Subcritical Contact Process Seen from the Edge: Convergence to Quasi-Equilibrium

The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.


Introduction
The contact process is a stochastic model for the spread of an infection among the members of a population. Individuals are identified with points of a lattice (Z in our case) and the process evolves according to the following rules. An infected individual will infect each of its neighbors at rate λ > 0, and recover at rate 1. This evolution defines an interacting particle system whose state at time t is a subset η t ⊆ Z, or equivalently an element η t ∈ {0, 1} Z . We interpret that individual x ∈ Z is infected at time t if η t (x) = 1, and is otherwise healthy.
The contact process is one of the simplest particle systems that exhibits a phase transition. There exists a critical value 0 < λ c < ∞ such that the probability that a single individual infects infinitely many others is zero when λ < λ c and is positive when λ > λ c . See [Lig85] for the precise definition of the model.
For A ⊆ Z, let (η A t ) t 0 denote the process starting from η 0 = A. When A is random and has distribution µ, we denote the process by η µ t . We also write η x t when A = {x}. Let Σ = {A ⊆ Z : A ∩ N is finite} and Σ * = {A ∈ Σ : A ∩ −N is infinite}.
Notice that both Σ and Σ * are invariant for the contact process dynamics. For A ∈ Σ, the contact process seen from the rightmost point is the Markov process on Σ defined by if η A t = ∅ and ζ A t = ∅ otherwise. In fact, defining Σ 0 = {A ∈ Σ : A = ∅ or max A = 0} and Σ * 0 = {A ∈ Σ 0 : A is infinite}, the state-space of the process (ζ t ) t is Σ 0 , and the subset Σ * 0 is invariant. Durrett [Dur84] proved the existence of an invariant measure for ζ t when λ λ c on Σ * 0 . In the supercritical phase, Galves and Presutti [GP87] proved that the invariant measure µ is unique for each λ, and that ζ A t converges in distribution to µ for any A ∈ Σ * 0 . Kuczek [Kuc89] provided an alternative proof and showed an invariance principle for the position of the rightmost infected site. Uniqueness of µ and convergence in distribution was extended to the critical case by Cox, Durrett, and Schinazi [CDS91]. While some of these results were stated for the contact process and some others for planar oriented percolation, the arguments in [GP87,Kuc89,CDS91] can be translated effortlessly from one model to the other, which is not always the case.
The behavior in the subcritical phase is quite different. Schonmann [Sch87] showed that in this phase, planar oriented percolation seen from its rightmost point does not have any invariant measure on Σ * 0 . This result was extended to the contact process by Andjel, Schinazi, and Schonmann [ASS90].
In this paper we show that, despite non-existence of stationary measures, subcritical contact process seen from the rightmost point does converge in distribution. The limiting measure is quasi-stationary and is supported on configurations that contain finitely many infected sites.
This extends an analogous result for subcritical planar oriented percolation [And14]. The proof in [And14] used quite heavily that in the discrete setting the speed of the propagation of the infection is bounded by 1 almost surely. Since this does not hold for the contact process, there is no simple adaptation of that proof for this process. The difficulty is mostly due to the fact that unlikely events may have considerable influence when we observe an event of small probability.
Hereafter we assume that 0 < λ < λ c is fixed.
Define τ A = inf{t 0 : ζ A t = ∅}, and define τ x and τ µ analogously. We say that µ is a quasi-stationary distribution on Σ 0 if for every t > 0 the law of ζ µ t satisfies If µ is quasi-stationary, τ µ is exponentially distributed with parameter α = 1 0. We say that ν is minimal if E[τ ν ] is minimal among all quasi-stationary distributions. Notice that stationary is a particular case of quasi-stationary with α = 0. Proposition 1. The subcritical contact process seen from the rightmost point (ζ t ) t 0 has a unique minimal quasi-stationary distribution ν. This measure ν is supported on finite configurations. Moreover it satisfies the Yaglom limit for any finite configuration A ⊆ Z.
An analogous result was obtained by Ferrari, Kesten, and Martínez [FKM96] for a class of probabilistic automata that includes planar oriented percolation. The main step of their proof is to show that the transition matrix is R-positive with left eigenvector ν summable. In our proof we show that the contact process observed at discrete times falls in that class, and then apply standard theory of α-positive continuous-time Markov chains to obtain the Yaglom limit. In Section 3 we state and prove a more general version of the above proposition, valid on Z d .
We finally state our main result.
Theorem 1. For every infinite initial configuration A ⊆ −N, the subcritical contact process seen from the rightmost point ζ A t converges in distribution to ν as t → ∞.
Theorem 1 is proved in Section 2 using Proposition 1. A natural attempt to get Theorem 1 would be to consider the rightmost site x ∈ −N whose infection survives up to time t, and simply apply Proposition 1 to the set ζ x t of sites infected by x at time t. However, the choice of x as the first surviving site brings more information than simply "τ x > t". We define a sort of renewal space-time point in order to handle this extra information, and finally show that such point exists with high probability.
Some of the main arguments in this paper come from the second author's thesis [Eza12].

