El e c t ro n ic
Jo ur
n a l o
f P
r o
b a b il 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
∗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.
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.
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 Pηk(y) (Pη−(ky)) and depending also on the number of particles on the other sites x 6= y with rate PxRη0,(kx),η(y)p(x,y) (PxR−η(ky,0),η(x)p(y,x)). We consider a death rate R−η(ky,0),η(x)p(y,x) instead of the more natural R−η(ky,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).
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
χαx,,βy(η)p(x,y)X k>0
Γαk,β(f(S−x,ky,kη)−f(η))+
+ (R0,α,kβ+Pβk)(f(Skyη)−f(η)) + (R−α,kβ,0+Pα−k)(f(Sx−kη)− f(η)) (2.1)
whereχαx,,βy is the indicator of configurations with values(α,β)on(x,y), that is
χαx,,βy(η) = ¨
1 ifη(x) =αandη(y) =β
0 otherwise
and S−x,ky,k, Sky, S−yk, where k > 0, are local operators performing the transformations whenever possible
(S−x,ky,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)
(Skyη)(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,kβp(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,kβ,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);
- Pβ±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,kβ :=Rα0,,kβ+Pβk; Πα−,kβ,0:=R−α,kβ,0+Pα−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:Γnα,β+ Πα0,,nβ+ Π−α,nβ,0>0}<∞,
X
k>0
Γkα,β+ Π0,α,kβ+ Π−α,kβ,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) ={Γαk,β,R0,α,kβ,R−α,kβ,0,Pβ±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
Ia:= IaK(j,m) = K [
i=1
{k∈X:mi≥k> δ−β+ji} (2.9)
Ib:= IbK(j,m) = K [
i=1
{k∈X:γ−α+mi≥k> ji} (2.10)
Ic:= IcK(h,m) = K [
i=1
{k∈X:mi≥k> γ−α+hi} (2.11)
Id:= IdK(h,m) = K [
i=1
{k∈X:δ−β+mi ≥k>hi} (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,kβ+X k∈Ia
e
Γαk,β ≤ X
l∈X:l>j1
Π0,γ,lδ+X l∈Ib
Γlγ,δ (2.13)
X
k∈X:k>h1
e
Π−α,kβ,0+X k∈Id
e
Γαk,β ≥ X
l∈X:l>γ−α+h1
Π−γ,lδ,0+X l∈Ic
Γlγ,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, Ia, Ib, Ic, Id result in the same rate inequality. Since eΓkα,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
Π0,1α,β+eΓ1α,β≤Π0,1γ,δ+ Γ1γ,δ ifβ=δandγ≥α, (2.15)
e
Π0,1α,β≤Π0,1γ,δ ifβ=δandγ=α, (2.16)
e
Π−α,1,0β +eΓ1α,β≥Πγ−,1,0δ + Γ1γ,δ ifγ=αandδ≥β, (2.17)
e
Πα−,1,0β ≥Πγ−,1,0δ 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,kβ+X k∈Ia
e
Γαk,β ≤X
l>0
Π0,γ,βl +X l∈Ib
Γlγ,β.
SinceN =1, the valueK =1 covers all possible setsIa and Ib, namelyIa ={k:m1 ≥k>0}and
Ib={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,kβ =
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
Γ1α,β ≤Γ1γ,β ifγ > α; Γ1α,δ≥Γ1α,β ifδ > β.
2.2.2 Multitype contact processes
If Γkα,β = Γekα,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,kβ≤X
l>j1
Π0,γ,lδ; ii) X
k>h1
e
Π−α,kβ,0≥ X
l>γ−α+h1
e
Π−γ,lδ,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:
Pβ1=φβ1l{0≤β≤M−1}, R0,1α,β =2dλα1l{β=0},
Pβ−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
R0,1α,β =2dλ11l{α=1,β=0}, R0,2α,β=2dλ21l{α=2,β=0}, Pβ−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};
2.2.3 Conservative dynamics
If Π0,α,kβ = 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 rateΓαk,β(y−x)has the formΓkα,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
Γkα,βp(y−x)≤X
l>j
Γlγ,δp(y−x) for each j≥0 (2.20)
X
k>h
Γkα,βp(y−x)≥ X
l>γ−α+h
Γlγ,δp(y−x) for eachh≥0 (2.21)
while(2.13)−(2.14)become
X
k∈Ia
Γαk,βp(y−x)≤X
l∈Ib
Γlγ,δp(y−x) (2.22)
X
k∈Id
Γαk,βp(y−x)≥X
l∈Ic
Γγl,δp(y−x) (2.23)
for allIa, Ib,IcandId 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=mi > δ−β+ji to (2.20) with j= ji
X
mi≥k>δ−β+ji
Γkα,βp(y−x)≤ X
γ−α+mi≥l′>ji
Γlγ′,δp(y−x) (2.24)
and we get (2.22) by summingK times (2.24) with different values of ji, mi, 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≥MA>0,M >N>0,φ,φA,λpositive real numbers, the transitions are
Pβ1=β1l{β≤M−1} Pβ−1=β φA1l{β≤MA}+φ1l{MA<β}
Γkα,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: eitherφAif the local population size is smaller than MA(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(kx),η(y)= ¨
2dλη(x) k=1 andη(y) =0
2dβη(x) k=1 and 1≤η(y)≤M−1, (2.25)
Pηk(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)
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),
(
R0,1ξ(x),ξ(y)+Pξ1(y)≤R0,1η(x),η(y)+Pη1(y);
Pξ−(1y)≥Pη−(1y).
