key: cord-0196011-rc91jhh6
authors: Seiler, Marco; Sturm, Anja
title: Contact process in an evolving random environment
date: 2022-03-30
journal: nan
DOI: nan
sha: 7b034dfbb6b757faa00ba1641ec5816e67ae8ba1
doc_id: 196011
cord_uid: rc91jhh6

In this paper we introduce a contact process in an evolving random environment (CPERE) on a connected and transitive graph with bounded degree, where we assume that this environment is described through an ergodic spin systems with finite range. We show that under a certain growth condition the phase transition of survival is independent of the initial configuration of the process. We study the invariant laws of the CPERE and show that under aforementioned growth condition the phase transition for survival coincides with the phase transition of non-triviality of the upper invariant law. Furthermore, we prove continuity properties for the survival probability and derive equivalent conditions for complete convergence, in an analogous way as for the classical contact process. We then focus on the special case, where the evolving random environment is described through a dynamical percolation. We show that the contact process on a dynamical percolation on the $d$-dimensional integer dies out a.s. at criticality and complete convergence holds for all parameter choices. In the end we derive some comparison results between a dynamical percolation and ergodic spin systems with finite range such that we get bounds on the survival probability of a contact process in an evolving random environment and we determine in this case that complete convergence holds in a certain parameter regime.

The contact process (CP) is a particularly simple example of an epidemiological model, which was first introduced by Harris [Har74] . This process models the spread of an infection over time in a spatially structured population, whose structure is given through a graph G = (V, E). The vertex set V labels the individuals and two individuals x, y ∈ V are considered neighbours, i.e. they have physical contact, if there exists an edge {x, y} ∈ E between them. If an individual is infected it can pass on its infection to its neighbours. In addition, it can recover spontaneously from the infection.

Many variations of the classical contact process exist which try to incorporate more realistic assumptions. One aspect is for example that the underlying spatial structure given by G is in applications often not explicitly known, but changes in G have a drastic impact on the long-term behaviour of the model. One way to model uncertainty in the spatial structure is to introduce a random environment. Probably the first to consider a contact process in a random environment were Bramson, Durrett and Schonmann [BDS91] . Since then there has been a lot of effort in this direction, see for example [Lig92] , [Kle94] , [YC12] , [Xue14] and [GM12] . They all consider contact processes in a static random environment, i.e. the random environment is random but fixed for the whole time horizon.

More recently, the contact processes has also been considered in a dynamical or time evolving random environments. One of the first to explicitly study a contact process with dynamical rates was Broman [Bro07] . Other works have been for example [SW08] , [LR20] and [Hil+21] . A related variation are multi type contact process see for example [DS91] , [DM91] , [Rem08] and [Kuo16] .

The general observation is that a (random dynamical) change of the underlying spatial structure can have a drastic impact on the system's long-term behavior such as survival of the infection. This has also been the observation in applications, be it from real data or simulations, most recently in the global Covid-19 pandemic, see for example [Deh+20] .

In this article we study contact processes in an evolving random environment (CPERE) in some generality. We assume that the graph G = (V, E) is a connected and transitive graph with bounded degree. The CPERE (C, B) = (Ct, Bt) t≥0 is a Feller process on P(V ) × P(E), where P(V ) and P(E) are the power sets of V and E. We call the process C with values in P(V ) the infection process. If x ∈ Ct, then we call x infected at time t. We call the process B with values in P(E) the background process, since it describes the evolving random environment. If e ∈ Bt we call e open at time t and closed otherwise. We assume that B is an autonomous Feller process. Given B currently in state B the transitions of the infection process C currently in state C are for all x ∈ V , C → C ∪ {x} at rate λ · |{y ∈ C : {x, y} ∈ B}| and C → C\{x} at rate r,

where λ > 0 denotes the infection rate and r > 0 the recovery rate. We equip P(V ) × P(E) with the topology which induces point wise convergence. This means if (Cn, Bn) n∈N is a sequence in P(V ) × P(E), then (Cn, Bn) → (C, B) as n → ∞ if and only if 1 {(x,e)∈(Cn,Bn)} → 1 {(x,e)∈(C,B)} as n → ∞ for every (x, e) ∈ V × E. Furthermore, we denote by " ⇒ " the weak convergence of probability measures on P(V ) × P(E).

We will indicate the initial configuration (C, B) by adding it as a superscript to the process, i.e. (C C,B , B B ) or, if more convenient, to the law P, i.e. P (C,B) . If the initial configuration is a distribution µ then we will also write P µ . Note that if µ = δC ⊗ µ2, where µ2 is a probability measure on P(E), then we abuse the notation slightly and write P (C,µ 2 ) . Furthermore we add the model parameter as subscripts to the law P, i.e. P λ,r whenever needed for clarity.

The CP can be constructed via the so called graphical representation, where one draws infection and recovery events according to a Poisson point process, which are respectively depicted by arrows pointing from an individual x to a neighbour y and by crosses at a vertex x, see Figure 1 (a). The CPERE is essentially constructed in the same way as the CP with the difference that an infection arrow from x to y can only transmit an infection at a time t if the edge is open, i.e. {x, y} ∈ Bt, see Figure 1 (b).

One of the key quantities for models describing the spread of infections is the survival probability of the infection process C, which we denote by θ(C, B) := θ(λ, r, C, B) := P (C,B) λ,r Ct = ∅ ∀t ≥ 0 for C ⊂ V and B ⊂ E. We are interested in a phase transition dependent on the parameters (λ, r) from a zero to positive survival probability. Since the survival probability is monotone in λ we can define the critical infection rate for survival by λc(r, C, B) := inf{λ > 0 : θ(λ, r, C, B) > 0}.

The CPERE is well defined for a fairly general class of interacting particles systems acting as the background process B. We focus on the case where the background is a spin system on P(E) with generator of the form

where q(e, B) is the flip rate of e with respect to the current configuration B ⊂ E and is the symmetric difference of sets. Note that this is a spin system on the edge set E. In order to use the usual notation for interacting particle systems in this setting we additionally equip E with a spatial structure by considering the line graph L(G). Here, the original edge set E is considered to be the vertex set and edges e1, e2 ∈ E are defined to be adjacent if they have a vertex in common, i.e. there exists x ∈ V such that x ∈ e1, e2. Along with the original graph the line graphs is also a connected and transitive graph with bounded degree. Let B L n (e) denote the ball with center e ∈ E of radius n with respect to the graph distance of L(G). We assume that the spin system satisfies the following three properties.

1. It is attractive, i.e. the spin rate q(·, ·) satisfies that if B1 ⊂ B2, then q(e, B1) ≤ q(e, B2) if e / ∈ B2 and q(e, B1) ≥ q(e, B2) if e ∈ B1.

2. It is translation invariant, i.e. if σ is a graph automorphism then q(e, B) = q(σ(e), σ(B)) for all B ⊂ E.

3. The spin system is of finite range, i.e. there exists a constant R ∈ N such that q(e, B) = q(e, B ∩ B L R (e)) for all e ∈ E and B ⊂ E. We call such a spin system of range R.

In order to formulate further assumption on the background process B we introduce the exponential growth of the graph G by ρ := lim

where Bn(x) denotes the ball of radius n with x ∈ V as centre with respect to the graph distance d(·, ·). Note that since G is transitive ρ does not depend on the choice of x. If ρ = 0 we call G of subexponential growth. Next we define the permanently coupled region of the background B at time t through

where t ≥ 0.

Assumption 1.1. The background B satisfies the following assumptions:

(i) B is ergodic, i.e there exists a unique invariant law π such that B B t ⇒ π as t → ∞ for all B ⊂ E.

(ii) There exist constants T, K, κ > 0 such that P(e / ∈ Ψ t ) < K exp(−κt) for every e ∈ E and for all t ≥ T .

