Contact process
6.2 Coupling and duality
The contact model can be constructed in terms of families of Poisson pro- cesses. This representation is both informative and useful for what follows. For each x∈V , we draw a ‘time-line’ [0,∞). On the time-line{x}×[0,∞)
6.2 Coupling and duality 129
time
space 0
Figure 6.1. The so-called ‘graphical representation’ of the contact process on the line L. The horizontal line represents ‘space’, and the vertical line above a point x is the time-line at x . The marks◦are the points of cure, and the arrows are the arrows of infection. Suppose we are told that, at time 0, the origin is the unique infected point. In this picture, the initial infective is marked 0, and the bold lines indicate the portions of space–time that are infected.
we place a Poisson point process Dx with intensityδ. For each ordered
pair x,y ∈ V of neighbours, we let Bx,ybe a Poisson point process with
intensityλ. These processes are taken to be independent of each other, and we can assume without loss of generality that the times occurring in the processes are distinct. Points in each Dx are called ‘points of cure’, and
points in Bx,yare called ‘arrows of infection’ from x to y. The appropriate
probability measure is denoted byPλ,δ.
The situation is illustrated in Figure 6.1 with G =L. Let(x,s), (y,t)∈
V×[0,∞)where s≤t. We define a (directed) path from(x,s)to(y,t)to be a sequence(x,s)=(x0,t0), (x0,t1), (x1,t1), (x1,t2), . . . , (xn,tn+1)= (y,t)with t0≤t1≤ · · · ≤tn+1, such that:
(i) each interval{xi} ×[ti,ti+1] contains no points of Dxi, (ii) ti ∈Bxi−1,xi for i =1,2, . . . ,n.
We write(x,s)→(y,t)if there exists such a directed path.
We think of a point(x,u)of cure as meaning that an infection at x just prior to time u is cured at time u. An arrow of infection from x to y at time
(x,s)→(y,t)means that y is infected at time t if x is infected at time s. Letξ0∈ 6 = {0,1}V, and defineξt ∈ 6, t ∈[0,∞), byξt(y)=1 if
and only if there exists x∈ V such thatξ0(x)=1 and(x,0)→(y,t). It is clear that(ξt : t ∈[0,∞))is a contact model with parametersλandδ.
The above ‘graphical representation’ has several uses. First, it is a geo- metrical picture of the spread of infection providing a coupling of contact models with all possible initial configurationsξ0. Secondly, it provides cou- plings of contact models with differentλandδ, as follows. Letλ1 ≤ λ2 andδ1≥δ2, and consider the above representation with(λ, δ)=(λ2, δ1). If we remove each point of cure with probabilityδ2/δ1(respectively, each arrow of infection with probabilityλ1/λ2), we obtain a representation of a contact model with parameters(λ2, δ2)(respectively, parameters(λ1, δ1)). We obtain thus that the passage of infection is non-increasing inδand non- decreasing inλ.
There is a natural one–one correspondence between6and the power set 2V of the vertex-set, given byξ ↔ Iξ = {x ∈ V :ξ(x)=1}. We shall frequently regard vectorsξas sets Iξ. Forξ ∈6and A⊆V , we writeξtA
for the value of the contact model at time t starting at time 0 from the set A of infectives. It is immediate by the rules of the above coupling that:
(a) the coupling is monotone in thatξtA⊆ξtBif A⊆B,
(b) moreover, the coupling is additive in thatξtA∪B=ξtA∪ξtB. 6.1 Theorem (Duality relation). For A,B⊆V ,
(6.2) Pλ,δ(ξtA∩B6=∅)=Pλ,δ(ξtB∩A6=∅). Equation (6.2) can be written in the form
Pλ,δA (ξt ≡0 on B)=Pλ,δB (ξt ≡0 on A),
where the superscripts indicate the initial states. This may be termed ‘weak’ duality, in that it involves probabilities. There is also a ‘strong’ dual- ity involving configurations of the graphical representation, that may be expressed informally as follows. Suppose we reverse the direction of time in the ‘primary’ graphical representation, and also the directions of the arrows. The law of the resulting process is the same as that of the original. Furthermore, any path of infection in the primary process, after reversal, becomes a path of infection in the reversed process.
Proof. The event on the left side of (6.2) is the union over a∈ A and b∈B
of the event that(a,0)→ (b,t). If we reverse the direction of time and the directions of the arrows of infection, the probability of this event is unchanged and it corresponds now to the event on the right side of (6.2).
