getdoc7e91. 407KB Jun 04 2011 12:04:34 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 16 (2011), Paper no. 4, pages 106–151.

Journal URL

http://www.math.washington.edu/~ejpecp/

Stochastic order and attractiveness for particle systems

with multiple births, deaths and jumps

Davide Borrello

∗

d.borrello@campus.unimib.it

Abstract

An approach to analyse the properties of a particle system is to compare it with different pro-cesses to understand when one of them is larger than other ones. The main technique for that is coupling, which may not be easy to construct.

We give a characterization of stochastic order between different interacting particle systems in a large class of processes with births, deaths and jumps of many particles per time depending on the configuration in a general way: it consists in checking inequalities involving the transition rates. We construct explicitly the coupling that characterizes the stochastic order. As a corollary we get necessary and sufficient conditions for attractiveness. As an application, we first give the conditions on examples including reaction-diffusion processes, multitype contact process and conservative dynamics and then we improve an ergodicity result for an epidemic model . Key words: Stochastic order, attractiveness, coupling, interacting particle systems, epidemic model, multitype contact process.

AMS 2000 Subject Classification:Primary 60K35; Secondary: 82C22.

Submitted to EJP on March 22, 2010, final version accepted December 18, 2010.

∗_{Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, 20125 Milano, Italy. Laboratoire de} Math-ématiques Raphaël Salem, CNRS UMR 6085, Université de Rouen, Saint-Étienne-du-Rouvray, France. Supported by Fondation Sciences Mathématiques de Paris. Supported by ANR grant BLAN07-218426

1

Introduction

The use of interacting particle systems to study biological models is becoming more and more fruit-ful. In many biological applications a particle represents an individual from a species which interacts with others in many different ways; the empty configuration is often an absorbing state and corre-sponds to the extinction of that species. An important problem is to find conditions which give either the survival of the species, or the almost sure extinction. When the population in a system is always larger (or smaller) than the number of individuals of another one there is astochastic orderbetween the two processes and one can get information on the larger population starting from the smaller one and vice-versa.

Attractiveness is a property concerning the distribution at time t of two processes with the same generator: if a process is attractive the stochastic order between two processes starting from different configurations is preserved in the time evolution (see Section 2.1).

The main technique to check if there is stochastic order between two systems is coupling: if the transitions are intricate an increasing coupling may be hard to construct. The main result of the paper (Theorem 2.4, Section 2.1) gives a characterization of the stochastic order (resp. attractiveness) in a large class of interacting particle systems: in order to verify if two particle systems are stochastically ordered (resp. one particle system is attractive), we are reduced to check inequalities involving the transition rates.

A first motivation is a general understanding of the ordering conditions between two processes. The analysis of interacting particle systems began with Spin Systems, that are processes with state space {0, 1}Zd

. We refer to[11]and[12]for construction and main results. The most famous examples are Ising model, contact process and voter model. These processes have been largely investigated, in particular their attractiveness (see[11, Chapter III, Section 2]). Many other models taking place onXZd, whereX ={0, 1, . . .M} ⊆N_{, that is with more than one particle per site, have been studied.}

Reaction-diffusion processes, for example, are processes with state spaceNZd _{(hence non compact),}

used to model chemical reactions. We refer to [4, Chapter 13] for a general introduction and construction. In such particle systems a birth, death or jump of at most one particle per time is allowed. But sometimes the model requires births or deaths of more than one particle per time. This is the case of biological systems with mass extinction ([17],[18]), or multitype contact process ([6],[7],[14],[16]). A partial understanding of attractiveness properties can be found in[20]for the multitype contact process. A system with jumps of many particles per time has been investigated in[8, Theorem 2.21], where the authors found necessary and sufficient conditions for attractiveness for a conservative particle system with multiple jumps onNZd with misanthrope type rates.

Those examples and the need for more realistic models for metapopulation dynamics systems ([9]) has led us to consider systems ruled by births, deaths and migrations of more than one individual per time with general transition rates to get an exhaustive analysis of the stochastic order behaviour and attractiveness. Our method relies on[8], that it generalizes.

The main applications allow to investigate the ergodic properties of a process. A process is ergodic if there exists a unique invariant measure to which it converges starting from any initial configuration: if the process is attractive, it is enough to check the convergence of the largest and the smallest initial configurations. This is a first application of Theorem 2.4. In

Section 2.2.5 we combine attractiveness and a technique calledu−criterion (see[4]) to get ergodic-ity conditions on a model for spread of epidemics, either if there is a trivial invariant measure or not.

For many biological models the empty configuration 0 is an absorbing state and the main question is if the particle system may survive, that is if there is a positive probability that the process does not converge to the Dirac measureδ0 concentrated on 0, which is a trivial invariant measure. In order to prove that a metapopulation dynamics model (see [3]) survives, we largely use comparison (therefore stochastic order) with auxiliary processes: this is a second application of the result. Instead of constructing a different coupling for each comparison, we just check that inequalities of Theorem 2.4 are satisfied on the transition rates. Moreover the main technique we use to get survival is a comparison with oriented percolation (see [6]), and attractiveness is a key tool in many steps of the proofs.

The survival of a process does not imply the existence of a non trivial invariant measure: one can have the presence of particles in the system for all times but no invariant measures. If the process is attractive and the state space is compact, a standard approach allows to construct such a measure starting from the largest initial configuration: this is the third application. Once we get survival, we use this argument to construct non trivial invariant measures for metapopulation dynamics models. In Section 2.2.4 we introduce a metapopulation dynamics model with mass migration and Allee effect investigated in[3].

The transition rates of the particle systems we analyse in this paper depend on two sites x, y, on the number of particles at x and y and on the number of particles k involved in a transition: they are of the form b(k,α,β)p(x,y), where αandβ are respectively the number of particles on

x and y and p(x,y) is a probability distribution onZd _{given by a bistochastic matrix (we require}

neither symmetry nor translation invariance). Moreover we allow birth and death rates on a site x

depending only on the configuration state onx.

In other words we work with three different types of transition rates: given a configuration η, on each site y we can have a birth(death) of k individuals depending on the configuration state

η(y) on the same site y with rate P_ηk_(y) (P_η−₍k_y)) and depending also on the number of particles on the other sites x 6= y with rate P_xR_η0,₍k_x),η(y)p(x,y) (P_xR−_η(k_y,0_),η(x)p(y,x)). We consider a death rate R−_η(k_y,0_),η(x)p(y,x) instead of the more natural R−_η(k_y,0_),η(x)p(x,y) to simplify the proofs and since we are interested in applications given by a symmetric probability distribution p(·,·). This represents a possible different interaction rule between individuals of the same population and individuals from different populations. We can have a jump of k particles from x to y with rate Γk

η(x),η(y)p(x,y), which represents a migration of a flock of individuals (see Section 2.1).

We require that the birth/death and jump rates differ only on the term b(k,α,β), that is the conservative and non-conservative rates depend on the same probability distributionp(x,y).

In Section 2.1 we recall some classical definitions and propositions needed in the sequel, we intro-duce the particle system with more details and we state the main result, Theorem 2.4. In Section 2.2 we derive the conditions on several examples (multitype contact processes, conservative dynam-ics and reaction-diffusion processes); we also detail the conditions on models with transitions of at most one particle per time. In Section 2.2.5 we apply the attractiveness conditions and the so-called

u-criterion technique to improve an ergodicity result for a model of spread of epidemics. Others ap-plications to the construction of non-trivial invariant measures in metapopulation dynamics models will be presented in a subsequent paper (see[3]). In Section 3 we prove Theorem 2.4: the coupling is constructed explicitly through a downwards recursive formula in Section 3.2, where a detailed analysis of the coupling mechanisms is presented. We have to mix births, jumps and deaths in a non trivial way by following a preferential direction. Section 4 is devoted to the proofs needed for the application to the epidemic model. Finally we propose some possible extensions to more general systems.