(iii) B is a reversible Feller process.

Loosely speaking since we assume that B is ergodic in (i) then the expansion speed of the permanently coupled in (ii) gives us a rough insight on how fast the background process convergences to the invariant law π.

We list now some examples of spin systems which satisfy these assumptions. Note that with the trivial example Bt ≡ E for all t ≥ 0 we recover the CP from the CPERE. One can show that a sufficient condition for (i) and (ii) to be satisfied is that ε − M > ρ, see [Sei21, Corollary 1.4.3 ]. On the other hand, the class of stochastic Ising models is a natural choice in our setting since models of this type satisfy reversibility by definition, see [Lig12, Section IV.2] ) for the general definition of a stochastic Ising model.

Our first aim is to deduce some sufficient conditions such that the critical infection rate for survival does not depend on the initial configuration (C, B) as long as we have finitely many initially infected vertices, i.e. C is finite and non-empty. We often use the case where the background is started stationary, i.e. B0 ∼ π, as a reference, and thus denote the survival probability in this case by θ π (λ, r, C) := P (C,π) λ,r Ct = ∅ ∀t ≥ 0 .

Furthermore we denote the associated critical infection rate by λ π c (r) := inf{λ > 0 : θ π (λ, r, {x}) > 0}.

Note that λ π c (r) is independent of the choice of x ∈ V by translation invariance. We denote by Nx the set of neighbours of x ∈ V . Let c1(λ, ρ) be the unique solution of cλ − 1 − log(cλ|Nx|)) = ρ

(3) which satisfies 0 < c1(λ, ρ) ≤ 1 λ , where λ > 0 and x ∈ V . We will see that the constant c1(λ, ρ) −1 is an upper bound of the asymptotic expansion speed of the set of all infections. On the other hand κρ −1 is a lower bound of the asymptotic expansion speed of the permanently coupled region. Thus, the inequality c1(λ, ρ) > κ −1 ρ implies that asymptotically the growth speed of the infection C is slower than the expansion of the permanently coupled region Ψ . Our result states that if this growth condition holds the critical infection rate is independent of the initial configuration.

Theorem 2.1. Suppose that Assumptions 1.1 (i) and (ii) are satisfied. If there exists a non-empty and finite set C ⊂ V and a B ⊂ E such that c1(λc(r, C , B ), ρ) > κ −1 ρ, then it follows that λc(r, C, B) = λ π c (r) for all non-empty and finite C ⊂ V and all B ⊂ E. In this case we denote the critical infection rate simply by λc(r).

Remark 2.2. If we consider graphs with subexponential growth, i.e. ρ = 0, the inequality c1(λ, ρ) > κ −1 ρ is satisfied for all λ > 0. Thus, on these graphs the critical infection rate is always independent of the initial configuration as long as Assumptions 1.1 (i) and (ii) are satisfied.

Next, we consider the relationship to the critical infection rate for non-triviality of the upper invariant law. As in the CP we can conclude by standard methods the existence of this upper invariant law ν = ν λ,r with (C V,E t , B E t ) ⇒ ν as t → ∞. Since the upper invariant law is the largest invariant law in the stochastic order the equality ν = δ ∅ ⊗ π, where the right hand side is the trivial invariant law, is equivalent to ergodicity of the system, i.e. that there exists a unique invariant law which is the weak limit of the process. Since the upper invariant law ν λ,r is monotone in the stochastic order with respect to λ (and also r) we can define the critical infection rate for non-triviality of ν by λ c (r) := inf{λ > 0 : ν λ,r = δ ∅ ⊗ π}.

The next result connects this phase transition to the already known phase transition between certain extinction and survival (with positive probability) of the infection in the population.

Theorem 2.3. Suppose that Assumption 1.1 is satisfied. Then λ c (r) = λ π c (r). If additionally c1 λ π c (r), ρ > κ −1 ρ, then λ c (r) = λc(r).

Next we state results on the continuity of the survival probability with respect to its parameters. This is important for subsequently studying the behavior at the critical point and (complete) convergence.

If C is empty then θ(C, B) = 0 and if |C| = ∞ then θ(C, B) = 1 for any B. Thus, the survival probability is obviously continuous. Therefore, we will only consider the case where C is non-empty and finite, and define for such initial configurations (C, B) the region of survival by

On the complement S(C, B) c we see that the survival probability is again 0, and thus obviously continuous. So the only interesting question is if θ(·, C, B) is continuous on S(C, B) and its boundary. Unfortunately, on general graphs we are not able to determine whether the survival probability is continuous on the whole survival region. However, we can assure continuity on the interior of the parameter set

which contains all parameters (λ, r) such that a λ ≤ λ exists for which survival is still possible and the aforementioned growth condition is satisfied. The set Sc 1 does not depend on the choice of the initial configurations (C, B) of the CPERE with C being non-empty and finite. Furthermore, if the graph G is of subexponential growth, i.e. ρ = 0, then S(C, B) = Sc 1 for all (C, B) with C finite and non-empty. In this case we drop the initial configuration and write the survival region as S. We denote byŮ the interior of a set U ⊂ R d , i.e. the largest open set which is contained in U . Proposition 2.4. Let C ⊂ V be finite and non-empty and B ⊂ E. Suppose Assumption 1.1 is satisfied, then (i) the survival probability θ(·, C, B) is continuous onSc 1 .

(ii) If additionally ρ = 0 and θ(λ, r, C, B) = 0 for all (0, ∞) 2 \S, then the survival probability θ(·, C, B) is continuous on (0, ∞) 2 .

With this result we are now able to determine two conditions which are equivalent to complete convergence of the CPERE. In order to formulate the result we abuse notation somewhat by writing

where i.o. is short for "infinitely often". Theorem 2.5. Let Assumption 1.1 be satisfied. Suppose there exists a λ ≤ λ such that c1(λ , ρ) > κ −1 ρ, and that P

for all x ∈ V , C ⊂ V and B ⊂ E as well as

for any x ∈ V . Then complete convergence is satisfied, i.e.

Conversely if (8) holds for (λ, r) ∈ Sc 1 and additionally ν = δ ∅ ⊗ π, then (6) and (7) are satisfied.

Thus, our last main results, for which we apply Theorem 2.5, concern our main example the contact process on a dynamical percolation (CPDP) for which the background is given by the dynamical percolation introduced in Example 1.2 (i). Here we have two additional parameters α and β to consider, which are the opening and closing rates of the edges. Note that the dynamical percolation B satisfies Assumption 1.1 for all α, β > 0. Since we have two additional parameters for the CPDP define, analogously to (4), the survival region for the CPDP by S(C, B) := {(λ, r, α, β) ∈ (0, ∞) 4 : θ λ, r, α, β, C, B > 0}.

(9)

In this special case we are able to show that even if c1(λ, κ) ≤ κ −1 ρ the interior of the survival region is independent of the initial configuration.

Proposition 2.6. Let x ∈ V and (C, B) be a CPDP. ThenS(C, E) =S({x}, ∅) for any non-empty and finite C ⊂ V .

In Section 7 we even show a slightly stronger but somewhat more technical statement, see Proposition 7.1. We already note here that only in the cases where the boundary ofS is parallel to the λ-axis for some (r, α, β) the lack of knowledge about survival on the boundary itself keeps us from concluding that this already implies that for any such C and arbitrary B ⊂ E we have λc(·, C, B) = λ π c (·, C, B) = λc(·). For the next results we assume the underlying graph to be the d-dimensional lattice, i.e.

where || · ||1 denotes the 1-norm. We denote by 0 ∈ Z d the d-dimensional vector of zeros. Note that the d-dimensional lattice is of subexponential growth, and thus the growth condition c1(λ, ρ) > κ −1 ρ is always satisfied. Thus, the critical infection rate is given through λc(r, α, β) = inf{λ > 0 : θ(λ, r, α, β, {0}, ∅) > 0} = inf{λ > 0 : θ(λ, r, α, β, C, B) > 0}, for any C non-empty and finite as well as arbitrary B. Likewise, the survival region does not depend on the initial condition, and thus we denote it simply by S.

