by a system of non-linear reaction-diffusion equations, with additionally Dirichlet boundary conditions.

More precisely, denote by T d´1

N the pd´1q-dimensional microscopic torus of length N , where N is a scaling parameter. The non-conservative system that we consider evolves either in a bounded cylinder Λ N " t´N, ¨¨¨, N u ˆTd´1 N or in Z d . The cylinder Λ N has length 2N `1 along the axis of direction e 1 , where pe 1 , . . . , e d q denotes the canonical basis of R d .

In the bulk of Λ N , resp. in Z d , particles evolve according to the superposition of an exchange dynamics representing the displacements of the populations in different states, and the above generalized contact process. In Λ N , the movements of populations at the boundary Γ N of the domain Λ N are modelled thanks to reservoirs from which populations in different states are created or annihilated.

The exchange of the occupation variable ηpxq in any site x with the one of a nearest neighbour site is performed with rate 1. This exchange dynamics satisfies a detailed balance condition with respect to a family of Gibbs measures, parametrized by the so-called chemical potential p ρ " pρ 1 , ρ 2 , ρ 3 q P R 3 . In the finite volume case, the reservoirs are modelled by a reversible generalized contact process with fixed density. More precisely, for a fixed smooth vector valued function p bp¨q " pb 1 p¨q, b 2 p¨q, b 3 p¨qq defined on the boundary of the domain, the rates of this contact process are chosen so that a Gibbs measure of varying chemical potential p bp¨q is reversible for it.

To deal with infinite volume, we establish bounds on the entropy production and on the Dirichlet forms valid for a boundary driven version of our process on Λ 8 N " t´N, ..., N u ˆZd´1 , hence on Z d . We also establish uniqueness of the weak solution to the system of equations corresponding to the boundary driven case in infinite volume. The same method gives uniqueness on Z d . In Section 5.2, we detail our model, and state our results, namely on the specific entropy (Theorem 5.2.1), the hydrodynamic limit of the boundary driven generalized process (Theorem 5.2.2), the hydrodynamic limit of the generalized process in Z d (Theorem 5.2.3), a law of large numbers for currents (Proposition 5.2.1), uniqueness results for the equations in Subsection 5.2.6.

In Section 5.3, we prove Theorem 5.2.1 , in Section 5.4 we prove Theorem 5.2.2, in Section 5.5 we prove Proposition 5.2.1, in Section 5.6 we prove Theorem 5.2.3, results on uniqueness of solutions are proved in Section 5.7 and finally Appendices 5.A-5.B-5.C contain useful computations.

Notation and Results

The model

Instead of studying the three different values ηpxq " 1, 2, 3 considered above, we introduce another interpretation for the model. The configuration space is now p Σ N :" `t0, 1u ˆt0, 1u ˘ΛN or p Σ :" pt0, 1u ˆt0, 1uq Z d ; elements of p Σ N (resp. p Σ) are denoted by pξ, ωq. The correspondence with pη t q tě0 is given by the following relations : ηpxq " 0 ðñ p1 ´ξpxqqp1 ´ωpxqq " 1 , ηpxq " 1 ðñ ξpxqp1 ´ωpxqq " 1 , ηpxq " 2 ðñ p1 ´ξpxqqωpxq " 1 , ηpxq " 3 ðñ ξpxqωpxq " 1 .

(5.2.1)

In other words, ξ-particles represent the wild screw worms, while ω-particles represent the sterile ones. On a site x, ξpxq " 1 if wild screw worms are present on x, and ωpxq " 1 if sterile screw worms are present on x. Both can be present, giving the state 3 for ηpxq or only one of them, giving the states 1 or 2 for ηpxq.

The boundary driven generalized contact process with exchange of particles is the Markov process on p Σ N whose generator L N :" L λ 1 ,λ 2 ,r, p b,N can be decomposed as

L N :" N 2 L N `LN `N 2 L p b,N , (5.2.2)
where L N is the generator of exchanges of particles, L N the generator of the generalized contact process, and L p b,N the generator of the boundary dynamics. We now detail both dynamics and their properties.

For the exchange dynamics, the action of L N on cylinder functions f : p

Σ N Ñ R is L N f pξ, ωq " ÿ x,yPΛ N }x´y}"1
rf pξ x,y , ω x,y q ´f pξ, ωqs , (

where for any ξ P Σ N :" t0, 1u Λ N , ξ x,y is the configuration obtained from ξ P Σ N , by exchanging the occupation variables ξpxq and ξpyq, i.e. pξ x,y qpzq :"

$ ' & ' % ξpyq if z " x , ξpxq if z " y , ξpzq if z ‰ x, y .
118

Notation and Results

Note that, since pξ, ωq P p Σ N , these exchanges can be interpreted as jumps between sites x to y for ξ-particles and ω-particles, which do not influence each other.

To exhibit invariant measures for L N , for any x P Λ N , according to (5.2.1), we define

$ ' & ' %
η 1 pxq " ξpxqp1 ´ωpxqq " 1 tηpxq"1u , η 2 pxq " p1 ´ξpxqqωpxq " 1 tηpxq"2u , η 3 pxq " ξpxqωpxq " 1 tηpxq"3u .

(5.2.4)

By a misuse of language, when η i pxq " 1 for i " 1, 2, 3, we say that there is a particle of type i at x. The invariant measures will be product measures parametrized by three chemical potentials, since the exchange dynamics conserves the three quantities ř xPΛ N η i pxq, 1 ď i ď 3. It is convenient to complete (5.2.4) by defining, for x P Λ N , η 0 pxq " p1 ´ξpxqqp1 ´ωpxqq " 1 tηpxq"0u " 1 ´η1 pxq ´η2 pxq ´η3 pxq .

(5.2.5)

We denote by Λ the macroscopic open bounded cylinder p´1, 1q ˆTd´1 where T k is the k-dimensional torus r0, 1q k . For a vector-valued function p m " pm 1 , m 2 , m 3 q : Λ Ñ R

ψ i p p mq " E νN x m " η i p0q ‰ .
Observe that the function p ψ defined on p0, `8q 3 by p ψp p mq " pψ 1 p p mq, ψ 2 p p mq, ψ 3 p p mqq is a bijection from p0, `8q 3 to p0, 1q 3 . We will therefore do a change of parameter : For every p ρ " pρ 1 , ρ 2 , ρ 3 q P p0, 1q 3 , we denote by ν N p ρ the product measure such that

ρ i " E ν N p ρ " η i p0q ‰ , i " 1, 2, 3 . (5.2.8)
From now on, we work with the representation ν N p ρp¨q of the measure νN p mp¨q .

Notation and Results

Λ N described by the generator L p b,N defined by

L p b,N f pξ, ωq " ÿ xPΓ N c x `p bpx{N q, ξ, σ x ω ˘"f pξ, σ x ωq ´f pξ, ωq ı `ÿ xPΓ N c x `p bpx{N q, σ x ξ, ω ˘"f pσ x ξ, ωq ´f pξ, ωq ‰ `ÿ xPΓ N c x `p bpx{N q, σ x ξ, σ x ω ˘"f pσ x ξ, σ x ωq ´f pξ, ωq ‰ ,
(5.2.12)

where the rates c x `p bpx{N q, ξ, ω ˘are given for x P Γ N and pξ, ωq P p Σ N by

c x `p bpx{N q, ξ, ω ˘" 3 ÿ i"0 b i px{N qη i pxq , (5.2.13)
where b 0 px{N q " 1 ´ř3 i"1 b i px{N q and η i pxq, i " 0, 1, 2, 3 are defined in (5.2.4)-(5.2.5). Using Lemma 5.A.2, note that the measure ν N p θ is reversible with respect to the generator L p b,N .

As we deal with the process in infinite volume, define the generator in Z d by omitting the subscript N in L N and L N to denote the sums are carried over Z d . In infinite volume, the process has generator : :

L " N 2 L `L (5.2.14)
Notice that in view of the diffusive scaling limit, the generator L N (resp. L) has been speeded up by N 2 . We denote by pξ t , ω t q tě0 the Markov process on p Σ N with generator L N (resp. on p Σ with generator L) and by P N, p b µ (resp. P N µ q its distribution if the initial configuration is distributed according to µ. Note that P N, p b µ (resp. P N µ ) is a probability measure on the path space DpR `, p Σ N q (resp. DpR `, p Σq), which we consider endowed with the Skorohod topology and the corresponding Borel σ-algebra. Expectation with respect to P N, p b µ is denoted by E N, p b µ (resp. E N µ ). We denote by M the space of finite signed measures either on Λ or R d , endowed with the weak topology. For a finite signed measure m and a continuous function F on Λ or R d , we let xm, F y be the integral of F with respect to m. For each configuration pξ, ωq, denote by p π N " p π N pξ, ωq " pπ N,1 , π N,2 , π N,3 q P M 3 , where for i " 1, 2, 3, the positive measure π N,i is obtained by assigning mass N ´d to each particle of type η i :

π N,i " N ´d ÿ x η i pxq δ x{N ,
where δ u is the Dirac measure concentrated on u, and the sum is carried either on Λ N or Z d . For any continuous function p G " pG 1 , G 2 , G 3 q, the integral of p G with respect to p π N , also denoted by xp π N , p Gy, is given by xp π N , p Gy "

3 ÿ i"1
xπ N,i , G i y .

Notation and Results

For a positive integer n, we denote by s n pµ n |ν N p θ,n q the relative entropy of µ n with respect to ν N p θ,n defined by

s n pµ n |ν N p θ,n q " sup U PC b p p Σ N,n q ! ż U pξ,
L x,y N , L N,n " ÿ xPΛ N,n L x N,n , L p b,N,n " ÿ xPΛ N,n XΓ N L x N . (5.2.22)
Here for a bond px, yq P Λ 2 N,n , L x,y N stands for the piece of generator associated to the exchange of particles between the two sites x and y, L x N,n corresponds to the flips at site x P Λ N,n for the generalized contact process restricted to Λ N,n , and for x P Γ N , L x N stands for the flips at site x due to the boundary dynamics. We have for x P Λ N,n , where C 1,2 0 `r0, T s ˆΛ; R 3 ˘is the space of functions from r0, T s ˆΛ to R 3 twice continuously differentiable in Λ with continuous time derivative and vanishing at the boundary Γ of Λ. Here n=pn 1 , . . . , n d q stands for the outward unit normal vector to the boundary surface Γ and dS for an element of surface on Γ. For G, H P L 2 pΛq, xGp¨q, Hp¨qy is the usual scalar product of L 2 pΛq : xGp¨q, Hp¨qy "

L x N,
3 ÿ i"1 ż Λ G i puqH i puqdu (B3) p ρp0, uq " p γpuq a.e.
Let M 1 `be the subset of M of all positive measures absolutely continuous with respect to the Lebesgue measure with positive density bounded by 1 :

M 1
`" π P M : πpduq " ρpuqdu and 0 ď ρpuq ď 1 a.e.

( .

Let Dpr0, T s, pM 1 `q3 q be the set of right continuous with left limits trajectories with values in pM 1 `q3 , endowed with the Skorohod topology and equipped with its Borel σá lgebra. For a probability measure µ on p Σ N denote by pξ t , ω t q tPr0,T s the Markov process with generator L N with initial distribution µ. Denote by P N, p b µ the probability measure on the path space Dpr0, T s, p Σ N q corresponding to the Markov process pξ t , ω t q tPr0,T s and by E N, p b µ the expectation with respect to P N, p b µ . We denote by p π N the map from Dpr0, T s, p Σ N q to Dpr0, T s, pM 1 `q3 q defined by p π N pξ ¨, ω ¨qt " p π N pξ t , ω t q and by Q N, p b µ " P N, p b µ ˝pp π N q ´1 the law of the process `p π N pξ t , ω t q ˘tPr0,Ts . We shall prove :

Notation and Results

Theorem 5.2.2. Let pµ N q N ě1 be a sequence of initial probability measures such that µ N is a probability measure on p Σ N for each N . The sequence of probability measures pQ N, p b µ N q N ě1 is weakly relatively compact and all its converging subsequences converge to some limit Q p b,˚t hat is concentrated on absolutely continuous paths whose densities p ρ P Cpr0, T s, pM 1 `q3 q satisfy (B1) and (B2). Moreover, if for any δ ą 0 and for any function p

G P C 0 pΛ; R 3 q lim N Ñ8 µ N !ˇˇˇx p π N pξ
) " 0 .
We will prove Theorem 5.2.2 in Section 5.4.

Currents.

In this subsection, we will study the evolution of the empirical currents, namely the conservative current (cf. [START_REF] Bertini | Large deviations of the empirical current in interacting particle systems[END_REF]) and the non-conservative current one (cf. [START_REF] Bodineau | Large deviations of the empirical currents for a boundary-driven reaction diffusion model[END_REF]).

For t ě 0, 1 ď i ď 3, 1 ď j ď d such that x, x `ej P Λ N , denote by J

x,x`e j t pη i q the total number of particles of type i that jumped from x to x `ej before time t and by W

x,x`e j t pη i q " J x,x`e j t pη i q ´Jx`e j ,x t pη i q the conservative current of particles of type i across the bond tx, x `ej u before time t. The corresponding conservative empirical measure W N t is the product finite signed measure on Λ N defined as W N t pη i q " pW N 1,t pη i q, . . . , W N d,t pη i qq P M d " tMpΛqu d , where for 1 ď j ď d, 1 ď i ď 3, W N j,t pη i q " N ´pd`1q ÿ

x,x`e j PΛ N W

x,x`e j t pη i qδ x{N .

For a continuous vector field G " pG 1 , . . . , G d q P C c pΛ; R d q the integral of G with respect to W N t pη i q, also denoted by xW N t pη i q, Gy, is given by

xW N t pη i q, Gy " d ÿ j"1
xW N j,t pη i q, G j y .

(5.2.35)

Finally, we introduce the signed measure y W N t pp ηq " pW N t pη 1 q, W N t pη 2 q, W N t pη 3 qq P pM d q 3 and for p G " pG 1 , G 2 , G 3 q P pC c pΛ; R d qq 3 the notation

x y W N t , p Gy "

3 ÿ i"1
xW N t pη i q, G i y .

(IB2) For every function p Gpt, uq " p G t puq " pG We prove these results in Section 5.7.

Proof of the specific entropy (Theorem 5.2.1)

In this section we prove Theorem 5.2.1 and Lemma 5.2.1.

Proof of Theorem 5.2.1. Integrate the expression (5.2.30) from 0 to t and use 5.2.29.

Proof of Lemma 5.2.1. For a measure µ n on p Σ N,n , denote by f t n the density of µ n ptq with respect to ν N p θ,n . For any subset A Ă Λ and any function f P L 1 pν N p θ q, denote by xf y A the function on pt0, 1u ˆt0, 1uq ΛzA obtained by integrating f with respect to ν N p θ over the coordinates tpξpxq, ωpxqq, x P Au. In the case where A " Λ N,n`1 zΛ N,n , we simplify the expectation by xf y n`1 . Following the Kolmogorov forward equation, one has

B t f t n " xL N,n`1 f t n`1 y n`1 , (5.3.1)
where L N,n stands for the adjoint operator of L N,n in L 2 pν N p θ,n q. From the convexity of the entropy (5.2.28) and by (5.3.1), Denote the last three integrals by Ω 1 , Ω 2 and Ω 3 respectively. Recall that ν N p θ,n stands for the measure associated to a smooth profile p θ : Λ 8 Ñ p0, 1q 3 such that p θ ˇˇ8

B t s n pµ n |ν N p θ,n q " B t ż f t n log f t n dν N p θp¨q,n " ż log f t n L N,n`1 f t n`1 dν N p θ,n`1 " N 2 ż log f t n L N,n`1 f t n`1 dν N p θ,n`1 `ż log f t n L N,n`1 f t n`1 dν N p θ,n`1 `N 2 ż log f t n L N,n`1 f t n`1 dν N p θ,n`1 . ( 5
Γ " p bp¨q. We now derive bounds on Ω 1 , Ω 2 and Ω 3 .