2

Main result and applications

2.1

Stochastic order an attractiveness

Denote by S = Zd _{the set of sites and let X ⊆} N _{be the set of possible states on each site of an}

interacting particle system(ηt)t≥0on the state spaceΩ =XS, with semi-groupT(t)and infinitesimal generatorL given, for a local function f, by

Lf(η) = X x,y∈S

X

α,β∈X

χ_αx_,,_βy(η)p(x,y)X k>0

Γ_αk_,β(f(S−_x,k_y,kη)−f(η))+

+ (R0,_α,k_β+P_βk)(f(Sk_yη)−f(η)) + (R−_α,k_β,0+P_α−k)(f(S_x−kη)− f(η)) (2.1) whereχ_αx_,,_βy is the indicator of configurations with values(α,β)on(x,y), that is

χ_αx_,,_βy(η) = ¨

1 ifη(x) =αandη(y) =β

0 otherwise

and S−_x,k_y,k, Sk_y, S−_yk, where k > 0, are local operators performing the transformations whenever possible

(S−_x,k_y,kη)(z) =   

η(x)−k ifz=x andη(x)−k∈X,η(y) +k∈X

η(y) +k ifz= y andη(x)−k∈X,η(y) +k∈X

η(z) otherwise;

(2.2)

(Sk_yη)(z) = ¨

η(y) +k ifz= y andη(y) +k∈X

η(z) otherwise; (2.3)

(S−_ykη)(z) = ¨

η(y)−k ifz= y andη(y)−k∈X

η(z) otherwise. (2.4)

The transition rates have the following meaning:

- p(x,y)is a bistochastic probability distribution onZd; - Γk

- R0,_α,k_βp(x,y)is the part of the birth rate ofkparticles in y such thatη(y) =β which depends on the value ofηinx (that isα);

- R−_α,k_β,0p(x,y)is the part of the death rate ofkparticles in x such thatη(x) =αwhich depends on the value ofηin y (that isβ);

- P_β±kis the birth/death rate ofkparticles inη(y) =βwhich depends only on the value ofηin

y: we call it anindependentbirth/death rate.

We calladditionon y (subtractionfromx) ofk particles the birth on y (death on x) or jump from

x to y ofkparticles. By convention we take births on the right subscript and deaths on the left one: formula (2.1) involves births uponβ, deaths fromαand a fixed direction, fromαtoβ, for jumps of particles. We define, for notational convenience

Π0,_α,k_β :=R_α0,_,k_β+P_βk; Π_α−_,k_β,0:=R−_α,k_β,0+P_α−k. (2.5) We refer to[11]for the classical construction in a compact state space. Since we are interested also in non compact cases, we assume that(ηt)t≥0 is a well defined Markov process on a subsetΩ0⊂Ω, and for any bounded local function f onΩ₀,

∀η∈Ω₀, lim

t→0

T(t)f(η)−f(η)

t =Lf(η)<∞. (2.6)

We will be more precise on the induced conditions on transition rates in the examples. We state here only a common necessary condition on the rates.

Hypothesis 2.1. We assume that for each(α,β)∈X2

N(α,β):=sup{n:Γn_α,β+ Π_α0,_,n_β+ Π−_α,n_β,0>0}<∞,

X k>0

Γk_α,β+ Π0,_α,k_β+ Π−_α,k_β,0<∞.

In other words, for eachα,βthere exists a maximal number of particles involved in birth, death and jump rates. Notice thatN(α,β)is not necessarily equal toN(β,α), which involves deaths fromβ, births uponαand jumps fromβtoα.

The particle system admits an invariant measure µ if µ is such that Pµ(ηt ∈ A) = µ(A) for each

t≥0,A⊆Ω, wherePµis the law of the process starting from the initial distributionµ. An invariant

measure is trivial if it is concentrated on an absorbing state when there exists any. The process isergodic if there is a unique invariant measure to which the process converges starting from any initial distribution (see[11, Definition 1.9]).

Given two processes(ξt)t≥0and(ζt)t≥0, a coupled process (ξt,ζt)t≥0is Markovian with state space

Ω₀×Ω₀, and such that each marginal is a copy of the original process. We define a partial order on the state space:

∀ξ,ζ∈Ω, ξ≤ζ⇔(∀x∈S,ξ(x)≤ζ(x)). (2.7) A setV ⊂Ωisincreasing(resp.decreasing) if for allξ∈V,η∈Ωsuch thatξ≤η(resp.ξ≥η) then

a decreasing one.

A function f :Ω→Rismonotoneif

∀ξ,η∈Ω, ξ≤η⇒f(ξ)≤ f(η).

We denote byM the set of all bounded, monotone continuous functions onΩ. The partial order onΩinduces a stochastic order on the setP of probability measures onΩendowed with the weak topology:

∀ν,ν′∈ P, ν≤ν′⇔(∀f ∈ M,ν(f)≤ν′(f)). (2.8) The following theorem is a key result to compare distributions of processes withdifferent generators

starting with different initial distributions.

Theorem 2.2. Let(ξt)t≥0 and(ζt)t≥0 be two processes with generators Lfand L and semi-groups

e

T(t)and T(t)respectively. The following two statements are equivalent: (a) f ∈ M andξ0≤ζ0impliesTe(t)f(ξ0)≤T(t)f(ζ0)for all t≥0.

(b) Forν,ν′∈ P,ν ≤ν′impliesνTe(t)≤ν′T(t)for all t≥0.

The proof is a slight modification of[11, proof of Theorem II.2.2].

Definition 2.3. A process (ζt)t≥0 is stochastically larger than a process (ξt)t≥0 if the equivalent

conditions of Theorem2.2are satisfied. In this case the process(ξt)t≥0 is stochastically smaller than

(ζt)t≥0 and the pair(ξt,ζt)t≥0 is stochastically ordered.

Attractiveness is a property concerning the distribution at time t of two processes with the same generatorwhich start with different initial distributions. By taking Te= T, Theorem 2.2 reduces to

[11, Theorem II.2.2]and Definition 2.3 is equivalent to the definition of an attractive process (see

[11, Definition II.2.3]). If an attractive process in a compact state space starts from the larger initial configuration, it converges to an invariant measure.

The main result gives necessary and sufficient conditions on the transition rates that yield stochastic order or attractiveness of the class of interacting particle systems defined by (2.1). It extends[8, Theorem 2.21].

We denote byS = S(Γ,R,P,p) ={Γ_αk_,β,R0,_α,k_β,R−_α,k_β,0,P_β±k,p(x,y)}_{α,β∈X,k>0,(x,y)∈S2_}, the set of pa-rameters that characterize the generator (2.1) of process(ηt)t≥0. We call the latter anS particle

systemand we writeηt∼ S. Given (α,β),(γ,δ)∈X2×X2, we write(α,β)≤(γ,δ)ifα≤γand

β≤δ.

LetS =S(Γ,R,P,p) andSf= S(eΓ,eR,eP,p) be two processes with different transition rates (but the same p). For all (α,β) ∈ X2, let Ne(α,β) be defined as N(α,β) in Hypothesis 2.1 using the transition rates ofSf.