The set infected by an infinite configuration
In this section we prove Theorem 1. Subsection 2.1 describes the graphical construction of the contact process, and in Subsection 2.2 we recall the FKG and BK inequalities for this construction.
In Subsection 2.3 we introduce the definitions of a good space-time point, and a break point, for fixed time t. The presence of a break point neutralizes the negative information mentioned at the end of Section 1, provided that all points nearby are good. Choosing some constants correctly, it turns out that most points are good, even when considering rare regions such as those where an infection happens to survive until time t. We conclude this subsection with the proof of Theorem 1.
In Subsection 2.4 we prove that a break point can be found with high probability as t → ∞. To that end we use again geometric properties of good points and exponential decay of subcritical contact process.

Graphical construction
Define L = Z + {±1/3} and let U be a Poisson point process in R 2 with intensity given by Given two space-time points (y, s) and (x, t), we define a path from (y, s) to (x, t) as a finite sequence ( Horizontal segments are also referred to as jumps. If all horizontal segments satisfy t i = t i−1 ∈ U x i−1 ,x i then such path is also called a λ-path. If, in addition, all vertical segments satisfy (t i−1 , t i ] ∩ U x i = ∅ we call it an open path from (y, s) to (x, t).
The existence of an open path from (y, s) to (x, t) is denoted by (y, s) (x, t). Also for two sets of the plane C, D we use if η A s,t = ∅ and ζ A s,t = ∅ otherwise. When s = 0 we omit it. We use (η t ) and (ζ t ) for the processes defined by (1), that is (η t ) is a contact process with parameter λ and (ζ t ) is this process as seen from the rightmost infected site. Both of them are Markov. Note that if A is finite, the same holds for η A t and ζ A t for every t 0 with total probability. Also note that ∅ is absorbing for both processes. When A is a singleton {y} we write η y t and ζ y t .

FKG and BK inequalities
We use ω for a configuration of points in R 2 and ω δ , ω λ for its restrictions to Z × R and L × R respectively. We write ω ω ′ if ω ′ λ ⊆ ω λ and ω δ ⊆ ω ′ δ . We slightly abuse the notation and identify a set of configurations Q with (U −1 Q) ⊆ Ω.
A minor topological technicality needs to be mentioned. Consider the space of locally finite configurations with the Skorohod topology: two configurations are close if they have the same number of points in a large space-time box and the position of the points are approximately the same. In the sequel we assume that all events considered have zero-probability boundaries under this topology. The important fact is that events of the form {E F } are measurable and satisfy this condition, as long as E and F are bounded Borel subsets of Z × R. See [BG91, Sect. 2.1] for a proof and precise definitions.
Definition 3. We way that Q 1 and Q 2 occur disjointly if there exist disjoint sets D 1 and D 2 such that Q 1 occurs on D 1 and Q 2 occurs on D 2 . This event is denoted by Q 1 Q 2 .
Theorem (BK Inequality). If Q 1 and Q 2 are increasing, and depend only on the configuration ω within a bounded domain, then P(Q 1 Q 2 ) P(Q 1 )P(Q 2 ).