Bound on Ω 1 . We shall decompose the generator L N,n`1 into a part associated to exchanges within Λ N,n and a part associated to exchanges at the boundaries, that is, denoting Λ

c N,n " Λ N zΛ N,n , Ω 1 " N 2 ż f t n`1 L N,n`1 plog f t n qdν N p θ,n`1 " N 2 ÿ px,yqPΛ N,n ˆΛN,n }x´y}"1 ż f t n`1 L x,y N plog f t n qdν N p θ,n`1 `N 2 ÿ px,yqPΛ N,n ˆΛc N,n }x´y}"1 ż f t n`1 L x,y N plog f t n qdν N p θ,n`1 " N 2 ÿ px,yqPΛ N,n ˆΛN,n }x´y}"1 Ω p1q 1 px, yq (5.3.3)
" ´pD 0 n q x,y pµ n |ν p θ,n q `ż L x,y N f t n pξ, ωqdν N p θ,n pξ, ωq (5.3.5)
where we used the fact that for any a, b ą 0, aplog b ´log aq ď ´`? b ´?a ˘2 `pb ´aq.

(5.3.6)

Now, by a change of variables pα, βq " pξ x,y , ω x,y q, using Lemma 5.A. For the part (5.3.4) associated to the boundaries, we shall write for each pair px, yq P Λ N,n ˆΛc N,n with }x ´y} " 1,

L x,y N " ÿ 0ďi "jď3 L x,y iØj (5.3.9)
where L x,y iØj stands for the exchange of values i and j at the boundaries.

L x,y iØj f pξ, ωq " η i pxqη j pyq ´f pξ x,y , ω x,y q ´f pξ, ωq ηj pxqη i pyq ´f pξ x,y , ω x,y q ´f pξ, ωq ¯.

( Let us detail the computation for i " 1 and j " 3, the other values would be deduced in a similar way. In this case, by a change of variables pξ 1 , ω 1 q " pξ x,y , ω x,y q in the integral corresponding to i " 1, j " 3 in the second term of the r.h.s. (5.3.11) using Lemma 5.A.1, and noticing for the integral corresponding to i " 1, j " 3 in the first term of the r.h.s. (5.3.11) that ξ x,y " ξ since i " 1, j " 3, we have

ż f t n`1 L x,y 1Ø3 log f t n pξ, ωqdν N p θ,n`1 pξ, ωq " ż η 1 pxqη 3 pyqf t n`1 pξ, ωq log f t n pξ, ω x,y q f t n pξ, ωq dν N p θ,n`1 pξ, ωq
`ż η 1 pxqη 3 pyq exp ´pϑ 3 py{N q ´ϑ3 px{N qq ´pϑ 1 py{N q ´ϑ1 px{N qq f t n`1 pξ x,y , ω x,y q log f pξ, ωq f pξ x,y , ω

" N A 1 N n`N ÿ m"n`1 ż E 1 p1,3qYE 2 p1,3q η 1 pxq ´bxη 3 pyqf t m pξ x,y , ω x,y qy Λ N,m zΛ N,n ´bxη 3 pyqf t m pξ, ωqy Λ N,m zΛ N,n ¯2dν N p θ,m pξ, ωq ď 1 A n`N ÿ m"n`1 ż E 1 p1,3qYE 2 p1,3q η 1 pxq A η 3 pyq ´af t m pξ x,y , ω x,y q ´af t m pξ, ωq ¯2E Λ N,m zΛ N,n dν N p θ,m pξ, ωq " 1 A n`N ÿ m"n`1 ż E 1 p1,3qYE 2 p1,3q
η 1 pxqη 3 pyq ´af t m pξ x,y , ω x,y q ´af t m pξ, ωq ¯2dν N p θ,m pξ, ωq (5.3.19)

ď 1 A n`N ÿ m"n`1 ż E 1 p1,3qYE 2 p1,3q
´af t m pξ x,y , ω x,y q ´af t m pξ, ωq ¯2dν N p θ,m pξ, ωq (5.3.20) which is of order OpN q. Now, to bound the second integral of the last r.h.s. in (5.3.18), we separate the integrations on E 1 p1, 3q and on E 2 p1, 3q. We first look at the integral on E 1 p1, 3q, to get The second term on the r.h.s. is of order OpN n d´1 q since the rates β N,n p., .q are bounded. is reversible with respect to the generator L p b,N,n , using inequality (5.3.6),

A N ż E 1 p1,3q η 1 pxq ´bxF p1q 1,3 pξ, ωqy n`1 `bxF p2q
Ω 3 " N 2 ÿ xPΛ N,n XΓ N ż f t n`1 L x N log f t n dν N p θ,n`1 " N 2 ÿ xPΛ N,n XΓ N ż xf t n`1 pξ, ωqy n`1 L x N log f t n dν N p θ,n ď ´N 2 D p b n pµ n ptq|ν N p θ,n q `N 2 ÿ xPΛ N,n XΓ N ż L x N f t n dν N p θ,n " ´N 2 D p b n pµ n ptq|ν N p θ,n q (5.3.26)
It is for the last equality that we needed ν N p θ,n to be reversible with respect to the generator L p b,N,n .

The estimate (5.3.26), together with (5.3.23) and (5.3.25), gives us

B t s n pµ n ptq|ν N p θ,n q ď ´N 2 D 0 n pµ n ptq|ν N p θp¨q,n q `pK 2 `C2 1 AqN n d´1 ´N 2 D p b n pµ n ptq|ν N p θ,n q
Therefore, multiplying by expp´n{N q and summing over n P N, one gets (5.2.30) with A 0 " 1 and A 1 " pK 2 `AC 2 1 q.

Hydrodynamics in a bounded domain

Let a ą 0 be decreasing to zero after ǫ, and a smooth function p θ a " pθ a,1 , θ a,2 , θ a,3 q : Λ Ñ p0, 1q 3 , equal in Λ p1´aqN " r´1 `a, 1 ´as ˆTd´1 N to some constant, say p α, and to p b at the boundaries. As sup k,pξ,ωq,x τ x V k pξ, ωq ă 8, we have 1

N d ÿ xPΛ N zΛ p1´aqN ż T 0 τ x V ǫN pξ s , ω s qds ď aT C 0 , for some positive constant C 0 . Therefore, E N, p b µ N ´1 N d ÿ xPΛ N ż T 0 τ x V ǫN pξ s , ω s qds ¯ď aT C 0 `EN, p b µ N ˜1 N d ÿ xPΛ p1´aqN ż T 0 τ x V ǫN pξ s , ω s qds ¸.
Denote by f T " T ´1 ż T 0 f s N ds, where f t N stands for the density of µ N ptq with respect to ν N p θa . Since Λ p1´aqN is finite, proceeding as in the proof of Theorem 5.2.1 for Ω 1 , there exists some positive constant C 1 paq such that the remaining expectation above is bounded by

T N d ż ÿ xPΛ p1´aqN τ x V ǫN pξ, ωq f T pξ, ωqdν N p θa pξ, ωq ´γT N 2´d D 0 N p f T q `γC 1 paq,
for all positive γ. Recall p θ a is equal to some constant p α within Λ p1´aqN . To prove (5.4.7), it thus remains to show that for every positive γ, a,

lim ǫÑ0 lim N Ñ8 sup f ˜1 N d ż ÿ xPΛ p1´aqN τ x V ǫN pξ, ωqf pξ, ωqdν N p α pξ, ωq ´γN 2´d D 0 N pf q ¸" 0,
where the supremum is carried over all densities f with respect to ν N p α such that D 0 N pf q ď CN d´2 . This result is a consequence of the one and two blocks estimates (5.4.2)-(5.4.3), for which we refer to Chapter 4 since we reduced ourselves to the interior of the domain. Conclude by letting γ Ñ 0, then, a Ñ 0. Now, let us prove the limit (5.4.8). Define W Ht i pξ t , ω t qpxq " B N e 1 H t px{N q ´ηi,t pxq ´bi px{N q ¯(5.4.11)

Decomposing the outward unit normal vector into both directions,

E N, p b µ N ´ˇˇż T 0 1 N d´1 ÿ xPΓ N
B N e 1 H i,s px{N qn 1 px{N q ´ηi,s pxq ´bi px{N q ¯ds ˇˇ¯.

ď E N, p b µ N ´ˇˇż T 0 1 N d´1 ÿ xPΓ Ǹ W Hs i pξ s , ω s qpxqds ˇˇ¯1 `EN, p b µ N ´ˇˇż T 0 1 N d´1 ÿ xPΓ Ń W Hs i pξ s , ω s qpxqds ˇˇ¯,
where Γ N " tpu 1 , . . . , u d q P ˘N ˆTd´1 N u. From now, consider the sum over Γ Ǹ as the proof will be similar for the negative part. By the entropy inequality, for any positive γ,

E N, p b µ N ´ˇˇż T 0 1 N d ÿ xPΓ Ǹ B N
e 1 H i,s px{N q ´ηi,s pxq ´bi px{N q ¯ds ˇˇ¯.

ď 1

γN d log E N, p b ν N p θ ˜exp ´γN d ˇˇż T 0 1 N d´1 ÿ xPΓ Ǹ W Hs i pξ s , ω s qpxqds ˇˇ¯`s N pµ N |ν N p θ q γN d
where s N pµ N |ν N p θ q was defined in (5.2.20). By (5.2.28), there exists some constant K 0 such that s N pµ N |ν N where the supremum is carried over all densities f with respect to ν N p ρ . By Lemma 5.C.1, .4.13) for some positive constant A 0 . We now consider the expression W Ht i pξ s , ω s qpxq between brackets in (5.4.12) and thanks to changes of variables given in Lemma 5.A.2, ż B N e 1 H i,t px{N q ´ηi pxq ´bi px{N q ¯f pξ, ωqdν N p θ pξ, ωq " ż B N e 1 H i,t px{N q ´ηi pxq ÿ j‰i b j px{N q ´bi px{N q ÿ j‰i η j px{N q ¯f pξ, ωqdν N p θ pξ, ωq (5.4.14)

xL N a f , a f y ď ´N 2 D p b N pf q `A0 N d . ( 5
We detail for instance the case i " 1, others follow the same way, this is equal to ż B N e 1 H 1,t px{N q ´η1 pxq ´b0 px{N q `b2 px{N q `b3 px{N q

b N q x pf q `C1 2AN }B N e 1 H 1,t px{N q 2 } 8
Now summing over tx P Γ N u and multiplying by N 1´d , (5.4.12) is bounded by

´AN 2´d 2 ´N 2´d γ ¯Dp b N pf q `C 2AN }B N e i H 1,t px{N q 2 } 8 `A0 γ
Choose A " 2{γ and conclude by letting tend γ Ñ 8, N Ñ 8.

Energy estimate.

We now deal with an energy estimate that allows us to exclude paths with infinite energy. For G P C

Q G pπ i q (5.4.15)
where

Q G pπ i q " d ř j"1 ż T 0 ż Λ π i t puqB e j G t puq dt du ´1 2 ż T 0 ż Λ G t puq 2 dt du.
Lemma 5.4.4. Fix a dense sequence pG ℓ q ℓě1 in C 8 c pr0, T s ˆΛ, Rq. For all i " 1, 2, 3, there exists a constant C 0 such that for any sequence tµ N : N ě 1u of probability measures on p Σ N , every k ě 1, lim

N Ñ8 E N, p b µ N ˜max 1ďℓďk ´ż T 0 `AG ℓ,s i pξ s , ω s q ´1 2N d ÿ xPΛ N G ℓ,s px{N q 2 ¯ds ¯¸ď C 0 . ( 5

.4.16)

where

A G ℓ,t i pξ t , ω t q :" N 1´d d ÿ j"1 ÿ x,
x`e j PΛ N pη i,t px `ej q ´ηi,t pxqqG ℓ,t px{N q.

Proof. By the entropy inequality, for all γ ą 0,

E N, p b µ N ´max 1ďℓďk ż T 0 A G ℓ,s i pξ s , ω s qds ď 1 γN d log E N, p b ν N p θ ˜exp ´max 1ďℓďk ´γN d ż T 0 A G ℓ,s i pξ s , ω s qds ¯¯¸`1 γN d s N pµ N |ν N p θ q,
where s N pµ N |ν N p θ q stands for the relative entropy of µ N with respect to ν N

N ´d log `ÿ 1ďℓďk a ℓ ˘ď max 1ďℓďk lim N N ´d log a ℓ , lim N Ñ8 E N, p b µ N ´max 1ďℓďk ż T 0 A G ℓ,s i pξ s , ω s qds ď max 1ďℓďk lim N Ñ8 1 γN d log E N, p b ν N p θ ˜exp ´γN d ż T 0 A G ℓ,s i pξ s , ω s qds ¯¸`C 0 γ .
By Feynman-Kac formula, 1

γN d log E N, p b ν N p θ ˜exp ´γN d ż T 0 A G ℓ,s i pξ s , ω s qds ¯ḑ ż T 0 sup f # ż A G ℓ
T 0 sup f ě0 # ż A G ℓ,s i pξ s , ω s q a f pξ, ωqdν N p θ pξ, ωq `1 γN d xL N a f , a f y + ds (5.4.18)
By Lemma 5.C.1, there exist positive constants K 0 and K 1 such that

N 2 xL N a f , a f y `xL N a f , a f y ď ´K0 N 2 D 0 N pf q `K1 N d . Therefore, 1 γN d log E N, p b ν N p θ ˜exp ´N d ż T 0 γA G ℓ,s i pξ s , ω s qds ¯ḑ ż T 0 sup f # ż A G ℓ,s i
To get rid of the second term, note that ż ´af pξ x,y , ω x,y q `af pξ, ωq ¯2dν N p θ pξ, ωq " ż f pξ x,y , ω x,y qdν N p θ pξ, ωq `ż f pξ, ωqdν N p θ pξ, ωq `ż a f pξ x,y , ω x,y q a f pξ, ωqdν N p θ pξ, ωq After a change of variable by Lemma 5.A.1, the first and second integrals are equal to a constant times the L 1 -norm of f . Use inequality (5.3.17) to divide the third integral into two similar terms. Then, since f is a density with respect to ν N p θ , for all positive A, (5.4.19) is bounded by

N 2´d D 0 N pf q `C1 A 1 N d ÿ xPΛ N G 2 ℓ,t px{N q
The expression between brackets in (5.4.17) is then bounded by

C 1 AN d ÿ xPΛ N G 2 ℓ,t px{N q
Choose 2C 1 " A to conclude.

µ N q N ě1 . Then, Q p b,˚´L2 pr0, T s, H 1 pΛq ¯" 1 (5.4.20) Proof. Fix 1 ď j ď d. Remark that lim N Ñ8 ż T 0 A Gt i pξ t , ω t qdt " d ÿ j"1 ż T 0 ż Λ B e j G ℓ,t puqπ i t puqdt du. If pQ N, p b µ N q N ě1 converges weakly to Q p b,˚, by Lemma 5.4.4, E p b,˚˜m ax 1ďℓďk ´ż T 0 ż Λ B e j G ℓ,s puqπ i s puqdu ds ´1 2 ż T 0 ż Λ G ℓ,s puq 2 du ds ¯¸ď C 0 .
Since pG ℓ q ℓě1 is dense in C 8 c pr0, T s ˆΛ; Rq, taking the limit as k goes to infinity, one has by monotone convergence theorem,

E p b,˚˜s up GPC 8 c pr0,T sˆΛ;Rq ´ż T 0 ż Λ B e j G s puqπ i s puqdu ds ´1 2 ż T 0 ż T d G s puq 2 du ds ¯¸ď C 0 .
Therefore, for all i, there exists some positive constant C so that for any smooth function

G P C 8 c pr0, T s ˆΛ, Rq, under Q p b,˚, d ÿ j"1 ż T 0 ds ż Λ du ρ i ps, uqB e j G s puq ď 1 2 ż T 0 ds ż Λ duG s puq 2 `C hence, p ρ P L 2 pr0, T s, H 1 pΛqq 3 . ´T ż 0 2dλ 1 N d ÿ Λ N G 1,s px{N qpπ N,1 s ˚ιǫ qpπ N,0 s ˚ιǫ qds ´T ż 0 2dλ 2 N d ÿ Λ N G 1,s px{N qpπ N,3 s ˚ιǫ qpπ N,0 s ˚ιǫ qds ´r T ż 0 xπ N,3 s , G 1,s yds `T ż 0 pr `1qxπ N,1 s , G
lim N Ñ8 Q p b,μ N ´sup 0ďtďT ˇˇB 1 ǫ pp π N t ˇˇą a ¯" 0.
Notice π ¨Þ Ñ B 1 ǫ pp π T q is continuous with respect to the Skorohod topology and let N go to infinity. We conclude using Lemma 5.4.6 and letting ǫ go to zero.