Theorem 2.4. Given K ∈N_{, j:={ji}i}

inN_{, andα,β,γ,δin X such thatα≤γ,β≤δ, we define}

I_a:= I_aK(j,m) = K [ i=1

{k∈X:m_i≥k> δ−β+j_i} (2.9)

Ib:= I_bK(j,m) = K [ i=1

{k∈X:γ−α+mi≥k> ji} (2.10)

I_c:= I_cK(h,m) = K [ i=1

{k∈X:m_i≥k> γ−α+h_i} (2.11)

I_d:= I_dK(h,m) = K [ i=1

{k∈X:δ−β+m_i ≥k>h_i} (2.12)

AnS particle system(ηt)t≥0isstochastically largerthan anSfparticle system(ξt)t≥0 if and only if

X k∈X:k>δ−β+j1

e

Π0,_α,k_β+X k∈Ia

e

Γ_αk_,β ≤ X

l∈X:l>j1

Π0,_γ,l_δ+X l∈Ib

Γl_γ,δ (2.13)

X k∈X:k>h1

e

Π−_α,k_β,0+X k∈Id

e

Γ_αk_,β ≥ X

l∈X:l>γ−α+h1

Π−_γ,l_δ,0+X l∈Ic

Γl_γ,δ (2.14)

for all choices of K≤Ne(α,β)∨N(γ,δ),h,j,m,α≤γ,β≤δ.

Remark 2.5. The restriction K ≤ Ne(α,β)∨N(γ,δ) avoids that an infinite number of K, I_a, I_b, I_c, I_d result in the same rate inequality. Since eΓk_α,β = 0for each k>Ne(α,β), if K > Ne(α,β)no terms are being added to the left hand side of (2.13), and adding more terms on the right hand side does not give any new restrictions. A similar statement holds for (2.14), with the corresponding condition K≤N(γ,δ).

Remark 2.6. IfS =Sf, Theorem2.4states necessary and sufficient conditions for attractiveness ofS.

We follow the approach in[8]: in order to characterize the stochastic ordering of two processes, first of all we find necessary conditions on the transition rates. Then we construct a Markovian increasing coupling, that is a coupled process(ξt,ζt)t≥0which has the property thatξ0≤ζ0 implies

P(ξ0,ζ0)_{ξ

t≤ζt}=1,

for allt≥0. HereP(ξ0,ζ0)_{denotes the distribution of(ξ}

t,ζt)t≥0with initial state(ξ0,ζ0).

2.2

Applications

We propose several applications to understand the meaning of Conditions (2.13)–(2.14). We think of(α,β)≤(γ,δ)as the configuration state of two processesηt ≤ξt on a fixed pair of sites x and

2.2.1 No multiple births, deaths or jumps LetN= sup

(α,β)∈X2

N(α,β):

Proposition 2.7. If N =1then a change of at most one particle per time is allowed and Conditions (2.13) and (2.14) become

e

Π0,1_α,β+eΓ1_α,β≤Π0,1_γ,δ+ Γ1_γ,δ ifβ=δandγ≥α, (2.15)

e

Π0,1_α,β≤Π0,1_γ,δ ifβ=δandγ=α, (2.16)

e

Π−_α,1,0_β +eΓ1_α,β≥Π_γ−_,1,0_δ + Γ1_γ,δ ifγ=αandδ≥β, (2.17)

e

Π_α−_,1,0_β ≥Π_γ−_,1,0_δ ifγ=αandδ=β. (2.18)

Proof. . Ifβ < δ, thenδ−β+ji≥δ−β+j1≥1 for allK>0, 1≤i≤Kso that 1∈/Iaby definition

(2.9). SinceN=1 the left hand side of(2.13)is null; ifβ=δthe only case for which the left hand side of(2.13)is not null is j₁=0, which gives

X k>0

e

Π0,_α,k_β+X k∈Ia

e

Γ_αk_,β ≤X

l>0

Π0,_γ,βl +X l∈Ib

Γl_γ,β.

SinceN =1, the valueK =1 covers all possible setsIa and Ib, namelyIa ={k:m1 ≥k>0}and

I_b={γ−α+m1 ≥l >0}. Ifm1 >0, we get (2.15). Ifγ=α andm1 =0 we get (2.16). One can prove(2.17)in a similar way.

If β = δ and γ ≥ α, Formula (2.15) expresses that the sum of the addition rates of the smaller process on y in state β must be smaller than the corresponding addition rates on y of the larger process on y in the same state. Ifβ =δandγ=αwe also need that the birth rate of the smaller process on y is smaller than the one of the larger process, that is (2.16). Conditions (2.17)–(2.18) have a symmetric meaning with respect to subtraction of particles from x.

Remark 2.8. IfSf=S, whenα=γandβ=δconditions (2.16), (2.18) are trivially satisfied, and we only have to check (2.15) whenα < γand (2.17) whenβ < δ.

Proposition 2.7 will be used in a companion paper for metapopulation models, see[3]. IfR0,_α,k_β =

0 for all α, β, k, the model is the reaction diffusion process studied by Chen (see [4]) and the attractiveness Conditions (2.15), (2.17) (the only ones by Remark 2.8) reduce to

Γ1_α,β ≤Γ1_γ,β ifγ > α; Γ1_α,δ≥Γ1_α,β ifδ > β.

In other words we need Γ1_α,β to be non decreasing with respect to α for each fixed β, and non increasing with respect to β for each fixedα. In[4], the author introduces several couplings in order to find ergodicity conditions of reaction diffusion processes. All these couplings are identical to the couplingH introduced in Section 3.2 (and detailed in Appendix A ifN=1), on configurations where an addition or a subtraction of particles may break the partial order, but differ fromH on configurations where it cannot happen.

2.2.2 Multitype contact processes

If Γk_α,β = Γek_α,β = 0, for all (α,β)∈ X2,k ≥ 0, that is when no jumps of particles are present, all rates are contact-type interactions. Such a process is called multitype contact process. Conditions (2.13)–(2.14) reduce to: for all(α,β)∈X2,(γ,δ)∈X2,(α,β)≤(γ,δ),h1≥0, j1≥0,

i) X k>δ−β+j1

e

Π0,_α,k_β≤X

l>j1

Π0,_γ,l_δ; ii) X

k>h1

e

Π−_α,k_β,0≥ X

l>γ−α+h1

e

Π−_γ,l_δ,0. (2.19) Many different multitype contact processes have been used to study biological models. We propose some examples with the corresponding conditions. Since the state spaceΩ ={0, 1, . . . ,M}Zd

(where

M<∞) is compact, we refer to the construction in[11].

Spread of tubercolosis model ([17]). HereM represents the number of individuals in a population at a sitex ∈Zd_{. The transitions are:}

P_β1=φβ1l_{{0≤β≤M−1}}, R0,1_α,β =2dλα1l_{β=0},

P_β−β =1l_{1≤β≤M}, p(x,y) =

1

2d1l{x∼y}.

where y∼x is one of the 2d nearest neighbours of site x.

Given two systems with parameters(λ,φ,M)and(λ,φ,M), the proof of[17, Proposition 1]reduces to check Conditions (2.19):

φβ1l_{0≤β≤M−1}+2dλα1l{β=0}≤φδ1l{0≤δ≤M−1}+2dλγ1l{δ=0}, ifβ=δ,j1=0 1l_{{1≤α≤M,α>h}

1}≥1l{1≤γ≤M,γ>γ−α+h1}, ifγ≥α,h1≥0 which are satisfied ifλ≤λ,φ≤φandM≤M.