Due to the independence of the dynamics of the edges in dynamical percolation we can explicitly state the invariant law π = π α,β of the background process. Namely, under π α,β the states of the edges are described by i.i.d. Bernoulli random variables with parameter α α+β , i.e. for every e ∈ E π({B ⊂ E : e ∈ B}) = α α + β and π({B ⊂ E : e / ∈ B}) = β α + β .

We adapt the techniques developed by [BG90] to the CPDP and use them show the following results.

Theorem 2.7. The CPDP on Z d goes a.s. extinct at criticality, i.e. θ λ, r, α, β, {0}, ∅ = 0 for all (λ, r, α, β) ∈ (0, ∞) 4 \S.

With the help of Theorem 2.7 and results analogous to Proposition 2.4 we can then extend the results on the continuity of the survival probability to the whole parameter regime (0, ∞) 4 . Proposition 2.8. Let C ⊂ V and B ⊂ E. For the CPDP on Z d the survival probability is continuous, i.e.

(λ, r, α, β) → θ(λ, r, α, β, C, B)

is continuous seen as function from (0, ∞) 4 to [0, 1].

Theorem 2.9. The CPDP on Z d satisfies complete convergence as stated in (8).

One reason for studying the CPDP is that the background is still simple enough so that we can adapt some techniques known for the CP and show the above results. But another reason is that we can compare some CPERE with more general background processes to it: Let (C, B) be a CPERE such that B is a spin system with rate q(·, ·). Assume that B is of range R, i.e. q(e, B) = q(e, B ∩ B L R (e)). Set N L e (R) := B L R (e)\{e} and define the minimal and maximal spin rates by

q(e, F ), βmin := min

q(e, F ) and βmax := max

q(e, F ∪ {e}).

Then there exist two CPDP with the same infection and recovery rates, which bound the CPERE from above and below.

Proposition 2.10. Let (C, B) be a CPERE with infection and recovery rate λ, r. Then there exist two CPDP (C, B) and (C, B) with the same infection and recovery rates λ, r as (C, B), for which the rates of B and B are respectively αmax, βmin and αmin, βmax, and the property that

As direct consequence of Proposition 2.10 we obtain bounds on the survival probabilities. In the following corollary θ denotes the survival probability of a general CPERE and θDP of the CPDP.

Corollary 2.11. Assume that there exist αmax,αmin, βmax, βmin ≥ 0 as in (10). Then θDP(λ, r, αmax, βmin, C, B) ≥ θ(λ, r, C, B) ≥ θDP(λ, r, αmin, βmax, C, B)

where C ⊂ V and B ⊂ E.

Furthermore, we are able to infer for a CPERE that complete convergence holds on a subset of its survival region. To be precise this subset will be the interior of the survival region of the CPDP (C, B), which lies "below" the CPERE. Since this is a consequence of Theorem 2.9 we need the underlying graph to be Z d .

Theorem 2.12. Let (C, B) be a CPERE on Z d and suppose that Assumption 1.1 is satisfied as well as that there exist αmin, βmax as in (10). If θDP(λ, r, αmin, βmax, {0}, ∅) > 0 then complete convergence as stated in (8) holds.

Remark 2.13. It is in fact possible to compare a CPERE on more general graphs G with a CPDP on a 1-dimensional integer lattice. For this we need to assume that G is distance transitive, i.e. for any two vertices x, y ∈ V exists a graph automorphism σ such that σ(x) = y. Note that distance transitivity is a stronger property than the transitivity which we assumed here since on transitive graphs only for neighboring vertices x, y ∈ V , i.e. {x, y} ∈ E, such a graph automorphism exists.

Then, for (C, B) a CPERE on G which satisfies Assumption 1.1 complete convergence as stated in (8) holds if c1(λ, ρ) > κ −1 ρ and θ Z DP (λ, r, αmin, βmax, {0}, ∅) > 0, where θ Z DP denotes the survival probability of a CPDP on the 1-dimensional integer lattice and αmin, βmax are defined as in (10). This is further discussed in Remark 7.15.

The rest of this paper is organized as follows. In Section 3 we discuss our main results and put them into context with the current state of research and cite the relevant literature. Furthermore, we state open problems and possible directions for future research.

In Section 4 we present briefly a graphical representation for a broad class of interacting particle systems as found in [Swa17] . We use this representation to first construct an arbitrary finite range spin system, and thereafter we construct the CPERE. We end this section by deriving some basic properties such as for example monotonicity of the process with respect to the rates λ and r.

In Section 5 we compare the asymptotic expansion speed of the set of all infections and the permanently coupled region and use this to prove our first main result Theorem 2.1

In the first subsection of Section 6 we derive a duality relation for the infection process C. With this duality we study the connection between survival and non-triviality of the upper invariant law and prove Theorem 2.3. Furthermore, in the subsequent subsections we prove Proposition 2.4, which states some continuity properties of the survival probability and we prove Theorem 2.5, which provides us with conditions which imply complete convergence of the CPERE.

In Section 7 we focus on our main example the CPDP. In the first part we prove Proposition 2.10 a comparison result between a general CPERE and a CPDP with adequately chosen rates. Furthermore, we show Proposition 2.6. This result yields that even if the growth condition used in Section 5 does not hold we still get that the positivity of the survival probability is independent of the initial configuration, at least in the non-critical parameter regime.

In the second part we focus on the CPDP on the d-dimensional integer lattice and adapt techniques developed by Bezuidenhout and Grimmett [BG90] . This enables us to prove that the CPDP dies out at criticality (Theorem 2.7) and as a consequence we are able to prove Proposition 2.8. We are also able to verify that complete convergence holds for the whole parameter regime of the CPDP (Theorem 2.9). We end this section by showing that complete convergence holds for the CPERE if a comparison to a suitable CPDP can be made. namely we show Theorem 2.12.

In this paper we study a contact process in a fairly general ergodic evolving random environment. Our main example, the CPDP with B dynamical percolation, was first considered by Linker and Remenik [LR20] . They studied the survival probability and the existence of a phase transition. Among other things they could prove that in this model there exists a so called immunization phase. This means that if we assume r > 0 to be fixed, then for certain choices of α and β there exists no infection rate λ such that survival is possible. Furthermore, they studied the asymptotic behavior for survival as the update speed of an edge α + β tends to 0 or ∞. Note that just recently [Hil+21] provide some further results in this direction for the CPDP on Z d . In all of these results the background is assumed to be stationary, i.e the initial distribution is the unique invariant law.

Thus, our first results concerning the influence of the initial configuration on the critical infection rate complement their results. We managed to show for general CPERE that if for a given r > 0 we find a λ > 0 with θ(λ, r, C, B) > 0 for some configuration (C, B), which satisfies the inequality c1(λ, ρ) > κ −1 ρ, then the survival of the infection process C is independent of the choice of the initial configuration, i.e. the critical infection rate λc(r) does not depend on (C, B). We note that as this condition is always fulfilled if ρ = 0 we have for subexponential growth graphs established independence of the initial condition in general.

With this we showed that in particular on subexponential growth graphs, as for example Z d , the results shown in [LR20] and [Hil+21] regarding the CPDP do not depend on their stationarity assumption of the background but hold in general.