Empirical currents

In this section, we derive the law of large numbers for the empirical currents stated in Proposition 5.2.1. Recall that for x P Λ N and j " 1, . . . , d, W

x,x`e j t pη i q stands for the conservative current of particles of type i across the edge tx, x `ej u, and Q x t pη i q the total number of particles of type i created minus the total number of particles of type i annihilated at site x before time t. We have the following families of jump martingales (see Lemma 5.B.1 for details) : for all 1 ď j ď d, x P Λ N , Ă W

x,x`e j t pη i q " W x,x`e j t pη i q ´N 2 ż t 0 ´ηi,s pxqp1 ´ηi,s px `ej qq ´p1 ´ηi,s pxqqη i,s px `ej q ¯ds (5.5.1)

with quadratic variation (because J

x,x`e j t pη i q and J

x`e j ,x t pη i q have no common jump)

x Ă W x,x`e j pη i qy t " x r J x,x`e j pη i qy t `x r J x`e j ,x pη i qy t (5.5.2)

" N 2 ż t 0 ´ηi,s pxqp1 ´ηi,s px `ej qq `p1 ´ηi,s pxqqη i,s px `ej q ¯ds and r Q x t pη i q " Q x t pη i q ´ż t 0 τ x f i pξ s , ω s qds (5.5.3) 146

Empirical currents

where p f " pf 1 , f 2 , f 3 q : p Σ N Ñ R 3 is defined by $ ' & ' % f 1 pξ, ωq " β N p0, ξ, ωqη 0 p0q `η3 p0q ´pr `1qη 1 p0q , f 2 pξ, ωq " rη 0 p0q `η3 p0q ´βN p0, ξ, ωqη 2 p0q ´η2 p0q , f 3 pξ, ωq " β N p0, ξ, ωqη 2 p0q `rη 1 p0q ´2η 3 p0q . x r Q x pη 1 qy t " ş t 0 τ x ´βN p0, ξ s , ω s qη 0,s p0q `η3,s p0q `pr `1qη where we did a Taylor expansion. Relying on (5.5.2), the expectation of x Ă M G y t vanishes when N Ñ 8, so that by Doob's martingale inequality, lim

N Ñ8 P N, p b µ N " sup 0ďtďT ˇˇz Ă G M t ˇˇą δ ı " 0 ,
for any δ ą 0. Using that the empirical density p π converges towards the solution of (5.2.31), this concludes the law of large numbers (5.2.36) for the current W N T .

Fix a smooth vector field p H " pH 1 , H 2 , H 3 q P C 8 c pΛ, R 3 q. Sum (5.5.

+ " 0.
This limit is a consequence of the one and two blocks estimates (5.4.2)-(5.4.3), for which we refer to Chapter 4 since we reduced ourselves to a finite volume and conclude by letting γ go to 0.

The hydrodynamic limit

To conclude to the hydrodynamic behaviour of our system, we still need to prove : tightness of the measures pQ N, p b µ N q N ě1 ; identification of the limit points of pQ N, p b µ N q N ě1 ; uniqueness of weak solutions of the hydrodynamic equation.

The two first steps are analogous to the proofs done in finite volume, we refer the reader to Chapter 4 for details. Though, we prove the uniqueness of weak solutions for the generalized contact process in infinite volume with stochastic reservoirs in Section 5.7, the method yields to prove the uniqueness of weak solutions of the system (5.2.38), this is given by Proposition 5.2.2 whose proof is postponed to Section 5.7.

Uniqueness of weak solutions

To conclude, we derive in this section the uniqueness of the weak solutions of Section 5.2. One concludes by letting δ Ó 0 and using Gronwall's inequality.

Uniqueness in finite volume

Uniqueness in infinite volume with boundaries

Proof of Proposition 5.2.2. The proof follows the arguments in [START_REF] Marra | Phase segregation dynamics for the Blume-Capel model with Kac interaction[END_REF] adapted to the our case. For u " pu 1 , ¨¨¨, u d q P Λ 8 , denote by ǔ " pu 2 , ¨¨¨, u d q P R d´1 , so that u " pu 1 , ǔq.

Uniqueness of weak solutions

Denote by L 2 pp´1, 1qq the Hilbert space on the one-dimentional bounded interval p´1, 1q equipped with the inner product, xϕ, ψy 2 " ż 1

´1 ϕpu 1 q ψpu 1 q du 1 ,

where, for z P C, z is the complex conjugate of z and |z| 2 " z z. The norm of L 2 pp´1, 1qq is denoted by } ¨}2 . Let H 1 pp´1, 1qq be the Sobolev space of functions ϕ with generalized derivatives B u 1 ϕ in L 2 pp´1, 1qq. H 1 pp´1, 1qq endowed with the scalar product x¨, ¨y1,2 , defined by xϕ, ψy 1,2 " xϕ, ψy 2 `xB u 1 ϕ , B u 1 ψy 2 , is a Hilbert space. The corresponding norm is denoted by } ¨}1,2 .

Consider the following classical boundary-eigenvalue problem for the Laplacian : " ´∆ϕ " αϕ , ϕ P H 1 0 pp´1, 1qq .

(5.7.1)

From the Sturm-Liouville theorem (cf. [START_REF] Zuily | Éléments de distributions et d'équations aux dérivées partielles[END_REF]), one can construct for the problem (5.7.1) a countable system tϕ n , α n : n ě 1u of eigensolutions which contains all possible eigenvalues. The set tϕ n : n ě 1u of eigenfunctions forms a complete orthonormal system in the Hilbert space L 2 pp´1, 1qq. Moreover each ϕ n belong to H 1 0 pp´1, 1qq and the set tϕ n {α 1{2 n : n ě 1u is a complete orthonormal system in the Hilbert space H for all ψ 1 , ψ 2 in H 1 0 pp´1, 1qq. One can easily check that in our case, α n " n 2 π 2 and ϕ n pu 1 q " sinpnπu 1 q, n P N.

Fix T ą 0, define the heat Kernel on the the time interval p0, T s defined by the following expression p 1 pt, u 1 , v 1 q " ÿ ně1 e ´αnt ϕ n pu 1 qϕ n pv 1 q , t P r0, T s , u 1 , v 1 P r´1, 1s .

Let g P C 0 c pp´1, 1q; Rq and denote by δ ¨the Dirac function. The heat Kernel p 1 is such that p 1 p0, u 1 , v 1 q " δ u 1 ´v1 , p P C 8 pp0, T s ˆp´1, 1q ˆp´1, 1q; Rq and the function defined via the convolution operator : ϕ 1 pt, u 1 q :" pp 1 ‹ gqpt, u 1 q " ż 1 ´1 p 1 pt, u 1 , v 1 qgpv 1 qdv 1 solves the following boundary value problem $ & % B t ϕ " B 2 u 1 ϕ , ϕp0, ¨q " gp¨q , ϕpt, ¨q P H 1 0 pp´1, 1qq for 0 ă t ď T .

(5.7.3)

Let p be the heat kernel for pt, ǔ, vq P p0, T q ˆRd´1 ˆRd´1 ppt, ǔ, vq " `4πt ˘´pd´1q{2 exp # ´1 4t It is known that ȟf t solves the equation B t ρ " ∆ρ, ρ0 " f , on p0, ts ˆRd´1 . Moreover ȟ P C 8 pp0, T s ˆRd´1 ; Rq.

For a positive time t P p0, T s, p f " pf 1 , f 2 , f 3 q P C c pΛ 8 ; R 3 q and ε ą 0 small enough, let H f t,ε : r0, ts ˆΛ8 ÝÑ R be defined by H p f t,ε ps, uq :"

3 ÿ i"1
H f i t,ε ps, uq :"

3 ÿ i"1
`p ˚fi ˘pt `ǫ ´s, uq, where p is the heat kernel on p0, T s ˆΛ8 ˆΛ8 given by ppt, u, vq " p 1 pt, u 1 , v 1 qppt, ǔ, vq.

It is easy to check that H f t,ε solves the equation B t ρ " ∆ρ on p0, ts ˆRd , ρ 0 " f . Consider p ρ p1q " pρ 3 q two weak solutions of (5.2.31) associated to an initial profile p γ " pγ 1 , γ 2 , γ 3 q : Λ 8 Ñ r0, 1s for some positive constant C 1 , where for a trajectory m : r0, ts ˆΛ8 Ñ R, }m} 8 " }m} L 8 pr0,tsˆΛ 8 q stands for the infinite norm in L 8 pr0, ts ˆΛ8 q.

On the other hand, from the fact that ρ p1q i , ρ p2q i , 1 ď i ď 3 are in L 8 pr0, T s ˆΛ8 q, it follows that there exists a positive constant C 2 such that, for almost every ps, uq P r0, ts ˆΛ8 , for every 1 ď i ď 3, Since p ρ p1q and p ρ p2q are two weak solutions of (5.2.31), we obtain by (5.7.4) that for all 0 ď τ ď t, 1 ď i, k ď 3 ˇˇ m i pτ, .q, H f k τ,ε pτ, .q ˇˇ" 3 ÿ i"1 ˇˇż τ 0 F i pp ρ p1q q ´Fi pp ρ p2q q, H f k τ,ε pτ, .q ˇď

C 1 1 t ´3 ÿ i"1 }ρ p1q i ´ρp2q i } 8 ¯}f k } 1 ,
for C 1 1 " C 1 C 2 . By observing that ppε, ¨, ¨q is an approximation of the identity in ε, we obtain by letting ε Ó 0, ˇˇ m i pτ, .q, f k ˇˇď C where we used the fact that pps, ¨, ¨q is a probability kernel in R d´1 for all s ą 0.

5.A Changes of variables formulas

In the following, one states useful formula concerning change of variables with respect to a varying smooth profile. It is convenient to use the form (5.2.17) of the reference measure. Lemma 5.A.1. For pi, jq P t0, 1, 2, 3u 2 , i " j, ż η i pxqη j pyqf pξ x,y , ω x,y qdν N p θ pξ, ωq " ż η j pxqη i pyqe pϑ j py{N q´ϑ j px{N qq´pϑ i py{N q´ϑ i px{N qq f pξ, ωqdν N p θ pξ, ωq

Proof. Let us detail the change of variable when pi, jq " p1, 2q, the other ones are similar.

Posing pξ 1 , ω 1 q " pξ x,y , ω x,y q one has, ż η 1 pxqη 2 pyqf pξ x,y , ω x,y qdν N p θ pξ, ωq " ż ξpxqp1 ´ωpxqqp1 ´ξpyqqωpyqf pξ x,y , ω x,y qdν N p θ pξ, ωq " ż p1 ´ξ1 pxqqω 1 pxqξ 1 pyqp1 ´ω1 pyqqf pξ 1 , ω 1 q dν p θ pξ 1x,y , ω 1x,y q dν p θ pξ 1 , ω 1 q dν N p θ pξ 1 , ω 1 q # ´ϑ1 px{N q ´ϑ3 px{N q ¯´η 3 pxq ´η1 pxq θ2 px{N q ´ϑ0 px{N q ¯´η 0 pxq ´η2 pxq ¯+.

5.B Quadratic variations computations

We compute here the quadratic variations of the two jump martingales appearing in Section 5.5. Using computations as in C. Cocozza and C. Kipnis [START_REF] Cocozza | Existence de processus Markoviens pour des systèmes infinis de particules[END_REF], Lemma 5.B.1. For t ě 0, 1 ď i ď 3, 1 ď j ď d such that x, x `ej P Λ N , r J

x,x`e j t pη i q " J x,x`e j t pη i q ´N 2 ş t 0 η i,s pxqp1 ´ηi,s px `ej qqds and r J

x`e j ,x t pη i q " J

x`e j ,x t pη i q ´N 2 ş t 0 η i,s px èj qp1 ´ηi,s pxqqds are two P N µ N -martingales whose quadratic variations are given by x r J x,x`e j pη i qy t " N 2 ż t 0 η i,s pxqp1 ´ηi,s px `ej qqds (5.B.1)

x r J x`e j ,x pη i qy t " N 2 ż t 0 η i,s px `ej qp1 ´ηi,s pxqqds (5.B.2)

Proof. Consider jumps over the bond px, x `ek q, by writing the generator of diffusion as in (5.3.9), we shall decompose the jumps associated to the exchanges of particles between each type i and j, i, j P t0, 1, 2, 3u. That is, J x,x`e 1 pη i q " ÿ j‰i J x,x`e 1 iÑj pξ, ωq and J x,x`e 1 pη i q " ÿ j‰i J x,x`e 1 jÐi pξ, ωq.

where for fixed i, J x,x`e 1 iÑj correspond to the exchanges of particles over the bond px, x`e 1 q when x is in state i and x `e1 is in state j.

For z, z `ek P Λ N , consider the function f z,z`e k iÑj pξ, ωq " η 3 pzqη 1 pz `ek q. Then, L N f z pη i q " ÿ x,yPΛ N }x´y}"1 ´ηx,y j pzqη x,y i pz `ek q ´ηj pzqη i pz `ek q " ÿ uPΛ N }u´z}"1,u‰z`e k ´ηj puqη i pz `ek q ´ηj pzqη i pz `ek q ηj pz `ek qqη i pzq ´ηj pzqη i pz `ek q ÿ vPΛ N }v´pz`e k q}"1,v‰z ´ηj pzqη i pvq ´ηj pzqη i pz `ek q

The martingale problem states that r f iÑj z,z`e 1 pξ t , ω t q :" f iÑj z,z`e 1 pξ t , ω t q ´ż t 0 L N f iÑj z,z`e 1 pξ s , ω s qds is a P N µ N -martingale. Consider the predictable process g iÑj z,z`e 1 pξ s , ω s q " η i,s´p zqη j,s´p z è1 qq. Since the set ts : η i,s´p zqη j,s´p z `ek ‰ η i,s pzqη j,s pz `ek qu is ds-negligible, ż t 0 g iÑj z,z`e k pξ s , ω s qd r f iÑj z,z`e k pξ s , ω s q " ż t 0 g iÑj z,z`e k pξ s , ω s qdf iÑj z,z`e k pξ s , ω s q ´ż t 0 g iÑj z,z`e k pξ s , ω s qL N ´f iÑj z,z`e k pξ s , ω s qds " ÿ sďt g iÑj z,z`e k pξ s , ω s q ´ηj,s pzqη i,s pz `ek q ´ηj,s´p zqη i,s´p z `ek q ż t 0 # ÿ uPΛ N }u´z}"1,u‰z`e k ´ηj,s puqη i,s pz `ek q ´ηj,s pzqη i,s pz `ek q ¯ηi,s pzqη j,s pz `ek qq ´´η j,s pz `ek qqη i,s pzq ´ηj,s pzqη i,s pz `ek q ¯ηi,s pzqη j,s pz `ek qq ´ÿ vPΛ N }v´pz`e k q}"1,v‰z ´ηj,s pzqη i,s pvq ´ηj,s pzqη i,s pz `ek q ¯ηi,s pzqη j,s pz `ek q + ds " J iÑj z,z`e j pξ t , ω t q ´ż t 0 η i,s pzqη j,s pz `ek qds.

Let V iÑj z,z`e k pξ t , ω t q " ´ηi,s pzqη j,s pz `ek q ¯2´1 ´f iÑj z,z`e k pξ s , ω s q ¯dV iÑj z,z`e k pξ s , ω s qds " ż t 0 η i,s pzqη j,s pz `ek q ¯dV iÑj z,z`e k pξ s , ω s q " ż t 0 η i,s pzqη j,s pz `ek qds By inverting the direction of the jumps, we compute similarly that A´r J iÐj z,z`e k pξ, ωq ¯Et " ż t 0 η j,s pzqη i,s pz `ek q ¯ds Lemma 5.B.2. For t ě 0, 1 ď i ď 3 and x P Λ N , r Q x t pη i q " Q x t pη i q ´şt 0 τ x f i pξ s , ω s qds is a P N µ N -martingale whose quadratic variations is given by Proof. As in previous lemma, one shall decompose the non-conservative dynamics according to interaction between each type of particles i, j P t0, 1, 2, 3u. That i,s

ş t 0 L N f iÑj z,
Q x t pη i q " ÿ j‰i Q iÑj z pξ, ωq ´QjÐi z pξ, ωq
where for fixed i, Q iÑj z pξ, ωq corresponds to state j when z is in state i and Q iÐj z pξ, ωq corresponds to flips to state i when z is in state j. It suffices to consider the case i " 0, j " 1 as others follow a similar way.