In the following examples we suppose Sf= S, that is we consider necessary and sufficient conditions for attractiveness.

2-type contact process ([14]). In this model M = 2. Since a value on a given site does not represent the number of particles on that site, we write the state space{A,B,C}Zd

. The value B

represents the presence of a type-B species,C the presence of a type-C species andAan empty site. IfA=0,B=1,C =2 then the transitions are

R0,1_α,β =2dλ11l{α=1,β=0}, R0,2α,β=2dλ21l{α=2,β=0}, P_β−β =1l_{1≤β≤2}, p(x,y) =

1

2d1l{x∼y}.

By takingh₁=0, Condition (2.19) is

X k>δ−β

1l_{k=1}2dλ11l{α=1,β=0}+1l{k=2}2dλ21l{α=2,β=0}

≤2dλ11l{γ=1,δ=0}+2dλ21l{γ=2,δ=0};

By takingβ=0,δ=1,α=γ=2 we get 2dλ2≤0, which is not satisfied sinceλ2>0. As already observed, see[20, Section 5.1], one can get an attractive process by changing the order between species: namely by takingA=1,B=0 andC =2 the process is attractive.

2.2.3 Conservative dynamics

If Π0,_α,k_β = 0, for all (α,β) ∈ X2,k ∈ N_{, we get a particular case of the model introduced in} [8], for which neither particles births nor deaths are allowed, and the particle system is conservative. Suppose that in this model the rateΓ_αk_,β(y−x)has the formΓk_α,βp(y−x)for eachk,α,β. Necessary and sufficient conditions for attractiveness are given by[8, Theorem 2.21]:

X k>δ−β+j

Γk_α,βp(y−x)≤X

l>j

Γl_γ,δp(y−x) for each j≥0 (2.20)

X k>h

Γk_α,βp(y−x)≥ X

l>γ−α+h

Γl_γ,δp(y−x) for eachh≥0 (2.21)

while(2.13)−(2.14)become

X k∈Ia

Γ_αk_,βp(y−x)≤X

l∈Ib

Γl_γ,δp(y−x) (2.22)

X k∈Id

Γ_αk_,βp(y−x)≥X

l∈Ic

Γ_γl_,δp(y−x) (2.23)

for allI_a, I_b,I_candI_d given by Theorem 2.4.

Proposition 2.9. Conditions (2.20)–(2.21) and Conditions (2.22)–(2.23) are equivalent.

Proof. . By choosing ji = j, hi = hand mi = m> N(α,β)∨N(γ,δ) for each i in Theorem 2.4,

Conditions (2.22)–(2.23) imply (2.20)–(2.21). For the opposite direction, by subtracting (2.21) withh=m_i > δ−β+j_i to (2.20) with j= j_i

X mi≥k>δ−β+ji

Γk_α,βp(y−x)≤ X

γ−α+mi≥l′>ji

Γl_γ′_,δp(y−x) (2.24)

and we get (2.22) by summingK times (2.24) with different values of j_i, m_i, 1≤i≤K. Condition (2.23) follows in a similar way.

2.2.4 Metapopulation model with Allee effect and mass migration

The third model investigated in[3]is a metapopulation dynamics model where migrations of many individuals per time are allowed to avoid the biological phenomenon of the Allee effect (see[1],

[19]). The state space is compact and on each site x ∈Zd _{there is a local population of at mostM}

individuals. GivenM≥M_A>0,M >N>0,φ,φA,λpositive real numbers, the transitions are

P_β1=β1l_{β≤M−1} P_β−1=β φA1l{β≤MA}+φ1l{MA<β}

Γk_α,β=

¨

λ α−k≥M−N andβ+k≤M, 0 otherwise.

for eachα,β ∈X, and p(x,y) = 1

2d1l{y∼x}. In other words each individual reproduces with rate 1,

but dies with different rates: eitherφ_Aif the local population size is smaller than M_A(Allee effect) orφ if it is larger. When a local population has more thanM−N individuals a migration of more than one individual per time is allowed. Such a process is attractive by[3, Proposition 5.1, where

2.2.5 Individual recovery epidemic model

We apply Theorem 2.4 to get new ergodicity conditions for a model of spread of epidemics.

The most investigated interacting particle system that models the spread of epidemics is the contact process, introduced by Harris[10]. It is a spin system(ηt)t≥0 on{0, 1}

Zd

ruled by the transitions

ηt(x) =0→1 at rate X y∼x

ηt(y)

ηt(x) =1→0 at rate 1

See[11]and[12]for an exhaustive analysis of this model.

In order to understand the role of social clusters in the spread of epidemics, Schinazi[17] intro-duced a generalization of the contact process. Then, Belhadji[2]investigated some generalizations of this model: on each site inZd _{there is a cluster ofM≤ ∞individuals, where each individual can}

behealthyorinfected. A cluster is infected if there is at least one infected individual, otherwise it is healthy. The illness moves from an infected individual to a healthy one with rateφif they are in the same cluster. The infection rate between different clusters is different: the epidemics moves from an infected individual in a cluster y to an individual in a neighboring cluster x with rateλif x is healthy, and with rateβif x is infected.

We focus on one of those models, theindividual recovery epidemic modelin a compact state space: each sick individual recovers after an exponential time and each cluster contains at mostM individ-uals. The non-null transition rates are

R0,_η(k_x),η(y)= ¨

2dλη(x) k=1 andη(y) =0

2dβη(x) k=1 and 1≤η(y)≤M−1, (2.25)

P_ηk_(y)=   

γ k=1 andη(y) =0

φη(y) k=1 and 1≤η(y)≤M−1

η(y) k=−1 and 1≤η(y)≤M,

(2.26)

p(x,y) = 1

2d1l{y∼x} for each(x,y)∈S

2_{. (2.27)}

andΓk

η(x),η(y)=0 for allk∈N. The rateγrepresents a positive “pure birth” of the illness: by setting

γ= 0, we get the epidemic model in[2], where the author analyses the system with M <∞and

M=∞and shows[2, Theorem 14]that different phase transitions occur with respect toλandφ. Moreover ([2, Theorem 15]), if

λ∨β < 1−φ

2d (2.28)

the disease dies out for each cluster sizeM.

By using the attractiveness of the model we improve this ergodicity condition. Notice that a depen-dence on the cluster sizeM appears.

Theorem 2.10. Suppose

λ∨β < 1−φ

2d(1−φM₎, (2.29)

withφ <1and either i)γ=0, or ii)γ >0andβ−λ≤γ/(2d). Then the system is ergodic.

Notice that ifγ >0 andβ≤λhypothesisii)is trivially satisfied.

If M =1 and γ=0 the process reduces to the contact process and the result is a well known (and already improved) ergodic result (see for instance[11, Corollary 4.4, Chapter VI]); as a corollary we get the ergodicity result in the non compact case asM goes to infinity.

In order to prove Theorem 2.10, we use a technique calledu-criterion. It gives sufficient conditions on transition rates which yield ergodicity of an attractive translation invariant process. It has been used by several authors (see[4],[5],[13]) for reaction-diffusion processes.

First of all we observe that

Proposition 2.11. The process is attractive for allλ,β,γ,φ, M .