For general CPERE it is not clear in the case ρ > 0 if independence of the initial condition may still hold in general, or if there are background dynamics for which additional assumptions (such as our sufficient condition c1(λ, ρ) > κ −1 ρ) are necessary: Problem 1. Let x ∈ V be arbitrary but fixed and suppose ρ > 0. Is the critical infection rate always independent of the initial conditions? In other words is λc(r, {x}, ∅) = λc(r, C, B) for all C ⊂ V finite and B ⊂ E?

For the CPDP we were able to show in Proposition 2.6 that even if the growth condition is not satisfied the interior of the survival region is still independent of the initial configuration. Thus, the only missing piece is the behaviour at criticality, i.e. on the boundary of the survival region. Since the CP dies out at criticality on a large class of graphs, we would expect the same to be true for the CPDP. Hence, we come to the following conjecture. We were also able to show that the survival probability is continuous on the interior of the subset Sc 1 of the survival region defined in (5). Furthermore, we conclude that the phase transition of survival with the background started stationary, i.e. θ π (λ, r, {x}) = 0 to θ π (λ, r, {x}) > 0, agrees with the phase transition of non-triviality of the upper invariant law, i.e. ν = δ ∅ ⊗ π to ν = δ ∅ ⊗ π. Thus, if additionally c1(λ π c (r), ρ) > κ −1 ρ holds the initial configuration of the background is of no importance to the question of non-triviality of ν. This in itself is an interesting observation. But of course one would like to characterize all invariant laws for the CPERE which follows from showing complete convergence.

For the CP Durrett and Griffeath [DG82] formulated equivalent conditions for complete convergence, which have been adapted to several variations of the CP, for example to the multi-type contact process by Remenik [Rem08] . We managed to provide a similar characterization for the CPERE based on the previously stated results: Under the assumption that the growth condition c1(λ, ρ) > κ −1 ρ is satisfied we could show that Condition (6) and (7) imply complete convergence of the CPERE, i.e. for which complete convergences holds. We then have that (λ, r) ∈ S cc c 1 if there exists a λ ≤ λ such that (λ , r) ∈ S({x}, ∅) and c1(λ , ρ) > κ −1 ρ, as well as that the conditions (6) and (7) hold.

We illustrated the survival region S(C, B) and the subsets Sc 1 and S cc c 1 in Figure 2 . On subexponential graphs, i.e. ρ = 0, the inequality c1(λ , ρ) > κ −1 ρ is trivially satisfied for all λ > 0 since c1(λ, ρ) > 0 for all λ > 0. Thus, Sc 1 = S(C, B) for all (C, B) with C non-empty and finite.

Theorem 2.5 only yields complete convergence of the CPERE if (λ, r) ∈ S cc c 1 . Some crucial steps of our proof rely on the assumption that the asymptotic expansion speed of the permanently coupled region is greater than that of the infection. But it is not clear that this assumption is necessary.

Let us for a moment consider the CP on a regular tree. In this setting it is possible to show that the CP has an intermediate phase, where complete convergence does not hold but the survival probability is positive. The reason for this occurrence is that in this phase global survival is possible, i.e. the survival probability is positive, but if we consider an arbitrary vertex x the probability that it gets reinfected infinitely often is 0, and thus local survival is not possible. Thus, (6) and (7) fail. If local survival is not possible the CP started in a finite initial configuration will converge to δ ∅ as t → ∞. But it is possible to construct several (in fact, infinitely many) non-trivial invariant laws, see [Lig13, Chapter I.4 ] for details. Now the question is if the background process can cause a similar phenomenon if the growth condition is not satisfied. Hence, the following question arises.

Problem 3. What is the limiting behaviour of (Ct, Bt) as t → ∞ if for a pair of rates (λ, r) ∈ (0, ∞) 2 which satisfies Condition (6) and (7), no λ ≤ λ with c1(λ , ρ) ≤ κ −1 ρ does exist which satisfies the (6) and (7)? In the special case that G is the d-dimensional integer lattice so that the growth condition always holds, we showed for our main example, the CPDP, that complete convergence holds for all (λ, r, α, β) ∈ (0, ∞) 4 . Furthermore, for general CPERE whose background process B satisfies Assumption 1.1 complete convergence holds on the survival region of a CPDP with suitably chosen parameters, see Theorem 2.12. Therefore, we might ask the following two questions. 

Then there exists a bounded and convex subset U ⊂ R d such that for every ε > 0

Let us briefly explain the three conditions mentioned in this conjecture. Condition (11) implies that if the infection process C goes extinct, then this will happen most likely early on. Condition (12) basically states that if C survives, the infection expands asymptotically at least according to some linear speed with high probability. Condition (13) has a similar interpretation for the permanently coupled region.

Garet and Marchand [GM12] proved an asymptotic shape theorem for the contact process on Z d in a static random environments. Deshayes adapted their techniques in [Des14] to a dynamical setting and showed an asymptotic shape theorem for a contact process with ageing. Furthermore, in [Des15] it was explained that this can also be extended to a broader class of time dynamical contact processes, which includes among others the contact process with varying recovery rates studied by [Bro07] and [SW08] . Since the latter model shares a lot of similarities with the CPERE constructed here we believe that Conjecture 6 should hold true.

Both works [GM12] and [Des14] have proven similar conditions to (11), (12) and (13), for the contact process in a static random environment and respectively for the contact process with ageing, by an adaption of the techniques developed in [BG90] . Since we also adapt these techniques for the CPDP, we believe the following conjecture to be true.

Conjecture 7. Let (C, B) be a CPDP with rates λ, r, α, β > 0 on the d-dimensional integer lattice. Suppose θDP(λ, r, α, β) > 0, then there exists C1, C2, M > 0 such that (11), (12) and (13) are fulfilled.

In this paper we tried to minimize the assumptions we pose on the underlying graph G such that it is as general as possible. But if we restrict the class of graphs we could still relax some assumption further.

For example, if we only consider subexponential growth graphs we could relax Assumption 1.1 (ii). Here, it would suffice to assume that P(e ∈ Ψ n ) decays polynomially with a high enough exponent in order to show that the critical infection rate is independent of the initial configuration. This can be seen in the proof of Proposition 5.3 where we compare the asymptotic expansion speed of the infected area and of the permanently coupled region.

One may still be able to prove continuity of the survival probability without assuming the reversibility of Assumption 1.1 (iii). There are interesting background dynamics that are not reversible, for example the noisy voter model on more general graphs than the 1-dimensional integer lattice. Depending on the underlying graph one may be able to use results from ergodic theory instead of duality combined with reversibility, for example if the graph automorphism group of G is finitely generated.

Furthermore, one could relax Assumption 1.1 (i) and consider non-ergodic systems as for example the ferromagnetic Ising model on Z d for d ≥ 2 and for a large enough inverse temperature β. In this case the setting would be fundamentally different since already the invariant distribution of the background process B would depend, by assumption, on the initial configuration.

Apart from these concrete questions there are a number of further research directions regarding the CPERE. Here, we mainly considered the behaviour of the CPERE in the supercritical regime, i.e. the parameter regime where the survival probability is positive. For the CP it is known that in the subcritical regime the infection dies out exponentially fast, i.e. P(Ct = ∅) ≤ e −C(λ)t , where C(λ) is a constant only depending on the infection rate λ see [BG91] . Linker and Remenik [LR20] showed on Z that an immunization region exists where the infection cannot survive no matter how large we choose the infection parameter. One heuristic explanation for this phenomenon is that if the update speed is slow enough then the background isolates the infection long enough in a bounded area such that the infection dies out. One could roughly say that the background dynamics dominates the survival behaviour. This may also have consequences for the subcritical behaviour of CPERE which has not been studied yet in any detail.