As in the proof of Lemma 5.B.1, for z P Λ N consider f 1Ð0 z pξ t , ω t q " η 1,s pzq and g 1Ð0 z pξ t , ω t q " η 0,s´p zq. Identical

5.C Estimates in bounded domain

Lemma 5.C.1. For a smooth profile p θ : Λ Ñ p0, 1q 3 such that θ ˇˇΓ " p b, there exist positive constants A 0 , A 1 0 and A 1 depending only on p θ such that for any c ą 0, for any f P L 2 pν N p θ q, xL p b,N f, f y " ´Dp b N pf 2 q, (5.C.1)

xL N f, f y " ´A0 D 0 N pf 2 q `A1 0 N d´2 }f } 2 L 2 pν N p θ
q , (5.C.2)

xL N f, f y " A 1 N d }f } 2 L 2 pν N p θ q .
(5.C.3)

Proof. Since ν N p θ is reversible with respect to the generator L p b,N , (5.C.1) is immediate. To prove (5.C.2), remark that for all A, B, c ą 0, ApB ´Aq " ´pB ´Aq 2 `BpB ´Aq and use (5.3.17) xL N f, f y " ÿ

x,yPΛ N ż f pξ, ωq ´f pξ x,y , ω x,y q ´f pξ, ωq ¯dν N p θ pξ, ωq " ´1 2 D 0 N pf q `1 2 ÿ

x,yPΛ N ż f pη x,y qpf pηq ´f pη x,y qqdν N p θ pξ, ωq `1 2 ÿ

x,yPΛ N ż f pηq ´f pη x,y q ´f pηq ¯dν N p θ pξ, ωq

" ´1 2 D 0 N pf q `1 2 ÿ x,yPΛ N ÿ i,j
´f pηq ´f pη x,y q ¯f pηqR x,y i,j p p θqdν N p θ pξ, ωq

ď ´p1 2 ´1 4c qD 0 N pf 2 q `c 2 N d´2 }f } 2 L 2 pν N p θ q `Op 1 N 2 q
where we did a Taylor expansion of R x,y i,j p p θq which was defined in 5. Let us deal with the first integral, the second will follow the same way. Since all the rates are bounded, we have

I 1 ď

Perspectives

So far, we have been concerned with a competition model for a population dynamics with random environment. Our results proved the existence of a unique phase transition on Z d within a dynamic random environment on one hand, and survival and extinction conditions on Z within a quenched random environment on the other hand. Assuming these stochastic dynamics are underlying a microscopic scale, the hydrodynamic equation of the system with stirring is given by a non-linear reaction-diffusion system, with additionally Dirichlet boundary conditions when in presence of stochastic reservoirs.

The following is an overview of possible guidelines.

Weak survival. Let T d be the homogeneous tree whereby each vertex has d `1 neighbours. A particular property that belongs to the basic contact process is that it exhibits two phase transitions on T d , meaning that according to Definitions (1.2) of Chapter 1, λ c and λ s are distinct. Following works on percolation by G. Grimmett and C. Newman [START_REF] Grimmett | Percolation in 8 `1 dimensions[END_REF], R. Pemantle [START_REF] Pemantle | The contact process on trees[END_REF] proved that in dimension 3, weak survival occurs and

λ c ă λ s , λ c ď 1 d ´1, λ s ě 1 2 ? d .
Extensions to dimension 2 and inhomogeneous trees were done by T.M. Liggett [START_REF] Liggett | Multiple transition points for the contact process on the binary tree[END_REF][START_REF] Liggett | Branching random walks and contact processes on homogeneous trees[END_REF] and A. Stacey [START_REF] Stacey | The existence of an intermediate phase for the contact process on trees[END_REF]. Still close to percolation behaviours [START_REF] Grimmett | Percolation in 8 `1 dimensions[END_REF], R. Durrett and R. Schinazi [START_REF] Durrett | Intermediate phase for the contact process on a tree[END_REF] proved the existence of infinitely many invariant measure in the intermediate phase.

See R. Schinazi [69, Chapter VII], T.M. Liggett [START_REF] Liggett | Stochastic interacting systems : contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]Part I.4] for further details on the contact process on the tree.

The existence of a weak survival arose interests in investigating the behaviour of the process within the intermediate phase. Biologically, a weak survival phase is thought of as being the tipping phase where the SIT program would fail or success.

Some observations lead to think the behaviour of our symmetric multitype process is similar to the basic contact process. Though, D. Griffeath showed that weak survival can occur for totally asymmetric contact processes on Z.

Random environment. We studied the contact process in a particular quenched random environment. Improved results would rely on finding conditions on the distribution of the environment for the survival or extinction of the process such as in C. Newman and S. Volchan [START_REF] Newman | Persistent survival of one-dimensional contact processes in random environments[END_REF] did in a 1-dimension case. Primarily based on percolation techniques, they proved the survival of the process with conditions on the tail of distribution of the environment, when the growth rate is small enough.

Studying the hydrodynamics of our system, it is foreseeable to investigate the process in the presence of a macroscopic random environment or disorder.

Stirring limits and Predator-prey systems. By scaling and stirring the particle system in Chapters 4 and 5, we proved it converges to the solution of a reaction-diffusion system. As stated by R. Durrett [START_REF] Durrett | Ten lectures on particle systems[END_REF]Chapter 9], it seems if one gets enough information

Theorem 1 . 1 . 3

 113 (T.M. Liggett (1972)). Assume that sup xPS ÿ T Qx sup ´cT pζ, F T q : ζ P Ω ¯ă 8 and sup xPS ÿ T Qx ÿ u‰x sup ´}c T pζ 1 , dαq ´cT pζ 2 , dαq} T : ζ 1 pyq " ζ 2 pyq for all y ‰ u ¯ă 8

Figure 1 . 1 :

 11 Figure 1.1: The graphical representation for the contact process on Z 1 ˆRF

p1q 1 ď λ p2q 1 ,

 1 if we constructed the process with rate λ p2q 1 and we keep each arrow with probability λ p1q 1 {λ p2q 1 , by the thinning property of the Poisson processes, we end up with the graphical representation of a contact process with growth rate λ p1q 1 .

Figure 1 . 2 :

 12 Figure 1.2: Average number of dengue cases in most highly endemic countries as reported to WHO 2004-2010.

Chapter 2 .Figure 2 . 1 :

 221 Figure 2.1: Graphical representation in the space-time picture Z ˆR`. Starting from η 0 " 1 t0u , following the arrows, if U 01 ă λ 2 λ 1 and U 0 2 ă λ 2 λ 1 , the wild population occupies at time t the set H t " t´1, 0, 1u in the asymmetric case and the set H t " t´2, ´1, 0, 1u in the symmetric case.

( 2 . 4 . 5 )ď λ p2q 1 . 1 ď λ p2q 1

 245111 All different possible scenarios provide the following necessary conditions :(I) j 1 P t0, 1u, δ " β P tA, Bu in (2.4.4) give (i) α " C, γ " D : λ p1q 2This is a consequence of conditions 1. and 3. or 2. and 4.(ii) α " γ " C : λ p1q 2 ď λ p2q 2 stated by condition 4. (iii) α " γ " D : λ p1q stated by condition 3. (II) j 1 " 0, β " B, δ " 1 `β " C in (2.4.4) give (i) α " D : λ p1q 1 ď 1 stated by condition 6. (ii) α " C : λ p1q 2 ď 1 stated by condition 7. (III) h 1 " 0, γ " α P tB, Du in (2.4.5) give r p1q ě r p2q stated by condition 5.

Chapter 2 .

 2 Phase transition on Z d

  for all x P Z d , pη p1q 0 pxq, η p2q 0 pxqq ‰ p0, 3q. In particular one could start from η

p2q 2 2 ď λ p2q 1 1 ď λ p2q 1

 22111 stated by condition 4. (ii) α " C, γ " D : λ p1q stated by conditions 1. and 3. (iii) α " γ " D : λ p1q stated by condition 3.

Chapter 2 .

 2 Phase transition on Z d

Figure 2 . 3 :

 23 Figure 2.3: Set d " 1. The targeted region goes upward-rightward (reflections are not drawn, but a symmetric draw going upward-leftward does exist as the dashed line suggests it). Occupied translated sets r´n, ns d stand in the gray regions and are reached by paths lying in the area delimited by the stair shaped plain lines.

3 Survival and extinction conditions in quenched environment Contents 3 . 1 72 3. 3 76 3. 4 3 . 5

 33172376435 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Settings and results . . . . . . . . . . . . . . . . . . . . . 70 3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random growth on vertices . . . . . . . . . . . . . . . . . 72 3.3.1 Extinction conditions . . . . . . . . . . . . . . . . . . . . . . 73 3.3.2 Survival conditions . . . . . . . . . . . . . . . . . . . . . . . . Random growth on oriented edges . . . . . . . . . . . . . 76 3.4.1 Extinction conditions . . . . . . . . . . . . . . . . . . . . . . 77 3.4.2 Survival conditions . . . . . . . . . . . . . . . . . . . . . . . . 78 Numerical bounds on the transitional phase . . . . . . . . 79 3.5.1 Back to the basic contact process . . . . . . . . . . . . . . . . 79 3.5.2 The transitional phase . . . . . . . . . . . . . . . . . . . . . . 80

Figure 3 . 1 :

 31 Figure 3.1: Quenched random environment

Figure 3 . 2 :

 32 Figure 3.2: Random environment on vertices

y:|y´x|" 1 ´λ1

 1 p1 ´ωpyqq `λ2 ωpyq ¯αpyq (3.3.3) 1 Ñ 0 at rate 1tn 0 px, αq ą 0u (3.3.4)

ρ e pk `1q λ e pk `1q ρ e pkq λ e pkq ρ e pk ´1q λ e pk ´1q ρ e pk ´2q Figure 3 . 3 :

 ´2q33 Figure 3.3: Random environment on oriented edges

  4.1), condition (3.4.2) from Theorem 3.4.1 gives us lower bounds on the phase transition. Moreover, by choosing parameters λ 2 and λ 1 satisfying (3.4.5), condition (3.4.6) from Theorem 3.4.2 gives us upper bounds.

  a bijection from R 3 `to p0, 1q3 given by (4.B.3). The measures tν N p ρ , p ρ P r0, 1s 3 u are invariant [see Lemma 4.B.1] with respect to the rapid-stirring process with generator N 2 L D N , and they are parametrized by the densities : " E N ν p ρ rη k pxqs " ν N p ρ pηpxq " kq " ρ k , 1 ď k ď 3, ν N p ρ pηpxq " 0q " 1 ´ρ1 ´ρ2 ´ρ3 . For any function φ on E N , denote by r φpp ρq the expectation of φ with respect to ν N p ρ : r φpp ρq " E N ν p ρ rφpηqs. (4.2.7)

Lemma 4 . 3 . 2 (Lemma 4 . 3 . 3 (

 432433 One block estimate). lim kÑ8 lim N Ñ8 sup f N :D D N pf N qďCN d´2 φpηq ´r φpp η k p0qq ¯ˇˇˇˇf N pηqdν N p ρ pηq " 0. (4.3.19) Two blocks estimate). For i P t0, hq ´ηǫN i pxq ˇˇf N pηqdν N p ρ pηq " 0. (4.3.20)

r 4 . 3 . 8 ,

 438 φpp η k pyqq ´r φpp η ǫN p0qq ˇˇˇˇf N pηqdν N p ρ pηq. (4.3.23) 92 The hydrodynamic limit The first expression (4.3.21) of the right-hand side can be decomposed into boxes of size p2k `1q d so that, which tends to zero when N goes to infinity. The second and third expressions (4.3.22) and (4.3.23) tend to zero as well as a consequence of the blocks estimates by translation invariance of ν N p ρ .

,s yds 4

 4

Following 4 . 3 . 2 .

 432 the uniqueness of weak solutions of non-linear parabolic equations done in[START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF] Appendix 2.4], one hasProposition There exists a unique weak solution to the reaction-diffusion system (4.2.14) satisfying (S1) and (S2).Proof. For each z P Z d , introduce ψ z : T d Ñ C defined by ψ z puq " exp ´p2πiqpz.uq ¯(4.3.26)

Chapter 4 . 2 )

 42 Hydrodynamic limit on the torus ℓ 3 " p2k `1q d ´ℓ0 , denote by m k p ℓ the measure ν k p ρ conditioned on the hyperplane tη : p2k `1q d p η k p0q " p ℓu, m k p ℓ p¨q " ν k p ρ `¨|p2k `1q d p η k p0q " p ℓ ˘(4.4.Note this measure does not depend on p ρ. It remains to show lim

p 4 . 4 .

 44 ρ pηq " 0 (4.4.5) 100 Proof of the replacement lemma By translation invariance of the measure, one can rewrite the integral as ż ˇˇη k i p0q ´ηk i phq ˇˇf N pηqdν N p ρ pηq

.4.8) 101 Chapter 4 .

 1014 Hydrodynamic limit on the torus

  Construction of the generator. It is about to construct a convenient transition function r c for which the measure ν N p ρ is invariant with respect to an auxiliary Markov process with generator r L R N , that is for any function f on E N ż r L R N f pηqdν N p ρ pηq " 0. (4.A.1) 102 4.A. Construction of an auxiliary process Let r rpx, ηq " r 0 1tηpxq " 0u `r1 1tηpxq " 1u, δ 1 , δ 2 and α be parameters associated to the generator r L R N to determine. By a change of variables [see Lemma 4

Proposition 4 .A. 2 .

 42 In a more suitable way, one can now prove the replacement lemma 4.3.1 for the process of generator Ă L N . After what we deduce the result for the reaction-diffusion process of generator L N by inequality entropy using 4.A.7 and 4.A.1. For all a ą 0, lim ǫÑ0 lim N Ñ8

P

  N pµ : ω 1 µ pγq ą ǫq " 0. On the other hand, condition (2) can by substituted by the following sufficient condition : Proposition 4.D.3 (D. Aldous (1978)). A sequence of probability measure tP N , N ě 1u in Dpr0, T s, Eq satisfies(2) of Theorem 4.D.1 if lim γÑ0 lim N Ñ8 sup τ PT T θďγ P N pδpµ τ , µ τ `θq ą ǫq " 0 (4.D.4)

.3.2) 130 5 . 3 .

 53 Proof of the specific entropy (Theorem 5.2.1)

1

 1 p θ q ď K 0 N d .Using that e |a| ď e a `e´a and lim N N ´d log `aN `bN ˘ď max ´lim N N ´d log a N , lim N N ´db N ¯, one can pull off the absolute value even if it means replacing H by ´H. By Feynman-Kac formula, 1 γN d log E N, H i,s px{N q ´ηi,s pxq ´bi px{N q Hs i pξ s , ω s qpxqf pξ, ωqdν N p θ pξ, ωq `1 γN d xL N

p θ defined in ( 5 . 2 .

 52 [START_REF] Durrett | Special invited paper : coexistence in stochastic spatial models[END_REF]. By (5.2.28), s N pµ N |ν N p θ q ď C 0 N d , for some constant C 0 . Using that exp

Lemma 5 . 4 . 5 (

 545 Energy estimate). Let Q p b,˚b e a limit point of the sequence pQ N, p b

d ÿ k" 2 pu k ´vk q 2 +.

 22 For each function f P C c pR d´1 ; Rqȟ f t pt, ǔq :" pp ‹ f qpt, ǔq " ż R d´1ppt, ǔ, vq f pvqdv .

3 ÿ

 3 ˇˇF i pρ p1q i ps, uqq ´Fi pρ p2q i ps, uqq ˇˇď C 2

3 . 8 .