Proof. . SinceM =1, thenN =1 and Conditions (2.13)–(2.14) reduce to (2.15), (2.17). Namely, given two configurations η∈Ω, ξ∈Ω withξ≤η, necessary and sufficient conditions for attrac-tiveness are

ifξ(x)< η(x)andξ(y) =η(y),

(

R0,1_ξ(x),ξ(y)+P_ξ1_(y)≤R0,1_η(x),η(y)+P_η1_(y);

P_ξ−₍1_y)≥P_η−₍1_y).

Ifξ(y) =η(y)≥1, 2dβξ(x)≤2dβη(x); ifξ(y) =η(y) =0, 2dλξ(x)≤2dλη(x). In all cases the condition holds sinceξ≤η.

The key point for attractiveness is thatR0,1_η(x),η(y)is increasing inη(x).

Givenε >0 and{ul(ε)}l∈X such thatul(ε)>0 for alll∈X, letFε:X×X →R+be defined by

Fε(x,y) =1l{x6=y} |y−_Xx|−1

j=0

u_j(ε) (2.30)

for all x,y ∈X. When not necessary we omit the dependence onεand we simply write F(x,y) =

1l_{x6=y}

|y−_Xx|−1

j=0

u_j. Sinceu_l(ε)>0, this is a metric onX and it induces in a natural way a metric on

Ω. Namely, for eachηandξinΩwe define

ρα(η,ξ):=

X x∈Zd

F(η(x),ξ(x))α(x) (2.31)

where{α(x)}x∈Zd is a sequence such thatα(x)∈R,α(x)>0 for eachx ∈Zd and

X x∈Zd

α(x)<∞. (2.32)

Denote by Ωm := {η ∈ Ω :η(x) = mfor each x ∈Zd_{} and let η}M 0 ∈Ω

M _{and η}0 0 ∈Ω

0 _{(which is} not an absorbing state ifγ >0). The key idea consists in taking a “good sequence"{ul}l∈X and in

looking for conditions on the rates under which the expected valueEe_{(·)(with respect to a coupled}

measurePe_{) of the distance betweenη}M

t and η

0

with respect to x ∈S. We will use the generator properties and Gronwall’s Lemma to prove that if there existsε >0 and a sequence{ul(ε)}l∈X,ul(ε)>0 for alll∈X such that the metricF satisfies

e

E

LF(η0_t(x),ηM_t (x))≤ −εEe

F(η0_t(x),ηM_t (x)) (2.33) for each x∈Zd_{, then}

e

E

lim

t→∞F(η

0

t(x),η M t (x))

=0 (2.34)

uniformly with respect to x ∈ S so that the distance ρα(·,·) between the larger and the smaller

process converges to zero, and ergodicity follows. Condition (2.33) leads to

Proposition 2.12(u-criterion). If there existsε >0and a sequence{u_l(ε)}l∈X such that for any l∈X 

 

φlu_l(ε)−lu_l−1(ε)≤ −ε

l−1

X j=0

u_j(ε)−u¯(ε)(λ∨β)2d l u_l(ε)>0

(2.35)

where¯u(ε):=max

l∈X ul(ε), u−1=0by convention and u0=U>0, then the system is ergodic.

Hence we are left with checking the existence ofε >0 and positive{ul(ε)}l∈X which satisfy(2.35).

Such a choice is not unique.

Remark 2.13. Given U = 1> 0, if ul(ε) = 1for each l ∈ X then Condition (2.35) reduces to the

existence ofε >0such thatφl−l ≤ −εl−(λ∨β)2d l for all l ∈X , that isε≤1−φ−(λ∨β)2d, thus Condition(2.28). Notice that in this case

F(x,y) =

|y−_Xx|−1

j=0

1=|y−x|.

Another possible choice is

Definition 2.14. Given ε > 0 and U > 0, we set u0(ε) = U and we define (ul(ε))l∈X recursively

through

u_l(ε) = 1

φl

−ε

l−1

X j=0

u_j(ε)−U(λ∨β)2d l+lu_l−1(ε)

for each l∈X,l6=0. (2.36)

Definition 2.14 gives a better choice of{ul(ε)}l∈X, indeed theu-criterion is satisfied under the more

general assumption (2.29). Proofs of Theorems 2.10 and 2.12 are detailed in Section 4.

3

Coupling construction and proof of Theorem 2.4

In this section we prove the main result: we begin with the necessary condition, based on [8, Proposition 2.24].

3.1

Necessary condition

Proposition 3.1. If the particle system (ηt)t≥0 is stochastically larger than (ξt)t≥0, then for all

(α,β),(γ,δ) ∈ X2×X2 with (α,β) ≤ (γ,δ), for all (x,y) ∈ S2, Conditions (2.13)–(2.14) hold.

Proof. . Let(ξ,η) ∈Ω×Ω be two configurations such that ξ≤ η. Let V ⊂ Ω be an increasing cylinder set. Ifξ∈V orη /∈V,

1l_V(ξ) =1l_V(η). (3.1)

Sinceηtis stochastically larger thanξt, for allt≥0, by Theorem 2.2 (or[11, Theorem II.2.2]if we

are interested in attractiveness)

(Te(t)1l_V)(ξ)≤(T(t)1l_V)(η)

sinceν≤ν′is equivalent toν(V)≤ν′(V)for all increasing sets. Combining this with(3.1),

t−1[(Te(t)1l_V)(ξ)−1l_V(ξ)]≤t−1[(T(t)1l_V)(η)−1l_V(η)], which gives, by Assumption (2.6),

(Lf1l_V)(ξ)≤(L1l_V)(η). (3.2)

We have, by using (2.1),

(Lf1l_V)(ξ) = X

x,y∈S

p(x,y) X

α,β∈X

χ_αx_,,_βy(ξ)X k>0

e

Γ_αk_,β(1l_V(S−_x,k_y,kξ)−1l_V(ξ))

+Πe0,_α,k_β(1l_V(S_x0,_,k_yξ)−1l_V(ξ)) +Πe_α−_,k_β,0(1l_V(S_x−_,k_y,0ξ)−1l_V(ξ))

=−1l_V(ξ) X

x,y∈S

p(x,y) X

α,β∈X

χ_αx_,,_βy(ξ)X k≥0

e

Γk_α,β1l_Ω\V(S−_x,k_y,kξ)

+Πe0,_α,k_β1l_Ω\V(S0,_x,k_yξ) +Πe−_α,k_β,01l_Ω\V(S−_x,k_y,0ξ)

+1l_Ω\V(ξ) X

x,y∈S

p(x,y) X

α,β∈X

χ_αx_,,_βy(ξ)X k≥0

e

Γ_αk_,β1l_V(S_x−_,k_y,kξ)

+Πe0,_α,k_β1l_V(S0,_x,k_yξ) +Πe_α−_,k_β,01l_V(S_x−_,k_y,0ξ). (3.3) We write(L1lV)(η)by using the corresponding rates ofS.

We fix y ∈S, (αz,γz,β,δ)∈X4 with(αz,β)≤(γz,δ)for allz ∈S,z 6= y, and two configurations (ξ,η)∈Ω×Ωsuch thatξ(z) =αz,η(z) =γz, for allz∈S,z6= y,ξ(y) =β,η(y) =δ. Thusξ≤η.

We define the setC+_y of sites which interact with y with an increase of the configuration on y,

C+_y =nz∈S:p(z,y)>0 and X

k>0

e Γ_αk

z,β+ Γ

k

γz,δ+Πe

0,k

αz,β+ Π

0,k

γz,δ

>0o. _(3.4)

Denote by x = (x₁, . . . ,x_d) the coordinates of each x ∈ S. We define, for each n ∈N_{, C}+

y(n) =

trivially satisfied. GivenK ∈N_{, we fix {p}i

z}i≤K,z∈C+

y such that for each i, z, p

i

z ∈X and p i

z ≤ξ(z).