Also, in this paper we assumed that the background process evolves autonomously, i.e. we do not allow feedback from the infection process to the background. Depending on the application of the model this may not be a realistic assumption. For example, in the current Covid-19 Pandemic testing, contact tracing, and isolation through quarantine are important measures for curbing the spread of the infection. This could be modeled through closing edges adjacent to infected individuals at a higher rate. There is great interest in the effectiveness of these measures, and this has been studied recently via simulations, see for example by Aleta et al. [Ale+20] and Kucharski et al. [Kuc+20] . Of course, allowing a feedback from the infection process to the background would lead to a dependency structure that is far more complex, and we thus leave the analysis of these models for future research.

The construction of the CPERE via a graphical representation is essential for our proofs. Here, we follow the construction via random maps of Swart [Swa17] .

For an interacting particle system on a space Λ with state space P(Λ) the changes are described by maps m : P(Λ) → P(Λ). Let M be the set which contains all relevant maps and let (rm)m∈M be non-negative constants. Let Ξ be a Poisson point set on (M × R with intensity measure ξ characterized via ξ({m} × (s, t]) = rm(t − s) for m ∈ M and 0 ≤ s < t. If Λ is finite and hence the set M is finite then for every t ≥ 0 we can enumerate the points in Ξ0,t := Ξ ∩ (M × (0, t]) consecutively, i.e. Ξ0,t = {(m1, t1), . . . , (mn, tn)} with t1 < · · · < tn a.s., and define Xt(A) := mn • · · · • m1(A), where A ∈ P(Λ). By [Swa17, Proposition 2.5] this is a Feller process with the generator

where f ∈ C(P(Λ)) and A ∈ P(Λ). If M is a countable set of local maps more care has to be taken so that the process X is well defined. Let

This set is the collection of all x ∈ Λ which can possibly be changed by m. Next, for a given 

then X is a well defined càdlàg Feller process with values in P(Λ) and generator A, i.e. X has paths in D [0,∞) (P(Λ)), the space of all càdlàg functions mapping [0, ∞) to P(Λ).

We consider spin systems as our background processes whose generators are generally written in the form

where f ∈ C(P(E)) and B ⊂ E. The interpretation of a spin system is that at an edge e ∈ E a spin flip takes places with a spin rate q(e, B) which depends on the current configuration B. We can rewrite this generator in the form (14). Since we only consider finite range systems we know that there exists a range R ∈ N such that q(e, B) = q(e, B ∩ B L R (e)) for any B ⊂ E. We distinguish between an up or down flip, i.e. if e ∈ B or e / ∈ B. Then we consider every possible configuration of the R-neighbourhood of e, which we denote by N L e (R) := B L R (e)\{e}. For every e ∈ E and F ⊂ N L e (R) we set

for B ⊂ E and choose the rates to be rup e,F = q(e, F ) and r down e,F = q(e, F ∪ {e}).

Note that e / ∈ F , since F ⊂ N L e (R) and that for every e there are finitely many F ⊂ N L e (R) and therefore finitely many relevant maps. We denote the sets of the two types of maps by (15) is satisfied, and thus the associated process X from the aforementioned graphical representation is a well defined Feller process with the generator as in (14).

Also, by plugging in the maps and rates it can be easily verified that this generator is the same as the generator stated in (16).

In this subsection we explicitly construct the CPERE. We assume that the maps and rates used to construct the (autonomous) background B via the graphical representation are known, i.e. M back is a countable set which contains local maps m : P(E) → P(E) with corresponding rates (rm)m∈M back as in Section 4.1. In order to be in the setting of the graphical representation we consider Λ = V ∪ E and thus the state space is P(V ∪ E) which can be identified with P(V ) × P(E) via a one-to-one correspondence.

First we extend the maps m ∈ M back to maps m * :

back denote the set of all maps m * and set rm * = rm.

and set the rates to be rcoop x,y = λ and rrec x = r. The map coop x,y is called the cooperative infection map. The name comes from the fact that for x to successfully infect y it needs the edge {x, y} to be open. In this sense x and {x, y} must cooperate such that the infection spreads to y.

Let us denote by Ξ = Ξ λ,r the Poisson point set with respect to M := MCP ∪ M * back and by (rm)m∈M the corresponding rates. Obviously, (15) is satisfied, and thus there exists a càdlàg Feller process X on P(V ∪ E) with generator

The process X is a combination of infection process and the background in one. But, it is far more convenient to treat these two parts as separate object. Therefore, we switch back to the state space P(V ) × P(E), which we achieve by setting Ct := Xt and Bt := Xt\V for all t ≥ 0. We obtain the CPERE as described in the beginning, i.e. (C, B) is a Feller process on the state space P(V ) × P(E) and C has jump rates (1).

We visualized this construction in Figure 3 for the contact process on a dynamical percolation, i.e. B is a dynamical percolation (see Example 1.2 (i)). Note that in the case of the dynamical percolation there is a much simpler choice of maps and rates since it can be constructed via the Remark 4.1. The Poisson point set Ξ used in the graphical representation can be represented via three independent Poisson point set. These are Ξ inf on M inf × R, which are in the graphical representation (see Figure 3 ), the infection arrows, Ξ rec on Mrec × R corresponding to the recovery symbols and Ξ back on M * back × R which are the maps used to construct the background process.

In this subsection we summarize some basic properties of the CPERE which are already known for the CP and which can be shown analogously through couplings of the involved processes via 

the graphical representation. Let P λ,r be the law of the Feller process (C, B) constructed via the graphical representation using the Poisson point set Ξ = Ξ λ,r By construction it is clear that (C, B) is a strong Markov process with respect to the filtration (Ft) t≥0 generated by (Ξ0,t) t≥0 . We equip P(V ) and P(E) with the inclusion ⊂ as a partial order and for elements in

, which also defines a partial order. We denote by (T (t)) t≥0 = (T λ,r (t)) t≥0 the corresponding Feller semigroup of the CPERE. If µ is the initial distribution of the CPERE we use µT (t) as a short notation for

Furthermore, we denote by " " the stochastic order.

Lemma 4.2 (Monotonicity and additivity). Let (C, B) be a CPERE with parameters λ, r constructed using Ξ λ,r .

(i) (Monotoniciy with respect to the initial condition) If C1 ⊂ C2 and B1 ⊂ B2 then for all t ≥ 0

This also implies that (C, B) is a monotone Feller process, i.e. let µ1 and µ2 be probability measures on P(V ) × P(E) with µ1 µ2. Then this implies µ1T λ,r (t) µ2T λ,r (t) for all t ≥ 0.

(ii) (Monotoniciy with respect to λ, r) Let λ ≥ λ. Then there exists a CPERE ( C, B) with infection rate λ, the same initial configuration and recovery rate r such that Ct ⊆ Ct for all t ≥ 0. In words C is monotone increasing in λ. On the other hand C is monotone decreasing in r.

Proof. All of these statements follow directly from the graphical representation. In (i) we use that B is an attractive spin system which implies

and subsequently, as for the regular CP,

. For (ii) we define ( C, B) via Ξ λ,r and add additional independent infection arrows with rate ( λ − λ). For (iii) the background processes are identical and so this follows from the graphical representation, just as for the classical CP.

Furthermore the probabilities of events where C depends only on a finite time horizon are continuous with respect to the infection and recovery rates if we consider finitely many vertices are initially infected.

Proof. This can be proven analogously as for the classical CP, see [Lig13, Chapter I.1 ]. To show continuity in λ one constructs as in the proof of Lemma 4.2 a CPERE ( C, B) with infection rate λ > λ and the same recovery rate and initial configuration as (C, B). This process ( C, B) is then coupled via the graphical representation to (C, B) such that Ct ⊂ Ct for all t > 0 and C0 = C0. It suffices to show that as | λ − λ| → 0 then P(Cs = Cs for some s ≤ t) → 0, which follows because up to time t a.s. only finitely many vertices are infected. The statement for r follows again analogously.