 38 xL N f, f y " I 1 `I2 " ÿ xPΛ N ż ´βN px, ξ, ωqp1 ´ξpxqq `ξpxq ¯f pξ, ωq ´f pσ x ξ, ωq ´f pξ, ωq ¯dν N p θ pξ, ωq 160 5.C. Estimates in bounded domain `ÿ xPΛ N ż ´rp1 ´ωpxqq `ωpxq ¯f pξ, ωq ´f pξ, σ x ωq ´f pξ, ωq ¯dν N p θ pξ, ωq

  . DpΩq "

	"	f P CpΩq : lim tÓ0	S t f	´f t	exists	*	, and
			Lf " lim

tÓ0 S t f ´f t , f P DpΩq.

  λ 1 ă λ c survives weakly if λ c ă λ 1 ă λ s survives strongly if λ 1 ą λ s

		tÑ8	ξ t p0q " 1q ą 0u.	(1.2.6)
	for which, the process
	dies out if Since
	t lim tÑ8	ξ t p0q " 1u Ă t@t ě 0 Ξ t ‰ Hu,

5) Chapter 1. Introduction and λ s " inftλ 1 : P λ 1 p lim

Chapter 2. Phase transition on Z d '

  β the growth rate of a type-1 individual in y such that ηpyq " β, depending only on the value of ηpxq " α. The state in y flips from β to β `k. P ´k α the jump rate of a site from state ηpxq " α to state α´k for k ď α, depending only on the value of ηpxq.

	Next, define
	Π 0,k α,β :" R 0,k α,β	`P k β and Π ´k,0 α,β :" P ´k α .	(2.2.14)
	Theorem 2.2.4.
	' P k β the jump rate of a site from state ηpyq " β to state β `k, depending only on the value of ηpyq.

with rate 2.3. Graphical construction

  . Think of the diagram Z d ˆR`. For each x P Z d , consider the arrival times of mutually independent families of Poisson processes : tA x

	r, tD 1,x n	: n ě 1u and tD 2,x n

n : n ě 1u

Chapter 2. Phase transition on Z d

	with the rates previously defined by (2.2.16)-(2.2.17). One then has 1tj 1 " 0u1tk " 2u1tδ ´β " 1u ´1tδ " C, β " Bu `R0,2,p1q D,B 1tα " Du `R0,2,p1q C,B 1tα " Cu	1tδ
	" B, β " Au 1 " 0u1tk " 2u1tδ ´β " 0u ´1tδ " β " Bu `R0,2,p1q D,A 1tα " Du `R0,2,p1q `R0,2,p1q C,A 1tα " Cu D,B 1tα " Du `R0,2,p1q ˘1tj C,B 1tα " Cu 1tδ
	" β " Au 1 " 0u1tk " 1u1tδ ´β " 0u ´1tδ " β " CuP `R0,2,p1q D,A 1tα " Du `R0,2,p1q C,A 1tα " Cu 1,p1q C `1tδ " β " AuP ˘1tj A 1,p1q 1tj
	´1tδ " β " Bu	1 " 1u1tk " 2u1tδ ´β " 0u `R0,2,p1q D,B 1tα " Du `R0,2,p1q C,B 1tα " Cu	1tδ
	1tj 1 " 0u1tl " 2u ´1tδ " Bu	" β " Au D,B 1tγ " Du `R0,2,p2q `R0,2,p2q `R0,2,p1q D,A 1tα " Du C,B 1tγ " Cu `R0,2,p1q C,A 1tα " Cu 1tδ ˘ď
	" Au 1 " 0u1tl " 1u ´1tδ " CuP `R0,2,p2q D,A 1tγ " Du `R0,2,p2q C,A 1tγ " Cu 1,p2q C 1,p2q ˘1tj 1tj `1tδ " AuP A 1 " 1u1tl " 2u ´1tδ " Bu `R0,2,p2q D,B 1tγ " Du `R0,2,p2q C,B 1tγ " Cu 1tδ
	" Au	`R0,2,p2q D,A 1tγ " Du	`R0,2,p2q C,A 1tγ " Cu	˘¯(2.4.2)
	and 1th 1 " 0u1tk " 1u ´1tα " DuP 1 " 0u1tk " 2u ´1tα " DuP ´1,p1q D ´2,p1q `1tα " BuP B ´1,p1q D ´2,p1q 1th 1th `1tα " CuP C ´2,p1q 1 " 1u1tk " 2u ´1tα " DuP D
				γ,δ	p2q	and	ÿ kąh 1	Π ´k,0,p1q α,β	ě	ląγ´α`h 1 ÿ	Π ´l,0,p2q γ,δ

Table 2 .2 34 2.4. Attractiveness and stochastic order

 2

	transition p1, 0q ÝÑ $ ' ' & ' ' % p1, 1q p3, 2q p3, 0q p0, 0q $ p1, 3q ' ' & p0, 3q p0, 2q ÝÑ p0, 0q ' ' % p2, 2q	λ	rate 1 n 1 px, η p2q q p2q `λp2q 2 n 3 px, η p2q q r p2q r p1q ´rp2q 1

Table 2 .

 2 4 in which case, Table 2.4 substitutes Table 2.2.

Remark 2.4.1. In view of the proof of Proposition 2.4.2, one can actually relax the admissible initial conditions : it is enough to assume η

Table 2 .6 40 2.4. Attractiveness and stochastic order

 2

	transition $	rate
	p1, 0q ÝÑ

1 Table 2 . 7

 127 Remark now that Tables 2.5 and 2.7 differ from Tables2.1 and 2.3 but Table 2.6 stays identical to Table 2.4. As for Proposition 2.4.2, under conditions 1. to 5., if the initial conditions given by η

  14) give λ 2 ď λ 1 . While relation (2.4.15) does not give further condition. Condition λ 2 ď λ 1 is sufficient and allows us to construct the following coupling. transition rate

Table 2 .

 2 11 For all x P Z d , one has if χ ď ξ

n 1 px, χq " |ty " x : χpyq " ξpyq " 1u| (2.4.16) n 3 px, χq " |ty " x : χpyq " 3, ξpyq " 1u| (2.4.17) n 1 px, ξq " |ty " x : χpyq " ξpyq " 1u| `|ty " x : χpyq " 3, ξpyq " 1u| `|ty " x : χpyq P t0, 2u, ξpyq " 1u| (2.4.18)

  .2.19). Necessary and sufficient conditions on the parameters are given by relations (2.4.2)-(2.4.3) applied to rates (2.4.19) i.e. (2.2.19), (2.2.16) and (2.2.17),

  1, Subsection 2.5.2, one proves by (A) and (B) of pη t q tě0 that

	Proposition 2.6.2. Let n ě 1 and N ě 1, then
		lim tÑ8	lim LÑ8	P r p| L H	r´n,ns d t	| ě N q " P r pH	r´n,ns d t	‰ H @t ą 0q	(2.6.2)
	Proof. Since H 2.4.1),	r´n,ns d t	"	Ť Lě0	L H t	r´n,ns d , for any fixed t, by monotonicity (see Corollary
			lim					r´n,ns d t	| ě N q.	(2.6.3)
	It is thus enough to show
		lim tÑ8	P r p|H

LÑ8 P r p| L H r´n,ns d t | ě N q " P r p|H r´n,ns d t | ě N q " P r pH r´n,ns d t

  3, for any ǫ ą 0, there exist L and t sufficiently large such that P r p| L H r´n,ns d t X O u d | ě N q ą 1 ´ǫ2 d . Before going on space-time conditions, consider the lateral parts of the box p´L, Lq d r0, T s. For this, define SpL, T q :" tpx, tq P Z d ˆr0, T s : |x| 8 " Lu, the boundary of the box p´L, Lq d ˆr0, T s and define L H :" Ť tě0 L H t ˆttu. For any

  r´n, ns d for some x P r0, Lq d ˙ą 1 ´ǫ6 Fix δ ą 0. By Proposition 2.6.1, choose n such that

		P r pH	r´n,ns d t	‰ H @t ě 0q ą 1 ´δ2
				(2.6.8)
	and
	P r ˆL`2n H	r´n,ns d t`1

Ą x `r´n, ns d for some px, tq P tL `nu ˆr0, Lq d´1 ˆr0, T q ˙ą 1 ´ǫ6 (2.6.9) 60 2.6. The critical process dies out Proof.

  r´n, ns d for some x P r0, Lq d ˙ě p1 ´δ1{d2 d qp1 ´δq Suppose (2.6.8)-(2.6.9) are satisfied. Then, for any ǫ 7 " ǫ 7 pǫ 6 q ą 0, there exist n, L, T such that P r `2L`3n H r´n,ns d t Ą x `r´n, ns d for some px, tq P rL `n, 2L `ns ˆr0, 2Lq d´1 ˆrT, 2T s ˘ą 1 ´ǫ7 . (2.6.10) Proof. For any ǫ 7 ą 0, choose n, L and T as in (2.6.8)-(2.6.9), by Proposition (2.6.6). With (2.6.9), with P r ´probability at least 1´ǫ 6 , there exists px, tq P tL`nuˆr0, Lq d´1 r0, Ą x `r´n, ns d .By the Markov property and (2.6.8), starting from T `1, with P r ´probability at least 1´ǫ 6 , there exists some y such that y ´x P r0, Lq d satisfying L`2n H Ą x `r´n, ns d for some px, tq P rL `n, 2L `ns ˆr0, 2Lq d´1 ˆrT `1, 2pT `1qs ˘ě p1 ´ǫ6 q d .

	T q such that L`2n H	r´n,ns d t`1
	Consequently,		r´n,ns d T `1	Ą y `r´n, ns d .
	P r `2L`3n H	r´n,ns d t
	and
	P r ˆL`2n H	r´n,ns d T `1

Ą x `r´n, ns d for some px, tq P tL `nu ˆr0, Lq d´1 ˆr0, T q ˙ě p1 ´δ1{2 d qp1 ´δq.

Conclude by choosing δ such that p1´δ 1{2d qp1´δq ě 1´ǫ and p1´δ 1{d2 d qp1´δq ě 1´ǫ. Proposition 2.6.7.

  Assume that ω is a fixed environment. The inhomogeneous contact process pχ t q tě0 dies out if for all n P Z,

	Theorem 3.2.1-(1) can be rewritten as follows.
	Theorem 3.3.1. ÿ kěn	k ź j"n	λ v pj `1q ă 8 and	ÿ kďn	n ź j"k	λ v pj ´1q ă 8.	(3.3.2)

  not reach the second part of the table if χ 0 ď α 0 . In other words, the natural order on t0, 1u is preserved and by [10, Proposition 2.7], pα t q tě0 is stochastically larger than pχ t q tě0 . Finally one gets, lim If the family tωpkq, k P Zu is random and i.i.d. then the family tλ v pkq, k P Zu is i.i.d. as well, one deduces the following criterion from Theorem 3.2.1-(2). The process in random environment pχ t q tě0 dies out if E ω r log λ v p0q ă 0. that is, if λ 2 ă 1 and r ą ´log λ 2 { log λ 1 .

	p0, 0q ÝÑ p1, 1q ÝÑ p0, 1q ÝÑ p1, 0q ÝÑ lim " " " " kÑ8 ÿ kě0 a k " Therefore, assumptions (3.3.2) are satisfied as soon as E ω p1, 1q p0, 1q ř y:|y´x|"1 λ v pyqχpyq ř y:|y´x|"1 λ v pyqpαpyq ´χpyqq p0, 0q p0, 1q 1tn 0 px, αq ą 0u 1 ´1tn 0 px, αq ą 0u p1, 1q p0, 0q ř y:|y´x|"1 λ v pyqχpyq 1tn 0 px, αq ą 0u p1, 1q p0, 0q ř y:|y´x|"1 λ v pyqαpyq 1 1 k k ÿ j"0 log λ v pjq " E ω r log λ v p0q. Denote by a k " ÿ kě0 expplog a k q " k8 ÿ kě0 expp´bkq. r log λ v p0q ă 0. Applying this result to our dynamics given by (3.3.1), E ω r log λ λ 2 ă 1. By (3.2.3) and (3.3.1), E ω r `log λ v p0q ˘" r log λ 2 `log λ 1 r `1 , one has under (3.3.7) the following extinction criterion from (3.3.6) whose dynamics does Corollary 3.3.1. Proof. By the ergodic theorem, r ą ´log λ 1 log λ 2 .	(3.3.7) (3.3.8)

tÑ8

P ω r pχ t pxq " 0q " 1 3.3. Random growth on vertices for all x P Z. k ś j"0 λ v pjq the general term of series (3.3.2). Since E ω r log λ v p0q ă 0, lim kÑ8 1 k log a k ă ´b, for some positive b. That is, lim kÑ8 log a k ă lim kÑ8 p´bkq, written log a k " k8 ´bk, and v p0q " p log λ 2 `p1 ´pq log λ 1 ă 0 (3.3.6)

i.e. p ą log λ 1 {plog λ 1 ´log λ 2 q. Since p ă 1, this implies

  Rq be the set of functions n times continuously differentiable in time and m times continuously differentiable in space. For any function G i P C 1,2 pr0, T s ˆTd ; Rq, denote the integral of G i,t with respect to πN,i

		p π N t pηq :" pπ N,1 t , π N,2 t , π N,3 t qpηq,	(4.2.11)
	where π N,i
						t	by
		xπ N,i t , G i,t y "	1 N d	ÿ N xPT d	G i,t px{N qη i pxq.	(4.2.12)
	For any function p G G t
	with respect to p π N t by	xp π t , p G t y "	3 ÿ i"1	xπ N,i

.2.10) where δ x N pdyq stands for the Dirac measure at x{N . And note for pη s q sPr0,T s , t pηq " π N,i pη t q. Let C n,m pr0, T s ˆTd ; t " pG 1,t , G 2,t , G 3,t q P C 1,2 pr0, T s ˆTd ; R 3 q, define the integral of p t , G i,t y.

  Proof. By Proposition 4.D.4, it is enough to show tightness for the real-valued process txp π t , p Gy, t P r0, T su for all functions p G P C 2 pT d ; R 3 q. By Prohorov's theorem 4.D.1, to get the tightness of txp π t , p Gy, t P r0, T su in Dpr0, T s, R 3 q with the uniform topology, one needs to check the two following assertions :

							ď 1, p π t puq " p π t puqdu, t P r0, T s ¯" 1.	(4.3.6)
	(i) boundedness :		lim mÑ8	lim N Ñ8	Q N µ N ´sup tPr0,T s	|xp π t , p Gy| ě m ¯" 0.	(4.3.7)
	(ii) equicontinuity :
		lim δÑ0	lim N Ñ8	Q N µ N ´sup
	defined in (4.3.1) that		t
		lim δÑ0	lim N Ñ8	Q N µ N ´sup |t´s|ďδ	|M N,i t	´M N,i s | ą ǫ ¯" 0, for any ǫ ą 0	(4.3.9)
	and
	lim δÑ0	lim N Ñ8	Q N µ N ´sup 0ďt´sďδ	ˇˇż s t	L N xπ N,i r , G i ydr ˇˇą ǫ ¯" 0, for any ǫ ą 0.	(4.3.10)
	To prove (4.3.9), one shows the quadratic variation xM N,i y t of the martingale M N,i t converges to zero as N goes to 8. Note that since p G is not time-dependent, the time derivative of p G is null in the expression (4.3.1). By the Doob-Meyer decomposition,

|t´s|ďδ |xp π t , p Gy ´xp π s , p Gy| ą ǫ ¯" 0, for any ǫ ą 0. (4.3.8)

The limit (4.3.7) is immediate since for each t P r0, T s and 1 ď i ď 3, the total mass of π N,i t is bounded by 1. To prove (4.3.8), it is enough to show for the martingale M N,i

xM N,i y t "

  We now deal with replacing local functions of η by functions of the empirical density within a macroscopic box, in other words, for any cylinder function φ and the function r φp¨q defined by (4.2.7), one shows for any continuous function G and ǫ ą 0 the following replacement lemma,

	4.3. The hydrodynamic limit
	and define the vector pp η k t qpxq " pη k 1 , η k 2 , η k 3 qpt, xq. Proposition 4.3.1. For all a ą 0,
	lim ǫÑ0	lim N Ñ8
	For any positive integer k and x P T d N , denote by η k i pt, xq the empirical density of type´i particles given by
		η k i pt, xq "	1 p2k `1q d	ÿ }y´x}ďk N yPT d	η i,t pyq,	(4.3.13)