Moreover we fix{pi_y}i≤K such thatpiy > δ, for eachi. Forn∈N, let

Iy(n) = K [ i=1

ζ∈Ω:ζ(y)≥pi_y andζ(z)≥pi_z, for allz∈C+_y(n) .

The union of increasing cylinder setsIy(n)is an increasing set, to which neitherξnorηbelong. We

compute, using (3.3),

(Lf1l_I

y(n))(ξ) =

X z∈C+

y(n)

p(z,y)X k>0

1lSK

i=1{αz−k≥piz,β+k≥piy}

e Γk_α

z,β+1l

SK

i=1{β+k≥piy}

e Π0,_αk

z,β

+ X

z∈/C+ y(n)

p(z,y)X k>0

1lSK

i=1{β+k≥piy}

e Γk_α

z,β+Πe

0,k

α,β

(L1l_I

y(n))(η) =

X z∈C+

y(n)

p(z,y)X l>0

1lSK

i=1{γz−l≥pzi,δ+l≥piy}Γ

l

γz,δ+1l

SK

i=1{δ+l≥piy}Π

0,l

γz,δ

+ X

z∈/C+ y(n)

p(z,y)X l>0

1lSK

i=1{δ+l≥piy}

Γl_γ

z,δ+ Π

0,l

γ,δ

.

So, by setting

J(pz,a,b):=J({piz}i≤K,a,b) = K [ i=1

{l:a−l≥pi_z,b+l≥pi_y},

and by (3.2), ifp1_y :=min_i{pi_y}, we get

X z∈C+

y(n)

p(z,y) X k∈J(pz,αz,β)

e Γk_α

z,β+

X k≥p1

y−β

e Π0,_αk

z,β

+ X

z∈/C+ y(n)

p(z,y) X k≥p1

y−β

e Π0,_αk

z,β+eΓ

k

α_z,β

≤ X

z∈C+ y(n)

p(z,y) X l∈J(pz,γz,δ)

Γ_γl

z,δ+

X l≥p1

y−δ

Π0,_γl

z,δ

+ X

z∈/C+ y(n)

p(z,y) X l≥p1

y−δ

Π0,_γl

z,δ+ Γ

l

γz,δ

Taking the monotone limitn→ ∞gives

X z∈S

X k∈J(pz,αz,β)

e Γk_α

z,β+

X k≥p1

y−β

e Π0,_αk

z,β

p(z,y)≤X

z∈S

X l∈J(pz,γz,β)

Γl_γ

z,δ+

X l≥p1

y−δ

Π0,_γl

z,δ

p(z,y)

By choosing the valuesαz ≡α,γz≡γ, pzi ≡pαi, since

P

zp(z,y) =1, X

k∈J(pα,α,β) e

Γk_α,β+ X

k≥p1 y−β

e

Π0,_α,k_β ≤ X

l∈J(pα,γ,δ)

Γl_γ,δ+ X

l≥p1 y−δ

Π0,_γ,l_δ. (3.5)

A similar argument with

C_x−={z∈S:p(z,y)>0 and X

k>0

e

Γ_αk_,γ(z)+ Γk_γ,δ

z+Πe −k,0

α,βz + Π −k,0

γ,δz

subsets C−_x(n),n∈N_{, {p}i

z}z∈C−

x(n),i≤K ∈ X, {p

i

x}i≤K ∈ X with pix < α, for each i, p i

z ≥ δz, for all

z∈C−_y(n),i≤K, decreasing cylinder sets

D_x(n) = K [ i=1

ζ∈Ω:ζ(x)≤pi_x andζ(z)≤pi_z, for allz∈C−_y(n)

to the complement of whichξandηbelong, and the application of inequality (3.2) toξ,η,Ω\D_x(n)

(which is an increasing set since it is the complement of an increasing one) leads to

X k∈J−_(p

γ,α,β) e

Γ_αk_,β+ X

α−k≤p1 x

e

Π−_α,k_β,0≥ X

l∈J−_(p

γ,γ,δ)

Γ_γl_,δ+ X

γ−l≤p1 x

Π−_γ,l_δ,0 (3.6)

wherep1_x =max_i{pi_x},

J−(p_γ,a,b):=J−({p_γi}i≤K,a,b) = K [ i=1

{l :a−l≤pi_x,b+l≤p_γi}.

Finally, taking pi_y =δ+j_i+1, pi_α =α−m_i in (3.5), pi_x =α−h_i−1, pi_γ =δ+m_i in (3.6) gives

(2.13)–(2.14).

3.2

Coupling construction

The (harder) sufficient condition of Theorem 2.4 is obtained by showing (in this subsection) the existence of a Markovian coupling, which appears to be increasing under Conditions(2.13)–(2.14)

(see Subsection 3.3). Our method is inspired by[8, Propositions 2.25, 2.39, 2.44], but it is much more intricate since we are dealing with jumps, births and deaths.

Let ξt ∼ Sfand ηt ∼ S such that ξt ≤ ηt. The first step consists in proving that instead of

taking all possible sites, it is enough to consider an ordered pair of sites (x,y) and to construct an increasing coupling concerning some of the rates depending onηt(x),ηt(y) andξt(x),ξt(y)

(remember that we choose to take births on y, deaths on x and jumps from x to y) and a small part of the independent rates (by this, we mean deaths from x with a rate depending only onηt(x)

andξt(x)and births upon y with a rate depending only onηt(y)andξt(y)). We do not have to

combine any “dependent reaction”Re·_ξ,·

t(x),ξt(y)or jumps rateeΓ ·

ξt(x),ξt(y)on y with any rateR ·,·

ηt(z),ηt(y) orΓ·_η

t(z),ηt(y)ifzis different from x.

Definition 3.2. For fixed(x,y)∈S2, for allη∈Ωand k∈N_let

h(P_ηk_(z)) = ¨

P_ηk_(y)p(x,y) z= y,

0 otherwise; h(P

−k

η(z)) =

¨

P_η−₍k_x)p(x,y) z= x,

0 otherwise.

and

q(z,w) = ¨

p(x,y) if z=x and w= y, 0 otherwise.

An ordered pair of sites(x,y) is an attractive pair for(Sf,S) if there exists an increasing coupling for(ξt,ηt)t≥0 whereξt∼Sf(eΓ,eR,h(eP),q),ηt ∼ S(Γ,R,h(P),q). For notational convenience we call

Notice thatS(x,y)6=S(y,x), because we take into account births on the second site, deaths on the

first one, and only particles’ jumps from the first site to the second one. The same remark holds for

f

S(x,y). In other words in order to see if a pair is attractive, we reduce ourselves to a system with only

part of the rates depending on the pair, and a part of the independent rates depending on p(x,y)

(P_ηk

t(y)p(x,y)andP −k

ηt(x)p(x,y)).

Proposition 3.3. The processηt ∼ S is stochastically larger thanξt∼Sfif all its pairs are attractive

pairs for(Sf,S).

Proof. . If for each pair(x,y)we are able to construct an increasing coupling for(Sf(x,y),S(x,y)), we

define an increasing coupling for(Sf,S)by superposition of all these couplings for pairs. Indeed: it is a coupling since by Definition 3.2 the sum of all marginals gives the original rates; it is increasing since each coupling for a pair is increasing.