6.3 Invariant measures and percolation 131 6.3 Invariant measures and percolation
In this and the next section, we consider the contact processξ =(ξt : t ≥0)
on the d-dimensional cubic latticeLd with d ≥ 1. Thus,ξ is a Markov process on the state space6 = {0,1}Zd. Let I be the set of invariant
measures ofξ, that is, the set of probability measuresµ on6 such that µSt = µ, where S = (St : t ≥ 0)is the transition semigroup of the
process.1It is elementary thatIis a convex set of measures: ifφ1, φ2∈I,
thenαφ1+(1−α)φ2∈ Iforα ∈[0,1]. Therefore,Iis determined by knowledge of the setIeof extremal invariant measures.
The partial order on6 induces a partial order on probability measures on 6 in the usual way, and we denote this by ≤st. It turns out thatI possesses a ‘minimal’ and ‘maximal’ element, with respect to ≤st. The minimal measure (or ‘lower invariant measure’) is the measure, denoted δ∅, that places probability 1 on the empty set. It is called ‘lower’ because
δ∅≤stµfor all measuresµon6.
The maximal measure (or ‘upper invariant measure’) is constructed as the weak limit of the contact model beginning with the setξ0=Zd. Letµs
denote the law ofξZd
s . Sinceξ
Zd
s ⊆Zd,
µ0Ss =µs ≤stµ0. By the monotonicity of the coupling,
µs+t =µ0SsSt=µsSt ≤stµ0St =µt,
whence the limit
lim
t→∞µt(f)
exists for any bounded increasing function f : 6 → R. It is a general result of measure theory that the space of probability measures on a compact sample space is relatively compact (see [39, Sect. 1.6] and [73, Sect. 9.3]). The space(6,F)is indeed compact, whence the weak limit
ν= lim
t→∞µt
exists. Sinceνis a limiting measure for the Markov process, it is invariant, and it is called the upper invariant measure. It is clear by the method of its construction thatνis invariant under the action of any translation ofLd. 6.3 Proposition. We have thatδ∅≤stν≤stνfor everyν∈I.
1A discussion of the transition semigroup and its relationship to invariant measures can
Proof. Letν∈I. The first inequality is trivial. Clearly,ν≤stµ0, sinceµ0
is concentrated on the maximal setZd. By the monotonicity of the coupling, ν=νSt ≤stµ0St =µt, t ≥0.
Let t → ∞to obtain thatν≤stν.
By Proposition 6.3, there exists a unique invariant measure if and only if ν=δ∅. In order to understand when this is so, we deviate briefly to consider
a percolation-type question. Suppose we begin the process at a singleton, the origin say, and ask whether the probability of survival for all time is strictly positive. That is, we work with the percolation-type probability (6.4) θ (λ, δ)=Pλ,δ(ξt06=∅for all t ≥0),
whereξt0 = ξt{0}. By a re-scaling of time, θ (λ, δ) = θ (λ/δ,1), and we
assume henceforth in his section thatδ=1, and we writePλ=Pλ,1. 6.5 Proposition. The density of ill vertices underνequalsθ (λ). That is,
θ (λ)=ν {σ ∈6:σx=1}, x∈Zd. Proof. The event{ξT0∩Zd6=∅}is non-increasing in T , whence
θ (λ)= lim T→∞Pλ(ξ 0 T∩Z d 6=∅). By Proposition 6.1, Pλ(ξT0∩Zd 6=∅)=Pλ(ξ Zd T (0)=1),
and by weak convergence, Pλ(ξ
Zd
T (0)=1)→ν {σ ∈6 :σ0=1}
.
The claim follows by the translation-invariance ofν.
We define the critical value of the process by λc=λc(d)=sup{λ:θ (λ)=0}. The functionθ (λ)is non-decreasing, so that
θ (λ) =0 ifλ < λc, >0 ifλ > λc. By Proposition 6.5, ν =δ∅ ifλ < λc, 6=δ∅ ifλ > λc.
The caseλ=λcis delicate, especially when d≥2, and it has been shown in [36], using a slab argument related to that of the proof of Theorem 5.17, thatθ (λc)=0 for d ≥ 1. We arrive at the following characterization of uniqueness of extremal invariant measures.