  Cpk d´1 and occupation variables η i pxq have mean ℓ i {p2k `1q d under m k

								p ℓ ,
	q ÿ ℓ"1	|Λ p | |Λ k |	ż	ˇˇ1 |Λ p |	ÿ yPB ℓ	τ y φpηq ´r φ ´p ℓ p2k `1q d	¯ˇˇd m k
	ż	ˇˇ1 p2p `1q d	ÿ }y}ďp	τ y φpηq ´r φ ´p ℓ p2k `1q d	¯ˇˇd m k p ℓ pηq `Opp{kq
	By the equivalence of ensembles (see next Lemma 4.4.1), letting k go to infinity and
	p ℓ{p2k `1q d tend to p ρ, this integral converges to ż ˇˇ1 p2p `1q d ÿ }y}ďp τ y φpηq ´r φ `p ρ ˘ˇˇd ν p ρ pηq	(4.4.4)

p ℓ pηq `Opp{kq Moreover, the distribution of the occupation variables tpξ, ωqpyq, y P B ℓ u do not depend on ℓ, this sum is hence equal to

  Let tP N , N ě 1u be a sequence of probability measures in Dpr0, T s, Eq. Then tP N , N ě 1u is relatively compact if and only if[START_REF] Andjel | Survival of multidimensional contact process in random environments[END_REF] For all t P r0, T s and ǫ ą 0, there exists a compact set Kpt, ǫq Ă E such that sup

	1 µ pγq " inf tt i u 0ďiďr	max 0ďiďr	sup t i ďsătăt i`1	δpµ s , µ t q.	(4.D.3)
	One can characterize the compact sets of Dpr0, T s, Eq thanks the modified modulus of continuity :
	Proposition 4.D.2. A set A in Dpr0, T s, Eq is relatively compact if and only if (1) tµ t : µ P A, t P r0, T su is relativement compact on E. (2) lim γÑ0 sup µPA ω 1 µ pγq " 0.
	One can now state Prohorov's theorem,
	Theorem 4.D.1. N ě1

  ( with constant parameters is reversible with respect to the generator L N . For p m P R 3 and 1 ď i ď 3, let ψ i p p mq be the expectation of η i p0q under νN

							3 ,
	we define νN p mp¨q as the product measure on Λ N with varying chemical potential p m,
	dν N p mp¨q pξ, ωq " p Z ´1 p m exp	! 3 ÿ i"1	ÿ xPΛ N	m i px{N qη i pxq ) ,	(5.2.6)
	where p Z p m is the normalization constant :
	p Z p m "	ź xPΛ N	! 1	`3 ÿ i"1	exppm i px{N qq ) .	(5.2.7)
	Notice that the family of measures νN p m , p m P R 3 p m :

  ωqdµ n pξ, ωq ´log . Define the Dirichlet form D n pµ n |ν p θ,n q of the measure µ n with respect to ν p where D n pµ n |ν p θ,n q is the restriction of the process to the box Λ N,n Let L N,n denote the restriction of the generator L N to the box Λ N,n :

				ż	e U pξ,ωq dν N p θ,n pη, ξq ) .	(5.2.19)
	In this formula C b p p Σ N,n q stands for the space of all bounded continuous functions on p Σ N,n . Since the measure ν N p θ,n gives a positive probability to each configuration, all the
	measures on p Σ N,n are absolutely continuous with respect to ν N p θ,n and we have an explicit formula for the entropy :
		ż
		s n pµ n |ν N p θ,n q "	log pf n pξ, ωqq dµ n pξ, ωq,	(5.2.20)
	where f n is the probability density of µ n with respect to ν N p θ,n θ,n in the box Λ N,n D n pµ n |ν p θ,n q " ´ż a f n pξ, ωq `LN,n a f n ˘pξ, ωqdν p θ,n pξ, ωq ,
		L N,n " N 2 L N,n `LN,n	`N 2 L p b,N,n ,	(5.2.21)
	with
	L N,n "	ÿ x,yPΛ N,n
		}x´y}"1

  n f pξ, ωq " ´rp1 ´ωpxqq `ωpxq ¯"f pξ, σ x ωq ´f pξ, ωq `p bpx{N q, ξ, σ x ω ˘´a f n pξ, σ x ωq ´af n pξ, ωq ¯2dν p θ,n pξ, ωq `ż c x `p bpx{N q, σ x ξ, ω ˘´a f n pσ x ξ, ωq ´af n pξ, ωq ¯2dν p θ,n pξ, ωq `ż c x `p bpx{N q, σ x ξ, σ x ω ˘´a f n pσ x ξ, σ x ωq ´af n pξ, ωq ¯2dν p θ,n pξ, ωq .

		Chapter 5. With stochastic reservoirs or in infinite volume
	Similarly, we define the corresponding Dirichlet forms,
	with	D n pµ n |ν p θ,n q " D 0 n pµ n |ν p θ,n q ÿ i"1 ż T 0 ds ´żΛ ∇ρ i ps, uq 2 du ¯ă 8 . `Dp b n pµ n |ν p θ,n q ,
	D 0 n pµ n |ν p θ,n q " (B2) For every function p Gpt, uq " p G t puq " pG 1,t puq, G 2,t puq, G 3,t puqq in C 1,2 ÿ x,yPΛ N,n pD 0 n q x,y pµ n |ν p θ,n q 0 `r0, T s R 3 ˘, we have }x´y}"1 D p b n pµ n |ν p θ,n q " ÿ xPΛ N,n XΓ N pD p b n q x pµ n |ν p θ,n q , where pD 0 n q x,y pµ n |ν p θ,n q " ż ´af n pξ x,y , ω x,y q ´af n pξ, ωq ¯2dν p θ,n pξ, ωq, pD p b n q x pµ n |ν p θ,n q " ż D n pµ n |ν p θ,n q " xPΛ N,n pD n q x pµ n |ν p θ,n q (5.2.25) Λ; ÿ c x We will also need xp ρ T p¨q, p G T p¨qy ´xp ρ 0 p¨q, p G 0 p¨qy ´ż T 0 ds xp ρ s p¨q, B s p G s p¨qy " ż T 0 ds xp ρ s p¨q, ∆ p G s p¨qy `ż T 0 ds x p F pρ s qp¨q, p i"1 0 ds Γ n 1 prq b i prqpB 1 G i,s qprq dSprq , ´3 ÿ ż T ż G s p¨qy (5.2.33)
	where
	pD n q x pµ n |ν p θ,n q " `ż β N,n px, ξ, ωqp1 ´ξpxqq `ξpxq ż ´rp1 ´ωpxqq `ωpxq ¯´a f n pξ, σ x ωq ´af n pξ, ωq ¯2dν p θ,n pξ, ωq ¯´a f n pσ x ξ, ωq ´af n pξ, ωq ¯2dν p θ,n pξ, ωq,
	Define the specific entropy Spµ|ν N p θ q and the Dirichlet form Dpµ|ν N p θ q of a measure µ on p Σ N with respect to ν N p θ as
		ı `´β N,n px, ξ, ωq `1 ´ξpxq ˘`ξpxq ¯"f pσ x ξ, ωq ´f pξ, ωq ı Spµ|ν N p θ q " N ´1 ÿ ně1 s n pµ n |ν N p θ,n qe ´n{N , Dpµ|ν N p θ q " N ´1 ÿ ně1 D n pµ n |ν N p θ,n qe ´n{N .	, (5.2.23) (5.2.26) (5.2.27)
	where Notice that by the entropy convexity and since sup xPΛ N tξpxq `ωpxqu is finite, for any β N,n px, ξ, ωq " λ 1 ÿ yPΛ N,n }y´x}"1 ÿ ξpyqωpyq . (5.2.24) positive measure µ on p Σ N and any integer n, we have ξpyqp1 ´ωpyqq `λ2 yPΛ N,n }y´x}"1 s n pµ n |ν N p θ,n q ď C 0 N n d´1 , (5.2.28)
			123

  Ñ r0, 1s3 , then the sequence of probability measures pQ N, p b µ N q N ě1 converges to the Dirac measure concentrated on the unique weak solution p ρp¨, ¨q of the boundary value problem(5.2.31). Accordingly, for any t P r0, T s, any δ ą 0 and any function p G P C 1,2 `r0, T s ˆΛ; R 3 lim

	, ωq, p Gp¨qy ´xp γp¨q, p Gp¨qy ˇˇě δ	)	" 0 ,	(5.2.34)
	for an initial continuous profile p γ : Λ N Ñ8 P N, p b µ N !ˇˇˇx p π N pξ t , ω t q, p Gp¨qy ´xp ρ t p¨q, p Gp¨qy ˇˇě δ

  1,t puq, G 2,t puq, G 3,t puqq in C (Uniqueness in infinite volume with stochastic reservoirs). For any T ą 0, the system of equations (5.2.40) has a unique weak solution in the class `L8 `r0, T s ˆΛ8 ˘3.

	; R 3 ˘, we have	1,2 0 `r0, T s	Λ8
	xp ρ T p¨q, p G T p¨qy ´xp ρ 0 p¨q, p G 0 p¨qy " ż T 0 ds xp ρ s p¨q, ∆ p G s p¨qy i"1 0 ds Γ 8 n 1 prq b i prqpB 1 G i,s qprq dSprq , ´ż T 0 ds xp ρ s p¨q, B s p G s p¨qy `ż T 0 ds x p F pρ s qp¨q, p ´3 ÿ ż T ż G s p¨qy	(5.2.41)
	(IB3) p ρp0, uq " p γpuq. a.e. We now state the following proposition :
	Proposition 5.2.2

  exp ´pϑ j py{N q ´ϑj px{N qq ´pϑ i py{N q ´ϑi px{N qq ¯´1 .

	where
		R x,y i,j p p θq " (5.3.8)
	By a Taylor expansion, (5.3.7) is of order OpN ´1q.
					1 we have
	ż	L x,y N f t n pξ, ωqdν N p θ,n pξ, ωq "	ÿ 0ďi "jď3	ż	η j pyqη i pxqR x,y i,j p p θqf t n pξ, ωqdν N p θ,n pξ, ωq	(5.3.7)

3. Proof of the specific entropy (Theorem 5.2.1)

	Chapter 5. With stochastic reservoirs or in infinite volume
	" N `ż R x,y ż η 1 pxqxη 3 pyqf t n`1 pξ, ωqy n`1 log 1,3 p p θqη 1 pxqxη 3 pyqf t n`1 pξ, ω x,y qy n`1 log f t n pξ, σ x ωq f t n pξ, ωq dν N p θ,n pξ, ωq f pξ, ωq f pξ, σ x ωq `ż η 1 pxqxη 3 pyqf t n`1 pξ, ω x,y qy n`1 log A ż E 1 p1,3qYE 2 p1,3q η 1 pxq ´bxF p1q 1,3 pξ, ωqy n`1 ´bxF p2q 1,3 pξ, ωqy n`1 ¯2dν N dν N p θ,n pξ, ωq p θ,n pξ, ωq `A N ż ´bxF p1q 1,3 pξ, ωqy n`1 `bxF ¯2 p2q 1,3 pξ, ωqy n`1 E 1 p1,3qYE 2 p1,3q f pξ, ωq f pξ, σ x ωq dν N p θ,n pξ, ωq , where R x,y ď i,j p p θq was defined in (5.3.8). By a Taylor expansion of R x,y θq, the second line ˆ´d f t n pξ, σ x ωq f t ´1¯2 dν N p θ,n pξ, ωq . n pξ, ωq (5.3.18) i,j p p on the last r.h.s. is of order OpN ´1q. We deduce that To bound the first integral of the last r.h.s. in (5.3.18) by a piece of Dirichlet form, apply ż f t Cauchy-Schwarz inequality so that n`1 L x,y 1Ø3 log f t n pξ, ωqdν N p θ,n`1 pξ, ωq " ż η 1 pxq ´xF p1q 1,3 pξ, ωqy n`1 ´xF p2q 1,3 pξ, ωqy n`1 ¯log f t n pξ, σ x ωq f t n pξ, ωq dν N p (5.3.12) θ,n pξ, ωq `OpN ´1q N A ż E 1 p1,3qYE 2 p1,3q η 1 pxq ´bxF p1q 1,3 pξ, ωqy n`1 ´bxF p2q 1,3 pξ, ωqy n`1 ¯2dν N p θ,n pξ, ωq
	where
	F
	p2q i,j pξ, ωqy n`1 , i,j pξ, ωqy n`1 ď xF n pξ, σ x ωq ě f t f t n pξ, ωqu E 2 pi, jq " tpξ, ωq : xF p1q p2q i,j pξ, ωqy n`1 , f t n pξ, σ x ωq ď f t n pξ, ωqu	(5.3.14) (5.3.15)
	the integral in the r.h.s. of (5.3.12) is non-negative on E 1 p1, 3q Y E 2 p1, 3q. Then, thanks to the inequalities (we shall make a high use of them)
	log a ď 2p ? a ´1q 2ab ď N A a 2 `A N	b 2	(5.3.16) (5.3.17)
	for any positive a, b, A, the integral in the r.h.s. of (5.3.12) is bounded by
	ż E 1 p1,3qYE 2 p1,3q ď 2 ż E 1 p1,3qYE 2 p1,3q η 1 pxq ´xF 1,3 pξ, ωqy n`1 ´xF p1q 1,3 pξ, ωqy n`1 ¯log p2q η 1 pxq ´xF p1q 1,3 pξ, ωqy n`1 ´xF 1,3 pξ, ωqy n`1 f t n pξ, σ x ωq f t n pξ, ωq p2q df dν N p θ,n pξ, ωq
	t n pξ, σ x ωq f t n pξ, ωq	´1¯d	ν N p θ,n pξ, ωq
				133

x,y q dν N p θ,n`1 pξ, ωq 132 5.p1q i,j pξ, ωq " η j pyqf t n`1 pξ, ωq , F p2q i,j pξ, ωq " η j pyqf t n`1 pξ x,y , ω x,y q . (5.3.13) If we now define E 1 pi, jq " tpξ, ωq : xF p1q i,j pξ, ωqy n`1 ě xF

With stochastic reservoirs or in infinite volume

  pxqη 1 pyqe pϑ 3 py{N q´ϑ 3 px{N qq´pϑ 1 py{N q´ϑ 1 px{N qq f t n`1 pξ, ωqdνN

	5.3. Proof of the specific entropy (Theorem 5.2.1) ď 4A N ż E 1 p1,3q η 1 pxq ´f t n pξ, σ x ωq ´f t n pξ, ωq ¯dν N p Chapter 5. Proceeding as in (5.3.5), we get θ,n pξ, ωq ď 4A N ż η 1 pxqf t n pξ, σ x ωqdν N p θ,n pξ, ωq " ż ż η 3 pxqe pϑ 1 px{N q´ϑ 3 px{N qq f t n pξ, ωqdν N p θ,n pξ, ωq f t n`1 L N,n log f t n dν N p θ,n`1 ď ´Dn pµ n ptq|ν p θ,n q `ż L N,n f t n dν N p θ,n (5.3.24)
	ď	AC 1 N	(5.3.21)
	for some positive constant C 1 . We have used the definition (5.3.14) of E 1 p1, 3q for the first and third inequalities, the definition (5.3.13) of F p1q 1,3 pξ, ωq with the bound xF p1q 1,3 pξ, ωqy n`1 ď xf t n`1 pξ, ωqy n`1 " f t n pξ, ωq for the second inequality, Lemma 5.A.2(iii) for the equality, (5.2.18), (5.2.11) and that f t n is a probability density to conclude.
	We now look at the integral on E 2 p1, 3q, to get A N ż E 2 p1,3q η 1 pxq ´bxF p1q 1,3 pξ, ωqy n`1 `bxF 1,3 pξ, ωqy n`1 p2q	¯2
		ď ď ď	4A N 8A N 8A N	ˆ´d p2q 1,3 pξ, ωqy n`1 f t n pξ, σ x ωq f t n pξ, ωq xF f t ´af t ´1¯2 n pξ, σ x ωq ´af t dν N p θ,n pξ, ωq n pξ, ωq ¯2dν N p θ,n pξ, ωq n pξ, ωq xF p2q 1,3 pξ, ωqy n`1 f t n pξ, ωq f t n pξ, ωqdν N p θ,n pξ, ωq η 3 p E 2 p1,3q ż η 1 pxq ż E 2 p1,3q η 1 pxq ż θ,n`1 pξ, ωq
		ď	AC 1 1 N	(5.3.22)
	for some positive constant C 1 1 . We have used the definition (5.3.15) of E 2 p1, 3q for the first and second inequalities, the definition (5.3.13) of F p2q 1,3 pξ, ωq with Lemma 5.A.1 for the third inequality, and (5.2.18), (5.2.11) and finally that f t n is a probability density.
	¯2 To conclude to an upper bound of Ω 1 , combining (5.3.5) with (5.3.20), (5.3.21), 1,3 pξ, ωqy n`1 (5.3.22)
	ˆ´d Ω 1 ď ´N 2 D 0 f t n pξ, σ x ωq f t n pξ, ωq n pµ n ptq|ν N ´1¯2 p θ,n q `C2 dν N p 1 AN n d´1 θ,n pξ, ωq Bound on Ω 2 . We decompose the generator of the reaction part into a part involving (5.3.23) ď 4A N ż E 1 p1,3q η 1 pxq xF p1q 1,3 pξ, ωqy n`1 f t n pξ, ωq ´af t n pξ, σ x ωq ´af t n pξ, ωq ¯2dν N p only sites within Λ N,n and a part involving sites in Λ N,n`1 zΛ N,n . Recalling (5.2.21), (5.2.22), we have θ,n pξ, ωq ď 4A N ż E 1 p1,3q η 1 pxq ´f t n pξ, σ x ωq ´2a f t n pξ, σ x ωq a f t n pξ, ωq `f t ż ż n pξ, ωq ¯dν N p θ,n pξ, ωq Ω 2 " f t n`1 L N,n`1 log f t n dν N p θ,n`1 " f t n`1 L N,n log f t n dν N p θ,n`1 `Ωp1q 2
	134			135