From now on, we work on a fixed pair of sites (x,y)∈S2 with p(x,y)> 0 and on two ordered configurations(ηt,ξt)∈Ω2for a fixedt≥0,ηt∼ S,ξt ∼Sf,ξt≤ηt and

ξt(x) =α, ξt(y) =β, ηt(x) =γ, ηt(y) =δ. (3.7)

We denote byN:=N(α,β)∨N(γ,δ)and by p:=p(x,y).

Remark 3.4. With a slight abuse of notation, since rates on(α,β)involveSfand rates on(γ,δ)involve

S, we omit the superscript∼on the lower system rates: we denote by

e

Π_α·,·_,β = Π_α·,·_,β; eΓ_α·_,β = Γ·_α,β; S := (Sf,S).

The rest of this section is devoted to the construction of an increasing coupling for (x,y) and

(α,β)≤(γ,δ).

Definition 3.5. There is alower attractiveness problemonβif there exists k such thatβ+k> δand

Π0,_α,k_β+ Γk

α,β >0;β is k-badand k is abad value(with respect toβ). There is ahigher attractiveness

problemonγif there exists l such thatγ−l< αandΠ_γ−_,l_δ,0+ Γl_γ,δ>0;γis l-badand l is abad value

(with respect toγ). Otherwiseβ is k-good (resp. γis l-good). There is anattractiveness problemon

(α,β),(γ,δ)if there exists at least one bad value.

In other words we distinguish bad situations, where an addition of particles allows lower states to go over upper ones (or upper ones to go under lower ones) from good ones, where it cannot happen. Notice that Definition 3.5 involves addition of particles uponβand subtraction of particles fromγ. If we are interested in attractiveness problems coming from addition uponαand subtraction from

δwe refer to(β,α),(δ,γ).

We choose to define a coupling rate that moves both processes only if we are dealing with an attractiveness problem, otherwise we let the two processes evolve independently through uncoupled rates. Conditions(2.13)−(2.14)do not involve configurations without an attractiveness problem, so a different construction for them does not change the result. Since N is finite, we can construct the

thenJ(k−1,ε∗)>0.

Indeed, ifk=l−1 then(4.7)is the assumption(λ∨β)2d(1−φl)<1−φ.

Suppose there exists 0< ε∗≤ε¯such thatJ(k,ε∗)>0. By Definition 2.14,(4.7)is equivalent to

−(λ∨β)2d U(1+. . .+φk) +−ε

∗Pl−k−2

j=0 uj(ε∗) + (l−k−1) −U(λ∨β)2d+ul−k−2(ε∗)

φ(l−k−2) >0

that is

−(λ∨β)2d U(l−k−1)(1+. . .+φk)φ+1−ε∗ l−_Xk−2

j=0

uj(ε∗) + (l−k−1)ul−k−2(ε∗)>0.

Therefore

J(k−1,ε∗)>−(λ∨β)2d U(1+φ+. . .+φk+1) +ul−k−2(ε∗)>

ε∗ l−k−1

l−_Xk−2

j=0

uj(ε∗). Since by hypothesisuj(ε)>0 for each 0< ε≤ε, then¯ J(k−1,ε∗)>0 and by induction(4.7)holds for each 0≤k≤l−1. By takingk=0 we get

−(λ∨β)2d U+ul−1(ε∗)>0

which is the claim. Proof. of Proposition4.2.

We prove by induction on l ∈ X that there exists ¯εl such that for each 0 < ε < ε¯l and for each 0≤ j≤l,u_j(ε)is positive, decreasing in j,U < uj(ε)

2d(λ∨β) and

0> d

dεuj(ε)≥ −CU(j) (4.8)

withCU(j)the solution of

C_U(1) = U

φ, CU(j) =

C_U(j−1) +U

φ . (4.9)

This gives

C_U(j) = U

φj(1+φ+φ

2_{+. . .+φ}j−1_{) =U} 1−φj

φj_(1−φ) (4.10)

henceuj(ε)>0 is also decreasing inε.

We prove the induction basis whenl=1. By Definition 2.14 u₁(ε) = −εu0−U(λ∨β)2d+u0

φ =u0

−ε−(λ∨β)2d+1

φ .

Since(λ∨β)2d >1−φ we can takeε < ε1 small enough to have 1−(λ∨β)2d−ε < φ, that is

u₁(ε) is positive; moreover notice that u₁(ε) is decreasing inε, and _dεd u₁(ε) =−U_φ =−CU(1) for eachε. Hence the induction basis as well as the hypothesis of Lemma 4.3 are satisfied for l =1: there exists ¯ε1=ε1∧ε1∗ such that ifε≤ε¯1 thenU >u1(ε)>0,u1(ε)is decreasing inε, and(4.6)

holds.

Suppose there exists ¯εl−1 >0 such that for each 0< ε≤ε¯l−1, then 0> dεd uj(ε)≥ −CU(j),uj(ε)is decreasing in jfor each j≤l−1,U< uj(ε)

2d(λ∨β) anduj(ε)>0 for each j≤l−1.

First of all we prove that there exists εl > 0 such that if 0 < ε < εl then ul(ε) < ul−1(ε). By

Definition 2.14

u_l(ε) =u_l−1(ε)−ε Pl−1

j=0uj(ε)−U(λ∨β)2d l+lul−1(ε)

φlul−1(ε)

. (4.11)

By induction hypothesis, ifε≤ε¯l−1 thenU >uj(ε)>0 for each j≤l−1 and we get −εPl_j=−1₀uj(ε)−U(λ∨β)2d l+lul−1(ε)

φlul−1(ε)

<−U(λ∨β)2d l+lul−1(ε) φlul−1(ε)

<1 φ−

(λ∨β)2d φ <1 that isu_l(ε)<u_l−1(ε)by(4.11)and we setε_l =ε¯_l−1.

By (4.10), C_U(j) is always positive and increasing in j. We prove that (4.8) holds for j = l. By Definition 2.14

d

dεul(ε) = 1 φl

−ε d dε

l−1 X

j=0

u_j(ε)− l−1 X

j=0

u_j(ε) +l d

dεul−1(ε)

. (4.12)

We begin with the right inequality involving the derivative ofu_l(ε)in (4.8). By induction hypothesis, if 0< ε≤ε¯l−1 by(4.9),(4.10)and(4.12)on the one hand

φl d

dεul(ε)>−l U−l CU(l−1) =−CU(l)lφ.

and on the other hand φl d

dεul(ε)≤ε l−1 X

j=0

CU(j)− l−1 X

j=0

uj(ε) +l d

dεul−1(ε)< ε l−1 X

j=0

U 1−φ j φj_(1−φ)−

l−1 X

j=0

uj(ε).

Since 0 < ε <ε¯l−1 anduj(ε) is positive and decreasing inε, asε approaches 0 the first sum on the right hand side goes to 0, while the second sum is positive and increasing: therefore for each

j≤l−1, we can takeεˆ_l small enough so that _dεd u_l(ε)<0 for each 0< ε≤εˆ_l.