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 thatR0,1η(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
uj(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
uj. Sinceul(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 ηM 0 ∈Ω
M and η0 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 betweenηM
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(η0t(x),ηMt (x))≤ −εEe
F(η0t(x),ηMt (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{ul(ε)}l∈X such that for any l∈X
φlul(ε)−lul−1(ε)≤ −ε
l−1
X
j=0
uj(ε)−u¯(ε)(λ∨β)2d l ul(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
ul(ε) = 1
φl
−ε
l−1
X
j=0
uj(ε)−U(λ∨β)2d l+lul−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
3.1
Necessary condition
We define the setC+y of sites which interact with y with an increase of the configuration on y,
trivially satisfied. GivenK ∈N, we fix {pi
The union of increasing cylinder setsIy(n)is an increasing set, to which neitherξnorηbelong. We
compute, using (3.3),
Taking the monotone limitn→ ∞gives
subsets C−x(n),n∈N, {pi
(which is an increasing set since it is the complement of an increasing one) leads to
X
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)
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)
(Pηk
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
Πα·,·,β = Πα·,·,β; eΓα·,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,kβ+ 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< αandΠγ−,lδ,0+ Γlγ,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).
coupling by a downwards recursion on the number of particles involved in a transition.
Our purpose now is to describe a coupling for(Sf(x,y),S(x,y)), that we denote byH(x,y)(or simply
H), which will be increasing under Conditions(2.13)−(2.14).
First of all we detail the construction on terms involving the larger number N of particles and we prove that under Conditions (2.13)–(2.14) none of these coupling terms breaks the partial order: this is the claim of Proposition 3.18.
Remark 3.6. By Hypothesis 2.1, at least one of the termsΠ0,α,Nβ, Πγ0,,δN, Π−α,Nβ,0, Π−γ,Nδ,0, ΓNα,β andΓNγ,δ is not null. We assume all these terms (and the smaller ones) positive. Otherwise the construction works in a similar way with some null terms.
Definition 3.7. Let
b
N+:=N−δ+β; Nb−:=N−γ+α. (3.8)
If there is a lower attractiveness problem on β, then β+N > δ (Nb+ > 0) and an addition of N
particles uponβ breaks the partial order. Such a problem comes both from birth (Π0,α,Nβp) and from jump (ΓN
α,βp) rates. IfK=1, j1=N−δ+β−1=Nb+−1 andm1=Nthen Condition (2.13) writes
Π0,α,Nβ+ ΓNα,β≤ X
l≥Nb+
(Π0,γ,lδ+ Γlγ,δ). (3.9)
Notice that ifl ≥Nb+, thenβ+N≤δ+l, and additions ofNparticles uponβand ofl particles upon
δdo not break the partial order on y. The construction consists in coupling the terms on the left hand side (which involveN particles and break the partial order) to the ones on the right hand side (in such a way that the final configuration preserves the partial order on y and on x) by following a basic coupling idea. We couple jumps on the lower configuration with jumps on the upper one and births on the lower configuration with births on the upper one. Only if this is not enough to solve the attractiveness problem, we mix births with jumps.