In this section we prove Theorem 2.1, i.e. that the critical infection rate for survival does not depend on the initial configuration of CPERE if a certain growth condition is satisfied. We assume throughout the whole section that the background B satisfies Assumption 1.1 (i) and (ii).

At first we need to study the asymptotic expansion speed of the infection process C and the permanently coupled region Ψ or to be precise the asymptotic expansion speed of a connected component of an arbitrary edge e separately. Then we can compare these two objects in terms of expansion speed.

The maximal number of infected vertices can be represented by a classical contact process C C = ( C C t ) t≥0 with infection rate λ > 0, recovery rate r = 0 and

Simply put, we ignore the background B, and thus consider every infection arrow to be valid regardless of the state of the edge at the time of the transmission and ignore every recovery event.

In Figure 4 We start with the asymptotic expansion speed of the set of all infections, i.e. the process C C . It is well known that asymptotically the infected area can grow at most at some linear speed in time. This is also illustrated in Figure 4 (a). Next we provide an explicit upper bound for this linear speed. To be precise, for a given infection parameter λ > 0 this upper bound will be (c1(λ, ρ)) −1 , where c1(λ, ρ) is the unique solution of

such that 0 < c1(λ, ρ) < λ −1 .

Lemma 5.1. Let λ > 0 and x ∈ V . There exists a unique solution 0 < c1(λ, ρ) < λ −1 of (18).

Proof. The solution c1(λ, ρ) can actually be stated explicitly with the help of the Lambert Wfunction, i.e. the inverse function of t → te t . As domain of the function we consider

, which can be verified by inserting our guess into (18). First wee see that cλ − 1 − log(cλ|Nx|) = ρ if and only if |Nx| −1 exp(−(1 + ρ)) = cλ exp(−cλ). Therefore, inserting our guess in the right-hand side and using that W is the inverse function of t → te t verifies that this is a solution of (18).

The function W is continuous and strictly increasing. Furthermore,

Obviously the function gρ is smooth on (0, ∞) and its derivative is g ρ (c) = λ − 1 c > 0 for all c < 1 λ which implies that gρ is strictly decreasing on (0, 1 λ ), and thus c1(λ, ρ) must be the unique solution of (18) on (0, 1 λ ). At last the two properties follow immediately by the properties of the Lambert W -function.

Next we define the first hitting time of y ∈ V for C with initial infections C ⊂ V as τy(C) := inf{t ≥ 0 : y ∈ C C t }.

Lemma 5.2. Let λ > 0 and set gρ(c) := cλ − 1 − log(cλ|Nx|)) − ρ for all c > 0. Then for every 0 < c < c1(λ, 0) we have g0(c) > 0 and

where x = y. This implies in particular for all c < c1(λ, ρ) that for any

To understand this result more clearly let us consider Figure 4 (a). In this figure we visualized that the set of all infection expands asymptotically linear in time with some slope c > 0. What Lemma 5.2 basically states is that for every slope c < c1(λ, ρ) from some time point s ≥ 0 onwards the boundary of the set of all infected individuals will expand with a steeper slope than c, and thus c < c .

Proof. With some minor changes one can show analogously as in [Dur88, Lemma 1.9] that for 0 < c < c1(λ, 0)

where z ∈ V is arbitrary and cλ − 1 − log(cλ|Nz|) > 0 since c < c1(λ, 0), and thus the first claim follows. For the second claim we conclude that

Note that if c < c1(λ, ρ), then g0(c) > ρ. Thus,

and since G is of exponential growth ρ the first factor is finite. Since gρ(c) > 0 by a comparison with the geometric sum we see that the right hand side is summable. Thus, applying the Borel-Cantelli Lemma we get that

Next we consider the speed of expansion of the permanently coupled region Ψ . Recall from (2) that the permanently coupled region at time t, Ψ t contains all edges e such that e ∈ B B 1 s B B 1 s for all B1, B2 ⊂ E and all s ≥ t. In words this means that e is in state open or closed regardless of the initial configuration and from t on wards the state will never depend on the initial configuration again. Thus, we call such an edge permanently coupled. Recall that B L k (e) denotes the ball of radius k ∈ N around an edge e ∈ E in the line graph L(G).

Proposition 5.3. Let e ∈ E and κ as in Assumption 1.1 (ii). If c > κ −1 ρ, then

Proof. Fix an arbitrary e ∈ E and recall that by Assumption 1.1 (ii) there exist T, K, κ > 0 such that P(e / ∈ Ψ t ) ≤ Ke −κt for all t > T . Thus, it follows that

By assumption we see that |B L n+1 (e)|e −ρn → 1 as n → ∞, if G has exponential growth ρ > 0. Therefore,

where we used κc > ρ. If ρ = 0 then it follows that |B L n+1 (e)|e −Cn → 0 as n → ∞ for all C > 0, and thus the right hand side of (19) is finite. Since we know that the left hand side of (19) is summable, the Borel-Cantelli Lemma yields that

Note that Ψ cn ⊂ Ψ ct for all t ≥ n, which proves the claim.

We use these two results such that we can compare the asymptotic expansion speed of the infection and the coupled region. Since one process has values in P(V ) and the other in P(E) we need to introduce the following notation. We denote by

the set of all vertices whose attached edges are already permanently coupled at time t.

Theorem 5.4. Let λ > 0, C ⊂ V be non-empty and finite, κ as in Assumption 1.1 (ii). If c1(λ, ρ) > κ −1 ρ, then P(∃s ≥ 0 :

Proof. Let x ∈ V and y ∈ Nx. First we consider C = {x}. Note that we assumed c1(λ, ρ) > κ −1 ρ, and thus there exists a c < c1(λ, ρ) such that cκ > ρ. Since c < c1(λ, ρ) by Lemma 5.2 we get that

On the other hand we know that cκ > ρ, and hence Proposition 5.3 implies that

Since B L t +1 ({x, y}) contains all edges attached to any vertex in B t (x), we see by definition of the random set Φct that

By combining (20) and (21) we get that

Now let C ⊂ V be an arbitrary non-empty and finite subset. Then we see with additivity that

But we already showed that P( s ≥ 0 : C {x} t ⊆ Φt ∀t ≥ s) = 0 for all x ∈ V and thus, the right hand side is already equal to 0. This proves the claim.

We are finally ready to prove the main results of this chapter. We begin to prove independence of initial configuration of the background process B.

Proposition 5.5. Let C ⊂ V be finite and non-empty. Suppose that c1(λ, ρ) > κ −1 ρ, then θ(λ, r, C, B1) > 0 if and only if θ(λ, r, C, B2) > 0 for all B1, B2 ⊂ E.

Proof. Recall that we assumed that Assumptions 1.1 (i) and (ii) are satisfied. Furthermore let x ∈ V be fixed. The proof strategy is to use θ π ({x}) as a reference, i.e. B0 ∼ π. Note that we omit the infection and recovery rate as variables since they are considered constant throughout the whole proof. By monotonicity it suffices to show that θ(C, ∅) > 0 if and only if θ(C, E) > 0.

Note that the CPERE with B0 ∼ π is translation invariant which implies that θ π ({x}) = θ π ({y}) for all y ∈ V . Furthermore, if A ⊂ V be an arbitrary finite non-empty set, then by additionally using additivity we see that θ π (C) > 0 if and only if θ π (A) > 0. Thus, it suffices to show:

The key idea is that we prove this by coupling the CPERE (C, B) to processes C and C, which act as a upper and lower bound, i.e. C 0 = C0 = C0 and C t ⊂ Ct ⊂ Ct for all t > 0. Note that all three infection processes will depend on the same background process B. Let s > 0, then we define C C,B,s as follows.

1. We set C C,B,s 0 = C. On [0, s] we only consider the recovery symbols caused by Ξ rec and ignore all infection arrows, i.e. coop x,y maps.