Now we prove that there existsε∗_l >0 such thatu_l(ε)>0 for each 0< ε≤ε∗_l. By Definition 2.14, ul(ε)>0 if

−U(λ∨β)2d l+lu_l−1(ε)> ε l−1 X

j=0

u_j(ε). (4.13)

By the induction hypothesis, assumptions of Lemma 4.3 are satisfied, hence there exists 0< ε∗ ≤ ¯

εl−1such that (4.6) is satisfied. Thus−U(λ∨β)2d l+lul−1(ε∗)>0 and we can chooseε∗l ≤ε

that

−U(λ∨β)2d l+lu_l−1(ε∗)> ε∗_ll U (4.14) Ifε≤ε∗_l(≤ε¯l−1)

−U(λ∨β)2d l+lul−1(ε)>−U(λ∨β)2d l+lul−1(ε∗)

>ε∗_ll U > ε∗_l l−1 X

j=0

uj(ε)> ε l−1 X

j=0

uj(ε)

which is(4.13). By taking 0< ε <ε¯0∧. . . ¯εl−1∧εˆl∧ε∗_l,ul(ε)>0 and is decreasing inl andε, and the claim follows.

Proof. of Theorem 2.10.

Let{u_l(ε)}_l∈X be given by Definition(2.36). By Proposition 4.2 we can choose 0< ε <ε¯such that u_l(ε)>0 for eachl∈X and ergodicity follows from Proposition 2.12.

Acknowledgments. I thank the referee for a careful reading of the manuscript, and many comments that improve the presentation of the paper. I am grateful to Ellen Saada, the French supervisor of my Ph.D. Thesis, which was done in joint tutorage between LMRS, Université de Rouen and Università di Milano-Bicocca. I thank Institut Henri Poincaré, Centre Emile Borel for hospitality during the semester “Interacting Particle Systems, Statistical Mechanics and Probability Theory”, where part of this work was done and Fondation Sciences Mathématiques de Paris for financial support during the stay. I acknowledge Laboratoire MAP5, Université Paris Descartes for hospitality and the support of the French Ministry of Education through the ANR BLAN07-218426 grant.

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A

Appendix: Explicit coupling construction if

N

=

1

We detail the coupling whenN=1 to understand the simplest construction. Note that we have to mix birth with jump rates also in this case.

There is a lower attractiveness problem (β+1> δ) only ifβ=δ, and a higher one (γ−1< α) only ifγ=α. Definition 3.15 becomes

Definition A.1. The non zero coupling rates ofH are given by: H1,1,1,1_α,β,γ,δ= (eΓ1_α,β∧Γ1_γ,δ)p ifβ=δorγ=α;

H_α0,1,0,1_,β,γ,δ= (Πe0,1_α,β∧Π0,1_γ,δ)p

H_α1,1,0,1_,β,γ,δ= [eΓ1_α,βp−(eΓ1_α,β∧Γ1_γ,δ)p]∧[Π0,1_γ,δp−(Πe0,1_α,β∧Π0,1_γ,δ)p]

H_α0,1,1,1_,β,γ,δ= [eΠ0,1_α,βp−(Πe0,1_α,β∧Π_γ0,1_,δ)p]∧[Γ1

γ,δp−(eΓ

1

α,β∧Γ

1

γ,δ)p] ifγ > α

 

 ifβ=δ

H_α−1,0,−1,0_,β,γ,δ = (eΠ−1,0_α,β ∧Π−1,0_γ,δ )p

H_α−1,0,1,1_,β,γ,δ = [eΠ−1,0_α,β p−(Πe−1,0_α,β ∧Π−1,0_γ,δ )p]+∧[Γ1_γ,δp−(eΓ1_α,β∧Γ1_γ,δ)p]

H_α1,1,−1,0_,β,γ,δ = [eΓ1

α,βp−(eΓ1α,β∧Γ1γ,δ)p]∧[Π

−1,0

γ,δ p−(Πe

−1,0

α,β ∧Π

−1,0

γ,δ )p]ifβ < δ

 

 ifα=γ;

H_α0,1,0,0_,β,γ,δ=Πe0,1_α,βp−H_α0,1,0,1_,β,γ,δ−H_α0,1,1,1_,β,γ,δ; H1,1,0,0_α,β,γ,δ=Γe1_α,βp−H_α1,1,1,1_,β,γ,δ−H_α1,1,−1,0_,β,γ,δ −H_α1,1,0,1_,β,γ,δ; H0,0,0,1_α,β,γ,δ= Π0,1_γ,δp−H_α0,1,0,1_,β,γ,δ−H_α1,1,0,1_,β,γ,δ; H_α0,0,1,1_,β,γ,δ= Γ1_γ,δp−H_α1,1,1,1_,β,γ,δ−H_α0,1,1,1_,β,γ,δ−H_α−1,0,1,1_,β,γ,δ ; H−1,0,0,0_α,β,γ,δ =Πe−1,0_α,β p−H_α−1,0,−1,0_,β,γ,δ −H_α−1,0,1,1_,β,γ,δ ; H_α0,0,−1,0_,β,γ,δ = Π−1,0_γ,δ p−H_α−1,0,−1,0_,β,γ,δ −H_α1,1,−1,0_,β,γ,δ .

Proposition A.2. Definition A.1 gives an increasing coupling.

Proof. . The definition of uncoupled terms ensures this is a coupling. We have to prove attractive-ness.

Supposeβ=δandγ≥α. ThereforeH_α−1,0,1,1_,β,γ,β =H_α−1,0,−1,0_,β,γ,β =H_α1,1,−1,0_,β,γ,β =0.

• Suppose eΓ1_α,β ≥ Γ1_γ,δ. By Condition (2.15) we must have Πe_α0,1_,βp ≤ Π0,1_γ,δp, then H_α0,1,0,0_,β,γ,δ = 0. Moreover

H_α1,1,1,1_,β,γ,δ= Γ1

γ,δp, H

0,1,0,1

α,β,γ,δ=Πe

0,1

α,βp, H

0,1,1,1

α,β,γ,δ=0

H_α1,1,0,1_,β,γ,δ= [(eΓ1_α,β−Γ1_γ,δ)p]∧[(Π0,1_γ,δ−Πe0,1_α,β)p] = (eΓ1_α,β−Γ1_γ,δ)p by Condition (2.15). ThereforeH_α1,1,0,0_,β,γ,δ=eΓ1

α,βp−Γ1γ,δp−Γe1α,βp+ Γ1γ,δp=0. •SupposeΓe1

α,β <Γ

1

γ,δ. ThenH

1,1,1,1

α,β,γ,δ =eΓ

1

α,βpand we getH

1,1,0,1

α,β,γ,δ=H

1,1,0,0

α,β,γ,δ =0. We have to prove thatH0,1,0,0_α,β,γ,δ=0.

Ifγ=α, by Condition (2.16) thenH_α0,1,0,0_,β,γ,δ= Π0,1_α,βp−Π0,1_α,βp∧Π0,1_γ,δp=0 and we are done. Supposeγ > α:

H0,1,0,0_α,β,γ,δ=Πe0,1_α,βp−Πe0,1_α,βp∧Π0,1_γ,δp−[eΠ_α0,1_,βp−Πe0,1_α,βp∧Π0,1_γ,δp]∧[Γ1_γ,δp−eΓ1_α,βp]

By using the relation

witha= Γ1_γ,δp−eΓ1_α,βp, b= Π0,1_γ,δpandc=Πe0,1_α,βp, we get by Condition (2.15),

H_α0,1,0,0_,β,γ,δ=Πe0,1_α,βp−Πe0,1_α,βp∧Π_γ0,1_,δp−[Γ1_γ,δp−eΓ1_α,βp+ Π0,1_γ,δp]∧Πe_α0,1_,βp+Πe0,1_α,βp∧Π0,1_γ,δp

=Πe0,1_α,βp−Πe0,1_α,βp=0.