Good points and break points
The definitions below are parametrized by t > 0 and β > 0, but we omit it in the notation. We write βt as a short for ⌈βt⌉. For simplicity we assume through this whole section that the initial configuration A ∈ Σ * 0 is fixed.
Definition 4 (Good point). We say that (z, s) is good, an event denoted by G s z , if every λ-path starting at (z, s) makes less than βt jumps during [s, s + t]. We also denotê G s z := G s z ∩ G s z+2βt . The time s is omitted when s = 0. We will say that (z, s) is (β, t)-good when we need to make β and t explicit.
As a motivation for the above definition, observe that even though the events {0 L t } are not independent, they are conditionally independent givenĜ 0 . Hereafter we write 0 for the space-times point (0, 0).
Proof. Given the Poisson process U, the λ-paths starting at 0 can be constructed by choosing at each jump mark whether or not to follow that arrow. This way each finite path is associated to a finite binary sequence a ∈ {0, 1} n for some n ∈ N.
The λ-path corresponding to a finite sequence a makes |a| := n i=1 a i jumps. Such path is performed in time T a , whose distribution is that of the sum of n independent exponential random variables with parameter 2λ. Since for every a ∈ {0, 1} n , choosing β > max{12λe, ρ}, we have by Stirling's approximation Definition 5 (Break point). We say that the space-time point (y, s) is a break point if L 0 (w, s) for y < w y + 2βt.
Let X = max{x ∈ A : (x, 0) L t } denote the first site whose infection survives up to time t and Γ : [0, t] → Z given by Γ(s) = max{y : (X, 0) (y, s) L t } denote the "rightmost path" from (X, 0) to L t . Take Lemma 2. For any s ∈ [0, t], y ∈ Z, and A ′ ∈ Σ * 0 , Remark 1. The random elements considered in this paper are a graphical construction U and sometimes a random initial condition η 0 , both given by locally finite subsets of an Euclidean space. Therefore we can assume that (Ω, F ) is a Polish space, and as a consequence regular conditional probabilities exist. In particular, conditioning on events such as {R = s}, {Γ = γ}, etc. is well defined.
Proof. Consider the regions and the random variables X y,s = max{x ∈ A : (x, 0) (y, s)}, the first site whose infection reaches (y, s) and Γ y,s : [0, s] → Z given by the rightmost path from (X y,s , 0) to (y, s). Before continuing with the proof, the reader may see Figure 1 to have a glance of the argument.
Lemma 3. If β is large enough, then, as t → ∞, 2βt for all u < s. Third, there are no connections from A × {0} to L t to the right of Γ y,s . Even though the third condition might not depend only on the region E − y,s , it is the case when G s y+2βt occurs, since it implies that the leftmost path starting from (y + 2βt, s), also depicted by a dash-dotted line, does not reach distance βt by time t. Finally, E + y,s and E − y,s are disjoint, and under the occurrence of G s y the configuration ζ A t = ζ A ′ s,t depends only on E + y,s . Therefore only the second part (G s y ∩ {(y, s) L t }) influences its distribution.
Proof. These two limits hold for similar reasons. First notice that the probability that (y, 0) L t by a straight vertical path is e −t and that these events are independent over y.
Finally, P(G c 0 ) e −ρt ≪ e −t e −s P(0 L s ), and therefore proving the second limit.