2. On (s, ∞) we use the same graphical representation as for the C C,B , i.e. the same infection arrows and recovery symbols generated by Ξ inf and Ξ rec and the same background B B .

Next we define C C,B,s as follows.

1. We set C 2. On (s, ∞) we again use the same graphical representation as for C C,B and we use the same background B B .

See Figure 5 for a illustration of C s , C and C s on the same realization of B.

Recall that C C is the classical contact process without recoveries which is coupled to the CPERE

Another reason why we consider these two processes is that by the construction of C s and C s it is clear that

since both processes are independent of the background B on [0, s] and in the time interval (s, ∞) all infection paths stay in the coupled region, i.e. the initial configuration of the background process has no influence. We start by proving a). To avoid clutter we set As := As({x}). We see that

for every s > 0 and by (22) we get that

The state ∅ is obviously an absorbing state for the infection. Hence,

Let C s be a process which is constructed analogously as C s with the difference that on [0, s] also no recovery symbols have an effect. Therefore, C is just a delayed CPERE. By construction it is clear 

By construction it follows that (C s t ) t≤s and (Bt) t≤s are independent. Also since π is the unique invariant law of the background process we see that

for every s ≥ 0, where the last inequality follows by assumption. As already mentioned C is just a delayed CPERE and if it is started stationary the survival probability is constant in s. By Theorem 5.4 for every θ π ({x}) > ε > 0 there exists a S > 0 such that P(As) > 1 − ε for all s > S, where we used that As ⊂ A s if s ≤ s . We can use this to conclude that

Now using (24)-(28) successively yields that θ({x}, ∅) ≥ P(T > s)(θ π ({x}) − ε) > 0, where we used that P(T > s) > 0 for all s ≥ 0. This proves a). It remains to show b). Here, it suffices to show that

for all s > 0. This is because Theorem 5.4 yields that

where we used in the first equality that As ⊂ A s if s ≤ s . Hence,

and therefore (29) implies that the right hand side is 0. By constructions of C we see that

Furthermore by (23) 

where we used that by assumption θ π (C) = 0 for all finite C and | C {x} s | < ∞ a.s.. Therefore,

for all s ≥ 0, which implies θ({x}, E) = 0.

Now we have shown that if the growth condition c1(λ, ρ) > κ −1 ρ holds, then the chance to survive is independent of the initial configuration of the background process. Next we finally show the main result which is that if for a given r there exist a non-empty and finite set C ⊂ V and a set B ⊂ E such that c1(λc(r, C , B ), ρ) > κ −1 ρ, then it follows that λc(r, C, B) = λ π c (r) for all non-empty and finite C ⊂ V and B ⊂ E.

Proof of Theorem 2.1. Let r > 0 and suppose there exists a non-empty and finite C ⊂ V and set B ⊂ E such that c1 λc(r, C , B ), ρ > κ −1 ρ. By Lemma 5.1 before λ → c1(λ, ρ) is continuous and strictly decreasing. Hence, there exists an ε > 0 such that all λ < λc(r, C , B ) + ε satisfy c1(λ, ρ) > κ −1 ρ. Now we consider λ < λc(r, C , B ) + ε. Proposition 5.5 implies in particular that

Furthermore, since the CPERE is translation invariant if B0 ∼ π we see that

for every non-empty and finite C ⊂ V . This, in particular implies that λc(r, C , B ) = λ π (r).

Next we use again that c1(λ, ρ) > κ −1 ρ such that Proposition 5.5 together with (30) and (31) for all finite and non-empty C ⊂ V and B ⊂ E.

In this section we mainly study the invariant laws of the CPERE. We assume through out the whole section that the background B satisfies Assumption 1.1 (i)-(iii).

In this subsection we formulate a duality relation for the infection process C and among other things we establish a connection between the survival probability and the upper invariant law. First we introduce the notion of duality. Let X = (Xu) 0≤u≤t and Y = (Yu) 0≤u≤t be two processes on the same probability space and let the Polish spaces SX and SY denote their respective state spaces. We call X and Y dual with respect to a function H :

Now we construct a dual process for the infection process C on a finite time interval [0, t] for any t ≥ 0. We first fix the background B in the time interval [0, t], i.e. we set B B,t s := B B (t−s)− for all 0 ≤ s ≤ t. We denote by G := σ(Bs : 0 ≤ s ≤ t) the σ-algebra generated from the background process until time t. Next we define the dual process ( C A,B,t s ) 0≤s≤t by reversing the time flow and start at t. Let A ⊂ V be the initial configuration at time t, i.e. C A,B,t 0 = A. We define this process analogously to C via the graphical representation using the same infection and recovery events just backwards in time and the direction of the infection arrows is reversed, i.e.

where x, y ∈ V such that {x, y} ∈ E. Now we see that we coupled C t to C in such a way that the conditional duality relation

holds a.s. for all s ≤ t. Note that the process C and C t are dual with respect to the functions H(A, B) := 1 {A∩B =∅} . Obviously ( C t , B t ) will in general not be CPERE, but this process will nevertheless prove useful. See Figure 6 for a illustration of the construction. Next we show amongst other things that we can recover a self duality in the case where we assume stationarity of B, i.e. B0 ∼ π. Proof. Let t ≥ 0. By using (32) we see that

for all s ≤ t follows by taking the expectation in (32), which proves the first claim.

For the second claim choose s = 0 and integrate both sides with respect to π, and thus

We assumed that B is reversible with respect to its invariant law π. Let us consider (Bs) s≤t with B0 ∼ π and as before set B π,t s := B (t−s)− for 0 ≤ s ≤ t, then by reversibility it follows that (Bs) s≤t d = ( B π,t s ) s≤t . Again define by the reversed graphical representation ( C A,π,t s ) s≤t with respect to the background ( B π,t s ) s≤t . Now the process ( C A,π,t s , B π,t s ) s≤t is again a CPERE with initial distribution δA ⊗ π. Hence, this fact together with (33) yields that

Recall that we denoted by T (t) = T λ,r (t) the Feller semigroup corresponding to the CPERE (C, B) with parameters λ and r and by the stochastic order. Since we assumed that B is an attractive spin system it follows immediately that (C, B) is also attractive and thus by standard methods follows that (δV ⊗ δE)T (t) ⇒ ν as t → ∞, which provides the existence of the upper invariant law ν of (C, B) . It is called like that since, if ν is an arbitrary invariant law of (C, B), then ν ν. Furthermore it is monotone in λ and r, i.e. if λ1 ≤ λ2, then ν λ 1 ,r ν λ 2 ,r and if r1 ≥ r2 then ν λ,r 1 ν λ,r 2 .

It is not necessary to start the background with every edge being open, i.e. B0 = E, to ensure convergence towards the upper invariant law. As long as the initial distribution of the background dominates π stochastically, this ensures convergence towards ν.

Lemma 6.2. Let µ be a probability measure with π µ then (δV ⊗ µ)T (t) ⇒ ν as t → ∞.

Proof. First of all it is clear that δV ⊗ π δV ⊗ µ, and therefore

if the limit exists. So its enough to prove convergence for π = µ. Since π is the invariant law of the background and the infection process can only occupy fewer vertices than all of V it follows that (δV ⊗ π)T (s) (δV ⊗ π) for all s ≥ 0 and by Lemma 4.2 we get that (δV ⊗ π)T (t + s) (δV ⊗ π)T (t) for all t, s ≥ 0.

Since (δV ⊗π)T (t) is bound from below and decreasing with respect to it follow that (δV ⊗π)T (t) ⇒ ν as t → ∞. Since ν is the upper invariant law we know that ν ν, i.e. it suffices to show that ν ν holds. By Assumption 1.1 (i) we know that π is the unique invariant law of B. Thus, the second marginal of any invariant law of (C, B) must be π. Hence, it is clear that for every invariant law ν, it must hold that ν δV ⊗ π. Therefore, by monotonicity and stationarity we know that

Since this is true for any invariant law ν it also holds for the upper invariant law, i.e. ν = ν.