  H 1,t px{N q ˜´η 1 pxqb 0 px{N q ´b1 η 0 pxq ¯f pξ, ωq `´η 1 pxqb 3 px{N q ´b1 η 3 pxq ¯f pξ, ωq `b2 px{N qf pξ, ωq ¸dν N H 1,t px{N q ˜b1 η 0 pxq ´f pσ x ξ, ωq ´f pξ, ωq b1 η 3 pxq ´f pξ, σ x ωq0f pξ, ωq ¯`b 2 px{N qf pξ, ωq ¸dν N

	5.4. Hydrodynamics in a bounded domain
		´b1 px{N q ´η0 pxq `η2 pxq `η3 pxq ¯¯f pξ, ωqdν N p θ pξ, ωq ż
	ď	B N e 1 p θ pξ, ωq
		ż
	" ď	B N e 1 p θ pξ, ωq ż b 1 η 0 pxqAN 2 ¯2 ´af pσ x ξ, ωq ´af pξ, ωq `b1 η 3 pxqAN 2 ´af pξ, σ x ωq ´af pξ, ωq ¯2dν N p θ pξ, ωq `C1 1 2AN	}B N e 1 H 1,t px{N q 2 } 8
			p b N q x appear, (5.4.14) is bounded by
		AN 2	pD p

where C

1

1 is some constant, we used (5.3.17) to get ´f pσ x ξ, ωq ´f pξ, ωq " ´af pσ x ξ, ωq ´af pξ, ωq ¯´a f pσ x ξ, ωq `af pξ, ωq ď AN 2 ´af pσ x ξ, ωq ´af pξ, ωq ¯2 `1 2AN ´af pσ x ξ, ωq `af pξ, ωq ¯2, and that f is a density while expanding the last term. Overall, dealing with the sum over i, since parts of the Dirichlet form pD

  1,s yds while other component are defined the same way. It is enough to treat the case i " 1. By Proposition 5.4.1, we may replace local functions of pξ, ωq in the martingale (5.4.1). Since occupations variables η i pxq are of mean η ǫN i , resp. b i px{N q, under the measure ν N

	p η ǫn , resp. ν N p b , one has
	lim ǫÑ0

  1,s p0q ¯ds x r Q x pη 2 qy t " ş t 0 τ x ´rη 0,s p0q `η3,s p0q `βN p0, ξ s , ω s qη 2,s p0q `η2,s p0q ¯ds x r Q x pη 3 qy t " ş t 0 τ x ´βN p0, ξ s , ω s qη 2,s p0q `rη 1,s p0q `2η 3,s p0q ¯ds (5.5.5) Proof of Proposition 5.2.1. Given a smooth continuous vector field G " pG 1 , ..., G d q P C 8 c pΛ, R d q, after definition (5.2.35), sum the martingale (5.5.1) over tx, x `ej P Λ N u to get the martingale Ă

		M G t , given by
	Ă M G t pη i q "	d ÿ j"1 ´N 2 ˜xW N j,t pη i q, G j y N d`1 ÿ ż t 0 x,x`e j PΛ N xW N t pη i q, Gy ´1 N d	G j px{N q ´ηi,s pxq ´ηi,s px `ej q ¯ds d ÿ j"1 ż t ÿ xPΛ N 0 B x	"

j G j px{N qη i,s pxqds `OpN ´1q " xW N t pη i q, Gy ´d ÿ j"1

xπ N,i s , B x j G j y `OpN ´1q

7. Uniqueness of weak solutions

  3) over x P Λ N to get the martingaler N H t pη i q " xQ N t pη i q, H i y ´1 N d ÿNow, V ǫN pξ, ωq depends on configurations only through occupations variables tpξ, ωqpxq : x P B M N u, by Fubini's theorem and Theorem 5.2.1, there exists some positive constant C 1 such that the expectation in (5.6.2) is bounded by x V ǫN pξ, ωq f T pξ, ωqdν N p θa pξ, ωq ´γT N 2´d D 0 pM `2qN p f T q `γC 1 , for all positive γ, where f T " T ´1 ż T 0 f s pM `2qN ds, with f t pM `2qN standing for the density of µ N ptq with respect to ν

	ż t , the restriction of ν p θ to the box B N M . It thus remains τ x V ǫN pξ, ωqf pM `2qN p θ ÿ xPB M N T pξ, ωqdν pM `2qN p θ pξ, ωq H T N d ż ÿ xPB M N to show that lim ǫÑ0 lim N Ñ8 sup f # ż 1 N d ´γN 2´d D 0 pM `2qN pf
	xPΛ N	0

i px{N qτ x f i pξ s , ω s qds 5.τ T q

  Proof of Lemma 5.2.2. Let p ρ p1q and p ρ p2q be two weak solutions of 5.2.31 satisfying (B1) and (B2), starting from the initial profile p γ. For a given δ ą 0, denote by A δ the regularized absolute value function Rq is dense in H 1 pΛq, by approximating A δ by smooth functions and using (B2), we get (cf.[START_REF] Farfan | Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes[END_REF]) uq ¯´F i pp ρ p1q pt, uqq ´Fi pρ p2q pt, uqq ¯du |F i pp ρ p1q q ´Fi pp ρ p2q q| ď Cpλ 1 , λ 2 , rq ÿ Cpλ 1 , λ 2 , rq

								A δ puq :"	u 2 2δ	1t|u| ď δu `´|u|	´δ 2	¯1t|u| ą δu.
	Since C 8 c pΛ; ÿ i B t ż " ÿ A δ i ż " ´ÿ i `ÿ i i pt, " ´ρp1q i pt, uq ´ρp2q i pt, uq ¯du A 1 δ ´ρp1q i pt, uq ´ρp2q i pt, uq ¯Bt ż ! ∇A 1 δ ´ρp1q i pt, uq ´ρp2q i pt, uq ¯´∇ρ ´ρp1q i pt, uq p1q ´ρp2q i pt, uq ¯du i pt, uq ´∇ρ p2q i pt, uq ) du ż ´ρp1q A 1 δ i pt, uq ´ρp2q ´1 δ ÿ i ż ∇ ´ρp1q i pt, uq ´ρp2q ´ρp1q i pt, uq ¯.∇ i pt, uq ´ρp2q
		ď	´1 δ	ÿ i	ż › › ∇	`ρp1q i pt, uq	´ρp2q	p1q i pt, uq	´ρp2q i pt, uq δ	δ ¯du 1 V δ `1V c
											i	|ρ	p1q i	´ρp2q
	Therefore,
	ÿ i	B t	ż	A δ	´ρp1q i pt, uq	´ρp2q	ÿ i	ż	ˇˇρ p1q i pt, uq	´ρp2q

i pt, uq ¯1V δ du `ÿ i ż ´Fi pp ρ p1q pt, uqq ´Fi pρ p2q pt, uqq ¯´ρ i pt, uq ˘› › 2 1 V δ du `ÿ i ż ˇˇF i pp ρ p1q pt, uqq ´Fi pρ p2q pt, uqq ˇˇdu where V δ " tpt, xq P r0, T s ˆΛ : |p ρ p1q ´p ρ p2q | ď δu. Remark now that p F is Lipschitz, i |, for all i " 1, 2, 3. i pt, uq ¯du ď i pt, uq ˇˇdu.

  1 0 pp´1, 1qq. Hence, a function ψ belongs to L 2 pp´1, 1qq if and only if xψ, ϕ k y 2 ϕ k in L 2 pp´1, 1qq. In this case, for each ψ 1 , ψ 2 P L 2 pp´1, 1qq xψ 1 , ψ 2 y 2 " xψ 1 , ϕ k y 2 xψ 2 , ϕ k y 2 . k xψ 1 , ϕ k y 2 xψ 2 , ϕ k y 2 (5.7.2)

		ψ " lim nÑ8	n ÿ k"1
			8 ÿ
			k"1
	Furthermore, a function ψ belongs to H 1 0 pp´1, 1qq if and only if
		ψ " lim nÑ8	n ÿ k"1	xψ, ϕ k y 2 ϕ k
	in H 1 0 pp´1, 1qq, and	xψ 1 , ψ 2 y 1,2 "	8 ÿ k"1

α

. Uniqueness of weak solutions 1

  ď i ď 3. We shall prove below that for any function mp¨, ¨q P L 8 pr0, T s ˆΛ8 q and each i ď i ď d, mps, uqH f i t,ε ps, vqdv ˇˇˇd s ď C 1 t }m} 8 }f i } 1 , (5.7.4)

	5.7ż t	ds ˇˇˇż
	0	Λ 8
		p1q i	´ρp2q i ,

3

. Set m i " ρ

Changes of variables formulas it

  We claim that m i P L 8 pr0, ts ˆΛ8 q and }m i } 8 ď C 1 , by(5.7.5), for any open set U of Λ 8 with finite Lebesgue measure λpU q, we have for all 0 ď τ ď t, ż remains to prove inequality(5.7.4). From Fubini's Theorem, we have

	ż t 0 ď	ˇˇˇż Λ 8 ż t 0 ds mps, uqH f i t,ε ps, uqdu ˇˇˇd s ż R d´1 dv ż R d´1 dǔ ˇˇˇˇÿ ně1 e ´n2 π 2 pt`ε´sq ˆż 1 ´1 du 1 ! sinpnπu 1 qppt `ε ´s, ǔ, vqmps, u 1 , ǔq ż 1 ´1 dv 1 ! ) sinpnπv 1 qf i pv 1 , vq ) ż t ż ż	ˇˇˇď
		0	ds	R d´1	dv	R d´1	dǔ ppt `ε ´s, ǔ, vq	ˇˇˇď
		ż t		ż	ż		ˆˇˇˇˇÿ
			ds	du
		0		Λ 8	Λ 8
							1 1 t	´3 ÿ i"1	}ρ	p1q i	´ρp2q i } 8 ¯}f k } 1 .	(5.7.5)
							1 t	´3 ÿ i"1	}ρ	p1q i	´ρp2q i } 8 ¯.	(5.7.6)
	Indeed (cf. [67], [61]), denote by Rptq " i } 8 U ř 3 i"1 }ρ p1q i ´ρp2q m i pτ, uqdu ď C 1 1 t RptqλpU q.	(5.7.7)

Fix 0 ă δ ă 1. For any open set U of Λ 8 with finite Lebesgue measure and for 0 ď τ ď t let B U δ,τ " ! u P U : m i pτ, uq ą C 1 1 t Rptqp1 `δq ) . 5.A. ně1 ϕ n , mps, p¨, ǔqq ˆ ϕ n , f i p¨, vq dv ! |mps, uq| |f i pvq| ppt `ε ´s, ǔ, vq ) ď 4 t }m} 8 }f i } 1 ,

  z`e k pξ s , ω s qds. By Itô's lemma, `e k pξ t , ω t qf iÑj z,z`e k pξ t , ω t q `e k pξ s , ω s q r f iÑj z,z`e k pξ s , ω s q `ż t 0 r f iÑj z,z`e k pξ s , ω s qf iÑj z,z`e k pξ s , ω s qds. `e k pξ t , ω t qV iÑj z,z`e k pξ t , ω t q´ş t 0 ´r f iÑj z,z`e k pξ s , ω s qV iÑj z,z`e k pξ s , ω s qds is a martingale and ´r f iÑj z,z`e k pξ t , ω t q ¯2 " r f iÑj z,z`e k pξ t , ω t q ´2´r f iÑj z,z`e k pξ t , ω t qV iÑj z,z`e k pξ t , ω t q `e k pξ s , ω s qdV iÑj z,z`e k pξ s , ω s q `e k pξ s , ω s qdV iÑj z,z`e k pξ s , ω s qds. ´1 ´2f iÑj z,z`e k pξ s , ω s q ¯dV iÑj z,z`e k pξ s , ω s qds. `e k pξ s , ω s qd r f iÑj z,z`e k pξ s , ω s qds " r J iÑj z,z`e k pξ t , ω t q,

	5.B. Quadratic variations computations
	r f iÑj z,z" z,zTherefore, r ż t ´f iÑj 0 f iÑj z,z´ż t 0 ´r f iÑj z,z¯`ż t 0 z,zBy Doob's decomposition, p1 ´f iÑj A r f iÑj z,z`e k pξ, ωq E t ş t " ş t ´giÑj 0 Hence, since 0 z,zA r ż t E J iÑj z,z`e k pξ, ωq t " 0
	158

With stochastic reservoirs or in infinite volume

  computations give A´QTo conclude the case Q x t pη 1 q, compute as well

	ż t 0 ż t η 3,s pzqds, xQ 1Ñ0 z pξ, ωq ¯yt " Chapter 5. xQ 1Ð3 z pξ, ωqy t " xQ 1Ñ3 z 0 rη s,1 pzqds.	pξ, ωqy t "	ż t 0	η s,1 pzqds,
	1Ð0 z	¯Et pξ, ωq	"	ż t 0	βpz, ξ, ωqη s,0 pzqds.
						159

  Cpλ 1 , λ 2 , rq ÿ xPΛ N ż ´f pξ, ωqf pσ x ξ, ωq ´f pξ, ωq 2 ¯dν N ωq 2 `1 2c f pσ x ξ, ωq 2 ´f pξ, ωq 2 ¯dν N p θ pξ, ωq for an arbitrary c ą 0 with use (5.3.17) for the last inequality. Choosing c " 2, Bp p θq depending on p θ through a change of variables related to Lemma 5.A.2(ii). Similarly, one getsI 2 ďCpλ 1 , λ 2 , rqB 1 p p θq 4 θq from a change of variables corresponding to Lemma 5.A.2(iii). Since f P L 2 pν p θ q, we havexL N f, f y ď A 1 N d }f } 2

		ď Cpλ 1 , λ 2 , rq f pξ, I 1 ď ÿ xPΛ N ż ´c 2 Cpλ 1 , λ 2 , rq 4 ÿ xPΛ N	ż	p θ pξ, ωq θ pξ, ωq f pσ x ξ, ωq 2 dν N p
		ď	Cpλ 1 , λ 2 , rqBp p θq 4	ÿ xPΛ N
					ÿ
					xPΛ N
					L 2 pν N p θ	q .
	with A 1 "	Cpλ 1 , λ 2 , rqBp p θq 4	`Cpλ 1 , λ 2 , rqB 1 p p θq 4	.