Proposition 2.
If β is large enough, then P R t 2 → 1 as t → ∞.
The proof of the above proposition is given in Section 2.4. We now prove Theorem 1 using the preceding results.
Proof of Theorem 1. For a signed measure µ on {0, 1} Z , we use µ = µ T V to denote the total variation norm. Denote H s y :=Ĝ s On the first equality we used Proposition 2 and Lemma 3. On the third equality we used Lemma 2. The last two inequalities are due to Lemma 3 and Lemma 4, respectively. The last lim sup vanishes by Proposition 1. Proof. We split time interval [0, r] into favorable and non favorable intervals in such a way that all non-favorable intervals have length less than √ t as follows: let t 0 = t and let v = sup{u t 0 : γ has more than β(t − u) jumps in time interval [u, t]}.

Existence of break points
if not we declare the interval [v, t) non favorable and let t 1 = v and then continue with t 1 playing the role of t 0 .
This algorithm is performed until we reach a t i < √ t and we let t i+1 = 0. Note that a non-favorable interval of length ℓ has at least 4βℓ jumps. We now consider the intervals I j = (t j+1 , t j ] for j = i − 1, . . . , 0. Let L be the sum of the lengths of the non-favorable intervals among the I j 's. Then 4βL βt. Therefore L Proof. On the one hand the existence of a QSD ν, Proposition 1, implies On the other hand and P U x ∩ [t, t + 1] = ∅ = e −1 , whence P L 0 (0, s) for some s t n P L 0 (0, s) for some s ∈ [t + n, t + n + 1] e · n P L 0 (0, t + n + 1) e −αt/2 for t large enough.
Consider the sets C t = {(x, 0) : x = 1, 2, . . . , 2βt} and shown in Figure 2. Proof. It is a simple consequence of Lemma 6. We give a full proof for convenience. If D t C t then either (x, −u) C t for some x ∈ Z and u = √ t, or (x, −u) L + 0 for some x = 0, 1, 2, . . . and u x/4β, where L + 0 = {1, 2, 3, . . . } × {0}. Using (3) and summing over y ∈ C t , the probability of the first event is bounded by 2βte −α √ t . For the second event, using FKG inequality and Lemma 6 we get q β := P (x, −u) L + 0 for any x = 0, 1, 2, . . . and u x/4β Proof. Let D γ be the closed set given by the union of the horizontal and vertical segments of γ. Then (R × [0, t]) \ D γ has two components: D + γ to the right and D − γ to the left. We note that Here the last event means that there is an open path starting and ending at different points of γ, whose existence is determined by the configuration ω ∩ D + γ , see Figure 3. On Finally, applying the FKG inequality to the last line, By definition of favorable interval and of the set D t , we have that z j + D t ⊆ D + γ ∪ D γ . On the other hand, if J z j occurs then z j + D t z j + C t , see Figure 5. γ z j Figure 5: Since these events depend on U ∩ R × (t j − √ t, t j ] , which are disjoint as j goes from 1 to k, we have that by Lemma 7, which finishes the proof.
Proof of Proposition 2. By Lemmas 5 and 8, as t → ∞, By Lemma 3, choosing β large enough we have P(G X ) → 1 and the result follows.

Yaglom limit for the set infected by a single site
In this section we prove Proposition 1, building upon Chapter 3 of the second author's PhD thesis [Eza12].
We start by recalling some properties of jump processes on countable spaces which are almost-surely absorbed but positive recurrent when conditioned on non-absorption, known as R-positive or α-positive processes. In the sequel we define the finite contact process modulo translations, extending to Z d the concept of "seen from the edge". We then discretize time appropriately to obtain some moment control using exponential decay, showing that it satisfies some probabilistic criteria for R-positiveness which ultimately implies the desired result.