If there is a higher attractiveness problem onγ, thenγ−N < α (Nb− >0) and a subtraction of N
particles fromγbreaks the partial order. In this case the problem comes fromΠ−γ,Nδ,0pandΓNγ,δp; we use a symmetric construction starting from Condition (2.14) withK=1,h1=N−γ+α−1=Nb−−1 andm1=N:
Π−γ,Nδ,0+ ΓγN,δ≤ X
k≥Nb−
(Π−α,kβ,0+ Γαk,β). (3.10)
We denote byHαk,,kβ,,·γ,·,δ(resp.Hα·,·,β,l,,lγ,δ) the coupling terms which involve jumps ofk(resp.l) particles from x to y on the lower (resp. upper) configuration; Hα0,,kβ,,·γ,·,δ (resp. Hα·,·,,0,β,γl,δ) are the coupling terms concerning births ofk (resp. l) particles on y on the lower (resp. upper) configuration and
Hα−,kβ,0,,γ,·,δ· (resp. Hα·,·,,β−,γl,0,δ) are the symmetric ones for death rates.
For instance Hαk,,βk,0,,γ,lδ combines the jump of kparticles from x to y on the lower configuration and the birth ofl particles on y on the upper one.
Step 1) Suppose both β and γ are N−bad; if this is not the case one of them is good and the construction works in an easier way.
We begin with jump rates. We couple the lower configuration N−jump rate ΓN
α,βp with jumps on
the upper configuration. We first couple it withΓN
γ,δp, becauseα≤γ,α−N ≤γ−N and the final
pairs of values are(α−N,β+N)≤(γ−N,δ+N). Let
HαN,,βN,γ,N,δ,N := (ΓNα,β∧ΓNγ,δ)p. (3.11)
Then if the lower attractiveness problem is not solved, that isHαN,,βN,γ,N,δ,N = ΓNγ,δp, we have a remainder of the lower configuration jump rate that we couple with the upper configuration jump rate with the largest change of particles left, N−1. We go on by coupling the new remainder ofΓαN,βp, if positive, withΓl
γ,δpatlthstep. The final pairs of values we reach are(α−N,β+N),(γ−l,δ+l),
which always preserve the partial order onx sinceα−N ≤γ−l whenl≤N. The partial order on
y is preserved only ifβ+N≤δ+l, that is ifl≥Nb+. For this reason we stop the coupling between jumps at stepNb+: this is the meaning of formula (3.9).
More precisely, whenΓNα,β >ΓNγ,δ andN−1≥Nb+, we get the second coupling rate
HαN,,βN,γ,N,δ−1,N−1:= (ΓαN,βp−HαN,,βN,γ,N,δ,N)∧ΓγN,−δ1p= (ΓαN,β−ΓγN,δ)p∧ΓNγ,−δ1p.
Then eitherHαN,,βN,γ,N,δ−1,N−1= (ΓαN,β−ΓNγ,δ)p, and with the two stepsl=N andl=N−1 we have no remaining part ofΓNα,βp, or HαN,,βN,γ,N,δ−1,N−1= ΓγN,δ−1pand we are left with a remainder(ΓNα,β−ΓNγ,δ−
ΓNγ,−δ1)p, to go on withl=N−2, . . . Therefore we can define recursively starting from (3.11)
HαN,,βN,γ,l,,δl :=(ΓNα,βp−X
l′>l
HαN,β,N,γ,l,′δ,l′)∧Γlγ,δp=:JαN,,βl+1∧Γlγ,δp (3.12)
whereJαN,,βl is the remainder of the jump (hence notation J ) rateΓNα,βp left over after the lthstepof the coupling construction; (3.12) means that we couple the remainder from the(l+1)thstep, JαN,,βl+1, withΓl
γ,δpat the lthstep.
We proceed this way until either we have no remainder ofΓN
α,βp, or we have reachedNb
+ with the
remainder
JαN,β,Nb+= ΓαN,βp− X
l≥Nb+
HαN,,βN,γ,l,,δl = ΓαN,βp− X
l≥Nb+
Γlγ,δp>0 (3.13)
Note that, sinceβ+N ≤ δ+l for l ≥ Nb+, none of the coupled transitions until this point have broken the partial ordering. Proceeding untilNb+−1 would break the partial ordering. Therefore we stop atNb+the construction in Step 1 and we will couple the remainder JαN,β,Nb+ with upper birth rates at Step 3a.
Step 1 is detailed in Tables 1 and 2, whereNd+ corresponds to the firstl (going downwards from
N) such that the minimum in (3.12) isJαN,,βl+1. We have to distinguish between two situations:
Table 1: Njump rate,Nd+≥Nb+(JαN,,βNb+=0)
l JαN,,βl+1 ∧ Γlγ,δp = HαN,,βN,γ,l,,δl
N ×
..