ż f pξ, ωq 2 dν N p θ pξ, ωq for some positive constant ż f pξ, ωq 2 dν N p θ pξ, ωq for some positive constant B 1 p p

This chapter is a joint work with M. Mourragui and E. Saada[START_REF] Kuoch | Hydrodynamic limits for a generalized contact process with stochastic reservoirs or in infinite volume[END_REF].

Remerciements

that admits ν N p ρ defined in (4.2.6) as unique invariant measure, this is the so-called auxiliary process.

Denote by r P N ν N p ρ the probability measure of the auxiliary process starting from the initial measure ν N p ρ and by r E N ν N p ρ the corresponding expectation. In view of the dynamics of the reaction part, there is no way to build a generator that is reversible with respect to the reference measure, this would though be possible for the symmetric CP-DRE, as we will discuss in the next chapter. Since the occupation variables are bounded, by convexity of the entropy (see [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF]Appendix I.8]),

Entropy of P

for some positive constant C 0 .

To study the entropy of P N µ N with respect to r

, begin by computing the associated Radon-Nikodym density. For this, introduce the following jump processes corresponding to each transition of the reaction part : ' D x,i t : number of deaths of type-i particles on site x up to time t, for i " 1, 2. ' B x t : number of births of type-1 particles on site x up to time t. ' I x,j t : number of arrivals of type-2 particles on site x in state j up to time t, for j " 0, 1.

5

Hydrodynamic limits of a generalized contact process with stochastic reservoirs or in infinite volume

According to (5.1.1), the generator L N :" L N,λ 1 ,λ 2 ,r of the generalized contact process is given by

(5.2.9)

with

where

, and for ξ P Σ N , σ x ξ is the configuration obtained from ξ by flipping the configuration at x, i.e.

The representation (5.2.1) sheds light on the fact that (5.2.9) corresponds to a contact process (the ξ-particles) in a dynamic random environment, namely the ω-particles. Indeed, the ω-particles move by their own and are not influenced by ξ-particles, while ξ-particles have birth rates whose value depends on the presence or not of ω-particles. Note that in [START_REF] Kuoch | Phase transition for a contact process with random slowdowns[END_REF] (see Chapter 3) a variant of the generalized contact dynamics in a quenched random environment was also considered, with the pξ, ωq-formalism. On the other hand, we noticed previously that ω-particles can also be considered as an environment for the exchange dynamics.

We now turn to the dynamics at the boundaries of the domain. We denote by Λ " r´1, 1s ˆTd´1 the closure of Λ, and by Γ " BΛ the boundary of Λ : Γ " tpu 1 , . . . , u d q P Λ : u 1 " ˘1u. For a metric space E, an any integer 1 ď m ď `8 denote by C m pΛ; Eq (resp. C m c pΛ; Eq) the space of m-continuously differentiable functions on Λ with values in E (resp. with compact support in Λ) .

Fix a positive function p b : Γ Ñ R 3 `. Assume that there exists a neighbourhood V of Λ and a smooth function p θ " pθ 1 , θ 2 , θ 3 q : V Ñ p0, 1q 3 in C 2 pV ; R 3 q such that 0 ă c ď min

for two positive constants c, C, and such that the restriction of p θ to Γ is equal to p b. The boundary dynamics acts as a birth and death process on the boundary Γ N of Denote respectively by ∆ N and ∆ the discrete Laplacian and the Laplacian defined for any functions G P C 2 pΛ; Rq (resp. G P C 2 pR d ; Rq), if x, x ˘ej P Λ N (resp. Z d ) for 1 ď j ď d and u P ΛzΓ (resp. Z d ). by

B 2 e j Gpuq.

We have now all the material to state our results.

Specific entropy and Dirichlet form

Denote by Λ 8 N " t´N, ¨¨¨, N u ˆZd´1 , the macroscopic space is Λ 8 " p´1, 1q ˆRd´1 and its boundary is Γ 8 :" tpx 1 , ..., x d q P Λ 8 : x 1 " ˘1u. In this subsection we consider the sub-lattice Λ N,n " t´N, ..., N u ˆt´n, ..., nu d´1 of Λ 8 N , for fixed n ě 1. Define p Σ N,n " pt0, 1u ˆt0, 1uq Λ N,n . We start by defining the two main ingredients needed in the proof of hydrodynamic limit in infinite box : the specific entropy and the specific Dirichlet form of a measure on p Σ N with respect to some reference product measure ν We fix as reference measure a product measure ν N p θ :" ν N p θp¨q , where p θ " pθ 1 , θ 2 , θ 3 q : Λ 8 Ñ p0, 1q 3 is a smooth function with the only requirement that p θp¨q ˇˇΓ 8 " p bp¨q. In other words (recall (5.2.6), (5.2.8)), introducing the function θ 0 p.q " 1 ´θ1 p.q θ2 p.q ´θ3 p.q, we have

To do changes of variables (detailed in Appendix 5.A), it is convenient to write (5.2.16) as follows :

ϑ j px{N qη j pxq ) (5.2.17) with ϑ j px{N q " log θ j px{N q.

(5.2.18)

for some constant C 0 that depends on p θ, λ 1 , λ 2 , r (see Chapter 4 Appendix 4.A). Moreover, by (5.2.26) and (5.2.28), there exists a positive constant C 1 0 " Cp p θ, λ 1 , λ 2 , rq such that for any positive measure µ on p Σ N ,

We need more notation. We denote by pS We first prove uniform upper bounds on the entropy production and the Dirichlet form.

Theorem 5.2.1. Let p θ : Λ 8 Ñ p0, 1q 3 be a smooth function such that p θp¨q ˇˇΓ 8 " p bp¨q. For any time t ě 0, there exists a positive finite constant

To get this result, one needs to consider our system in large finite volume and bound the entropy production in terms of the Dirichlet form. This is given by the following lemma.

Lemma 5.2.1.

Hydrodynamics in a bounded domain.

Suppose in this subsection that Λ N " t´N, ¨¨¨, N u ˆTd´1 N , the macroscopic space is Λ " p´1, 1q ˆTd´1 . Fix T ą 0. We shall prove in Theorem 5.2.2 below that the macroscopic evolution of the local particles density p π N is described by the following system of non-linear reaction-diffusion equations

(5.2.32)

where ρ 0 " 1 ´ρ1 ´ρ2 ´ρ3 . By weak solution of (5.2.31) we mean a function p ρp¨, ¨q : r0, T s ˆΛ Ñ R 3 satisfying

Chapter 5. With stochastic reservoirs or in infinite volume

For x P Λ N , we denote by Q x t pη i q the total number of particles of type i created minus the total number of particles of type i annihilated at site x before time t. The corresponding non-conservative empirical measure is

We introduce the signed measure p Q N t " pQ N t pη 1 q, Q N t pη 2 q, Q N t pη 3 qq P M 3 and for p H " pH 1 , H 2 , H 3 q P pC c pΛ; Rqq 3 the notation

We can now state the law of large numbers for the current :

Proposition 5.2.1. Fix a smooth initial profile p γ : Λ Ñ R 3 . Let pµ N q be a sequence of probability measures satisfying (5.2.34) and p ρ be the weak solution of the system of equations (5.2.31). Then, for each T ą 0, δ ą 0, p

lim

We shall prove Proposition 5.2.1 in Section 5.5.

Hydrodynamics in infinite volume.

In this subsection we derive the hydrodynamic limit to the generalized contact process in Z d with generator L defined in (5.2.14). For a fixed density profile p θ, denote by ν p θ the product measure on p Σ such that

Theorem 5.2.3. Consider a sequence of probability measures pµ N q N ě1 on p Σ associated to a continuous profile p γ : R d Ñ r0, 1s 3 , that is, for all continuous function

Gpx{N qη i pxq ´xp γ, p Gy ˇˇą δ ¯" 0 for all δ ą 0. Then for all t ě 0, lim

Uniqueness of weak solutions

In this subsection, we state the results concerning the uniqueness of the weak solution to the equations of the boundary driven case in finite volume case and in infinite volume. Begin with the finite volume case : Lemma 5.2.2 (Uniqueness in finite volume). For any T ą 0, the system (5.2.31) has a unique weak solution in the class `L2 `r0, T s; H 1 pΛq ˘3.

Fix T ą 0. Let p γ " pγ 1 , γ 2 , γ 3 q : Λ 8 Ñ r0, 1s 3 be a smooth initial profile, and denote by p ρ " pρ 1 , ρ 2 , ρ 3 q : r0, T s ˆΛ8 Ñ r0, 1s 3 a typical macroscopic trajectory. The macroscopic evolution of the local particles density p π N over Λ 8 N is described by the system of the following non-linear reaction-diffusion equations with Dirichlet boundary conditions :

where p F " pF 1 , F 2 , F 3 q : r0, 1s Ñ R 3 was given in (5.2.32). By weak solution of (5.2.40) we mean a function p ρp¨, ¨q : r0, T s ˆΛ8 Ñ R 3 satisfying (IB1) For any i P t1, 2, 3u, ρ i P L 8 pp0, T q ˆΛ8 q.

Hydrodynamics in a bounded domain

Hydrodynamics in a bounded domain

We now turn into the proof of the hydrodynamics in bounded domain. Denote by ν N p θp¨q , the reference measure restricted to Λ N . Let us consider, for any function p G P pC 2 0 pr0, T s ˆΛ; Rqq 3 ,

which is a martingale for all i " 1, 2, 3. Our goal will be to make this martingale converge, and for this, first we compute :

where Γ N " tpu 1 , . . . , u d q P Λ N : u 1 " ˘N u amd B N e 1 stands for the discrete gradient : B N e 1 Gpx{N q " N ´Gppx `e1 q{N q ´Gpx{N q ¯, with x, x `e1 P Λ N , as well as

L N η 2 p0q " rη 0 p0q `η3 p0q ´βN p0, ξ, ωqη 2 p0q ´η2 p0q, (

L N η 3 p0q " β N p0, ξ, ωqη 2 p0q `rη 1 p0q ´2η 3 p0q, (5.4.5)

Note that since p G vanishes at the boundaries on Λ, L p b,N xπ N,i t , G i y " 0. To get to the system of equations (5.2.31), we shall need to replace the local functions appearing in (5.4.3)-(5.4.5) by functions of the empirical measures, thanks to the replacement lemma.

Replacement lemma.

One main step in proving the hydrodynamic limit of a system lies in being able to replace local functions by functions of the density fields to close the equations. For any ℓ P N, define the empirical mean densities in a box of size p2ℓ `1q d centred at x by p η ℓ pxq " pη ℓ 1 pxq, η ℓ 2 pxq, η ℓ 3 pxqq :

For any cylinder function φ, r φp p θq stands for the expectation of φpξ, ωq with respect to ν N p θ , so that we can define for any ǫ ą 0,

for any δ ą 0, and

B N e 1 H t px{N qn 1 px{N q ´ηi,t pxq ´bi px{N q ¯ds ˇˇ¯" 0. (5.4.8)

Before proving the replacement lemma, let us state the so-called one and two blocks estimates. The one block estimate ensures the average of local functions in some large microscopic boxes can be replaced by their mean with respect to the grand-canonical measure parametrized by the particles density in these boxes. While the two blocks estimate ensures the particles density over large microscopic boxes are close to the one over small macroscopic boxes : Lemma 5.4.2 (One block estimate). Fix a constant profile p ρ " pρ 1 , ρ 2 , ρ 3 q P p0, 1q 3 ,

where for k P N, V k pξ, ωq was defined in (5.4.6).

Lemma 5.4.3 (Two blocks estimate). Given a constant profile p ρ " pρ 1 , ρ 2 , ρ 3 q P p0, 1q 3 , for all i " 1, 2, 3,

Here, D N 0 denotes the Dirichlet form associated to the generator of exchanges of particles in Λ N .

Proof of Proposition 5.4.1. First deal with the proof of (5.4.7). By Markov's inequality,

Now use the change of variables pξ 1 , ω 1 q " pξ x,y , ω x,y q so that ż

The first term of the right-hand side is of order OpN ´1q by expanding R x,y i,u , while by inequality (5.3.17), the second term is bounded by

ℓ px{N qη u px `ej qη i pxq ´af pξ x,y , ω x,y q `af pξ, ωq ¯2dν N p θ pξ, ωq (5.4.19)

The hydrodynamic limit.

To derive the hydrodynamic behaviour of our system, we divide the proof into several steps :

(1) tightness of the measures pQ N, p b µ N q N ě1 ; (2) uniqueness of a weak solution to the hydrodynamic equation (5.2.31) ;

(3) identification of the limit points of pQ N, p b µ N q N ě1 as unique weak solutions of (5.2.31).

The identification of the limit points is provided by the following Proposition and Lemmata.

Lemma 5.4.6 (Tightness). The sequence pQ N, p b µ N q N ě1 is tight and all its limit points Q p b,å re concentrated on To show Q p b,˚i s supported on densities p ρ that are weak solutions of (5.2.31).

Lemma 5.4.7 (Identification of limit points). All limit points Q p b,˚o f the sequence pQ N, p b µ N q N ě1 are concentrated on trajectories that are weak solutions of system (5.2.31).

For further details on the proof, we refer to Chapter 4. The difference here is we need to highlight the replacement of local functions at boundaries. Define the functional p B ǫ for any function p G P C 1,2 0 pr0, T s ˆΛ; R 3 q whose first component is given by

Relying on (5.5.5), the expectation of its quadratic variation vanishes as N Ñ 8 as well. Use the Replacement lemma to express r N H t pη i q with functionals of the density fields and conclude to (5.2.37) by Doob's martingale inequality having for any δ ą 0, lim

Hydrodynamics in infinite volume

In this section, we derive the hydrodynamic limit in infinite volume of Theorem 5.2.2.

Replacement lemma

To close the equations in the expression of martingales, we state here the replacement lemma for the infinite volume. It relies on uniform upper bounds on the entropy production and the Dirichlet form given by Theorem 5.2.1 and proved in Section 5.3.

We shall make use of Theorem 5.2.1, with a slight difference : we consider here for any n ě 1, a large finite box B n " t´n, ..., nu d (instead of Λ N,n " t´N, ..., N u t´n, ..., nu d´1 ), since we do not require boundary conditions. Indeed, to reach Z d , in the proof of Lemma 5.2.1 we need to expand the box B n over B n`1 in each direction pe 1 , ..., e d q so that in our estimates : n d´1 is replaced by n d . Therefore, the result of Theorem 5.2.1 still holds. Lemma 5.6.1 (replacement lemma). For any G P C 8 c pr0, T s ˆΛ, Rq,

where V ǫN pξ, ωq was defined in (5.4.6)

Proof. Let M ą 0 so that G has compact support contained in r´M, M s d . Therefore,

Suppose that λpB U δ,τ q ą 0, there exists an open set V , such that, B U δ,τ Ă V and λ `V zB U δ,τ ˘ď λpV q δ 2 and we have λpV q `C1 1 t Rptq ˘ă λpV q `C1 1 t Rptq ˘p1 `δqp1 ´δ{2q " `C1

1 t Rptq ˘p1 `δq `λpV q ´λpV qδ{2 ď `C1 and concludes the proof of (5.7.6) by the arbitrariness of τ P r0, ts.

We now turn to the proof of the uniqueness, from (5.7.6),

and then

Rptq ď 3C 1 1 t Rptq .

Choosing t " t 0 such that 3C 1 1 t 0 ă 1, this gives uniqueness in r0, t 0 s ˆΛ8 . To conclude the proof we have just to repeat the same arguments in rt 0 , 2t 0 s, and in each interval rkt 0 , pk `1qt 0 s, k P N, k ą 1. on the limiting differential system, one would be able to derive the existence of stationary distributions for the system with stirring.

Here, we studied a system evolving in a bulk in contact with stochastic reservoirs, creating a flow of particles through the volume. The macroscopic system has been investigated in a more intricate way than the microscopic one used to be. Going back to a microscopic scale, thus to the dynamics of population, it is relevant to ask ourselves how it alters the survival and extinction phases of the process.

Hydrostatics. In finite volume, e.g. when Λ N " t´N, ¨¨¨, N u ˆTd´1 N , the Markov process pξ t , ω t q tě0 on Λ N is irreducible : for each N ě 1, there exists a unique invariant measure µ stat N . In this case, we may derive the hydrostatic limit of the system.