Positive recurrence of conditioned processes
Let Λ be a countable set and consider a Markov jump process (ζ t ) t 0 on Λ ∪ {∅} such that Λ is an irreducible class and ∅ is an absorbing state which is reached almost-surely. The sub-Markovian transition kernel restricted to Λ is written as P t (A, A ′ ) = P(ζ A t = A ′ ), a matrix doubly-indexed by Λ and continuously parametrized by t.
A measure µ on Λ is seen as a row vector, and a real function f as a column vector, so that µP t f = Ef (ζ µ t ). With this notation, µ is quasi-stationary if and only if µP t = e −α(µ)t µ. By [Kin63, Theorem 1] there exists α > 0 with the property that t −1 log P t (A, In this case, by [Kin63, Theorem 4] there exist a measure ν and a positive function h, both unique modulo a multiplicative constant, such that Moreover, νh < ∞. If in addition ν is summable, then it can be normalized to become a probability measure on Λ, and the Yaglom limit follows from the result below. Theorem 2. If an irreducible sub-Markovian standard semi-group (P t ) t 0 on a countable space Λ is α-positive with summable normalized left-eigenvector ν, then Proof. We reprove this classical result [SVJ66,VJ69] for the reader's convenience.
Let 1 denote the unit column vector, and choose ν and h so that Let H denote the diagonal matrix corresponding to h. The h-transform of P t is Since νHQ t = νH, Q t 1 = 1 and (Q t ) t is a multiplicative semi-group, it defines a Markov process on Λ with invariant measure νH.
It follows from the α-positiveness of (P t ) t that Q t → 0, thus it is positive recurrent and hence Q t → 1νH as t → ∞. Therefore, e αt P t → hν as t → ∞. Summing over the second coordinate we have e αt P t 1 → h. That is, Therefore we get It remains to justify that summation over the second coordinate preserves the limit. Since e αt νP t = ν we get for every t 0 and A ′ ∈ Λ which is summable over A ′ . The limit thus follows by dominated convergence.

Finite contact process modulo translations
For the contact process on Z d in arbitrary dimension d 1, the concept of "seen from the edge" is generalized by considering the process modulo translations. We say that two configurations η and η ′ in the space {A ⊆ Z d : A is non-empty and finite} are equivalent if η = η ′ + y for some y ∈ Z d . Let Λ denote the quotient space resulting from this equivalence relation. We will denote by ζ the equivalence class of a configuration η, or indistinguishably any representant of such class when there is no confusion.
Since the evolution rules of the contact process are translation-invariant, the process (ζ t ) t 0 obtained by projecting (η t ) t 0 onto Λ ∪ {∅} is a homogeneous Markov process with values on Λ∪{∅}. Moreover, the subset Λ is an irreducible class, and the absorbing state ∅ is almost-surely reached if λ < λ c . We call (ζ t ) t 0 the contact process modulo translations.
For d = 1 this is the same as taking Λ = {A ⊆ −N 0 : A is finite and 0 ∈ A}. Therefore, Proposition 1 is the specialization to d = 1 of the next result.
Proposition 3. Let (ζ t ) t 0 denote the contact process modulo translations on Z d with subcritical infection parameter λ. This process has a unique minimal quasi-stationary distribution ν. Moreover the Yaglom limit L (ζ A t | τ A > t) → ν as t → ∞ holds for any finite non-empty initial configuration A.
Let (ξ n ) n denote an irreducible, aperiodic, discrete-time Markov chain on the statespace Λ ∪ {∅}, with transition matrix p(·, ·) such that the absorbing time τ A = inf{n : ξ A n = ∅} is a.s. finite. As for continuous-time chains discussed above, there is R such that p n (A, A) = R −n+o(n) , and we say that p is R-positive if lim sup R n p n (A, A) > 0. The proof of Proposition 3 will be based on the following criteria for R-positiveness.
The following proposition provides a set of configurations Λ ′ and an appropriate time discretization that satisfy the above criteria. It is analogous to Theorem 2 in [FKM96], but since the range of interaction of the contact process is infinite for any positive period of time, we cannot apply the latter directly. We give a simpler proof instead. By Theorem 3, the matrix P ψ is R-positive with summable left-eigenvector ν. Therefore the semi-group (P t ) t 0 is α-positive with the same left-eigenvector. By Theorem 2