. ... Γlγ,δp
Nd++1 ×
Nd+ × ΓαN,βp− X
l′>Nd+
Γlγ′,δp≥0
Nd+−1 × ..
. ... 0
b
N+ ×
Table 2:N jump rate,Nd+=Nb+−1 (JαN,,βNb+>0)
l JαN,,βl+1 ∧ Γl
γ,δp = H
N,N,l,l
α,β,γ,δ
N ×
..
. ... Γlγ,δp
b
N+ ×
rates (Step 3a) in order to solve the attractiveness problem (Table 2 whenJαN,,βNb+>0) and we put
Nd+=Nb+−1.
• if Nd+ ≥ Nb+ then HαN,,βN,γ,N,δd+−1,Nd+−1 = 0 by (3.12). Therefore there is no need to continue a coupling involving ΓNα,βp, since the attractiveness problem is solved. In this case HαN,,βN,γ,l,,δl = 0 for
b
N+≤l<Nd+by definition, we do not need Step 3aand we define
HαN,,βN,γ,0,,δl:=0 for eachl>0. (3.14) In both cases, we define
HαN,,βN,γ,l,,δl :=0 for 0<l<Nb+. (3.15) since these terms could break the partial ordering. Moreover, sinceβ isN−bad, we define
If there is a higher attractiveness problem, we repeat the same construction for the coupling terms involving the jump rate ΓNγ,δp starting from (3.11) and we define a value Nd− analogous to the previousNd+. The recursive formula symmetric to (3.12) ifΓγN,δ>ΓNα,β is
Hαk,,kβ,,Nγ,,δN:=Γαk,βp∧(ΓNγ,δp−X
k′>k
Hαk′,,βk,′γ,N,δ,N) =:Γαk,βp∧JγN,δ,k+1 (3.17)
where Jγk,,δN represents the remainder of the jump rateΓNγ,δp left over after the kth step of the coupling construction. We need to couple the remainder ofΓNγ,δpwith the lower death rates in Step 3aif
JγNb,δ−,N= ΓγN,δp− X
k≥Nb−
Hαk,,kβ,,Nγ,,δN = ΓNγ,δp− X
k≥Nb−
Γkα,βp>0. (3.18)
In this case we put Nd− = Nb−−1. By symmetry we define Hαk,,kβ,,Nγ,,δN = 0 for each 0 < k < Nb−,
Hα−,kβ,0,,γ,Nδ,N=0 for eachk>0 ifNd−≥Nb−andHα0,,βk,,Nγ,,δN =0 for eachk>0 ifγisN−bad.
Remark 3.8. By (3.11), either Nd+=N or Nd−=N , that is either the lower or the higher attractive-ness problem given by the jump of N particles is solved by the first coupling term.
If there is no lower (higher) attractiveness problem we putNd+=N+1 (Nd−=N+1).
If eitherβ or γ is N−good, we use only one of the previous constructions. Suppose for instance thatγisN−good: then the construction involvingΓN
α,βpworks in the same way, but the symmetric
one is not required and we use the coupling terms H·α,·,,βN,γ,N,δ only to solve the lower attractiveness problems induced by eitherΓNα,βp(at Step 1) or byΠ0,α,Nβp(at Step 3b). ThereforeH0,α,Nβ,,γN,δ,N (defined at Step 3a) might be non null, but we define
Hα−,lβ,0,,γN,δ,N:=0 for eachl>0. (3.19) Ifβ isN−good a symmetric remark holds.
Step 2)Supposeβ isN−bad. The birth rateΠα0,,Nβpcould break the partial order onβ. We work as in Step 1 and we begin with the coupling term
Hα0,,Nβ,,0,γ,δN :=(Π0,α,Nβ∧Π0,γ,Nδ)p. (3.20)
If the attractiveness problem is not solved we couple the remainder ofΠ0,α,Nβp with the birth rate of the upper configuration with the largest change of particles,Π0,γ,Nδ−1p and going down we couple it withΠ0,γ,δlpatlth-step, untill =Nb+. We define recursively starting from (3.20) the terms
Hα0,,Nβ,,0,γ,δl :=(Π0,α,Nβp−X
l′>l
Hα0,,βN,,0,γ,δl′)∧Π0,γ,lδp=:BαN,,βl+1∧Π0,γ,lδp (3.21)
whereBαN,,βl is the remainder of the birth (hence notation B) rateΠ0,α,Nβp left over after the lthstep of the coupling construction.