This enables us to deduce a connection between the survival probability θ π of the infection process C started with stationary background and the upper invariant law ν in the next result. Proof. By the self duality relation from Proposition 6.1 we get for C ⊂ V P (V,π) (Ct ∩ C = ∅) = P (C,π) (Ct = ∅) → P (C,π) (Ct = ∅ ∀t ≥ 0) as t → ∞, where we used continuity of the probability measure. On the other hand, since C is finite we get

as t → ∞, where we used Lemma 6.2. Now we can conclude that

which yields the first claim. By translation invariance we know that θ π ({x}) = θ π ({y}) for all x, y ∈ V . This yields in particular that the second claim does not depend on the choice of x. Now choose C = {x} for some x ∈ V . Suppose that θ π ({x}) > 0, then we see by (34) that

This implies that ν = δ ∅ ⊗ π. For the converse direction we assume that θ π ({x}) = 0, and hence and thus it follows that ν = δ ∅ ⊗ π. This provides the second claim.

As a consequence we can show that the critical value λ c (r) of the phase transition between triviality and non-triviality of the upper invariant law indeed agrees with the critical value for survival λ π c (r), where the background is assumed to be stationary. If we additionally assume that c1 λ π c (r), ρ > κ −1 ρ, then we know that the critical infection rate of survival does not depend on the initial configuration.

Proof of Theorem 2.3. Let r > 0, then as a direct consequence of Proposition 6.3 follows that λ c (r) = λ π c (r). If we assume additionally c1 λ π c (r), ρ > κ −1 ρ by Theorem 2.1 follows that there exists a λc(r) such that λc(r) = λc(r, C, B) for every C ⊂ V non-empty and finite and every B ⊂ E, and thus in particular λ c (r) = λc(r).

For the remainder of this section we provide some results which we need in the subsequent sections. where we consider continuity properties of the survival probability and complete convergence.

Proposition 6.4. The measure ν has the property that ν({∅} × P(E)) ∈ {0, 1}.

Proof. This can be shown analogously as for the CP. See [Sei21, Proposition 6.4] for a proof of this result.

where x ∈ V is arbitrary and we used Proposition 6.3 in the first equality. We want to extend this result to lim n→∞ θ(Bn(x), ∅) = 1.

Recall that B is an autonomous Feller process. Thus, we denote by (S(t)) t≥0 the Feller semigroup associated with the background process. Let s > 0, then we set πs := δ ∅ S(s) and

θ πs (C) := P(C C,B t = ∅ ∀t ≥ 0)πs(dB).

By Assumption 1.1 (i) there exists a unique invariant law π of the background process B such that πs ⇒ π as s → ∞. Recall that C denotes a classical contact process with infection rate λ > 0 without recovery, i.e. only infection arrows are taken into account and the background as well as recovery symbols are completely ignored.

Lemma 6.5. Let t > 0, ε > 0 and A ⊂ V finite. Then there exists a finite D = D(t, ε, A) ⊂ V such that

Proof. Let t > 0 and A ⊂ V finite and fixed. We know that for every finite initial configuration A the random set | C A t | < ∞ a.s.. This implies that for some x ∈ A,

} if m ≥ n and because of continuity of P, it follows that for every ε > 0 there exists an N ∈ N such that P C A t ⊂ Bn(x) > 1 − ε for all n > N , which proves the claim.

Recall that B L n (e) denotes the ball in the line graph L(G) of radius n ∈ N with e ∈ E as centre.

Lemma 6.6. Let e ∈ E and k ∈ N. There exists a probability law µs on P(E 2 ) with marginals π and πs such that for every ε > 0 there exists a s > 0 such that

Proof. Let B π be the background process such that B π 0 ∼ π. Now let B π be coupled to B ∅ via the graphical representation. Recall that the coupled region was defined by

Choose c > 0 such that cκ > ρ. By Theorem 5.3 we know that P(∃s ≥ 0 : B L t+1 (e) ⊂ Ψct ∀t ≥ s) = 1.

By continuity of the law P and monotonicity of the event, there exists an s > k such that P(B L t+1 (e) ⊂ Ψct ∀t ≥ s) > 1 − ε, which in particular implies that

Now set s = cs and let µ s be the joint probability distribution of (B π s , B ∅ s ). This distribution satisfies the claim.

Lemma 6.7. Let B be the background process with spin rate q(·, ·), B ⊂ E and e ∈ B. Furthermore, let u > 0 and n ∈ N, then for every ε > 0 there exists a k > n such that for all sets

Proof. The idea of the proof is that for a finite range spin systems the state of some edge e can only be influenced by the state of a second edge a, with high probability, after an at most linear amount of time proportional to the graph distance between e and a. See [Sei21, Proposition 3.2.5] for a detailed proof.

With these three lemmas we are able to show the following useful approximation result of the survival probability. Recall that c1(λ, ρ) is the solution of (3), κ is the constant from Assumption 1.1 (ii) and ρ denotes the exponential growth of the graph G.

Lemma 6.8. Let λ, r > 0 and suppose that c1(λ, ρ) > κ −1 ρ. Then for any C ⊂ V , lim s→∞ θ πs (λ, r, C) = θ π (λ, r, C).

Proof. Note that if |C| = ∞ or C = ∅ the statement is trivial, since either both sides are 1 or 0. Thus, we assume that C is a finite non-empty subset of V . Fix x ∈ C and y ∈ Nx. Since c1(λ, ρ) > κ −1 ρ by Proposition 5.4 we know that

We see that for every ε > 0 there exists a T > 0 such that P(A 1 u (C)) ≥ 1 − ε for all u ≥ T , where we used that A 1 u (C) ⊂ A 1 u (C) for u ≤ u and continuity of the law P.

Next we fix u ≥ T and define A 2 u,m (C) := { C C t ⊂ Bm(x) ∀t ≤ u} for m ∈ N. By Lemma 6.5 we can choose a m = m(u) large enough such that P(A 2 u,m (C)) > 1 − ε. This yields that

for any B ⊂ E. By Lemma 6.7 we can choose a k = k(m) > m + 1 large enough such that 

By Lemma 6.6 there exists a distribution µs on P(E 2 ) with marginals π and πs, such that for s > 0 large enough

Note that by choice of these events

since on the event Am,u (C, (B, D) ) the infection stays in Bm(x) until time u and afterwards only travels along edges already contained in the permanently coupled region. But, for any of the initial configuration B or D the background does not differ in the ball B L m+1 ({x, y}) at any time t ∈ [0, u] and thus, we can interchange B and D on Am,u(C, (B, D)). Finally we can conclude that

where we used (39) and (40) On the other hand we have that πs = δ ∅ S(s). Since B is by assumption a monotone Feller process we get that πs π for all s ≥ 0, and thus by monotonicity of the survival probability it follows that θ πs (C) ≤ θ π (C) ≤ θ πs (C) + 4ε, which proves the claim.

With this approximation result we are able to show the desired result.

Lemma 6.9. Let x ∈ V and r > 0. Suppose that c1(λ π c (r), ρ) > κ −1 ρ, then for all λ > λc(r) = λ π c (r) lim n→∞ θ(λ, r, Bn(x), ∅) = 1.

Proof. Let us fix x ∈ V . By Lemma 5.1 we know that λ → c1(λ, ρ) is continuous and strictly decreasing. Thus, if c1(λ π (r), ρ) > κ −1 ρ, then there exists an ε > 0 such that c1(λ, ρ) > κ −1 ρ for all λ ∈ (λ π c (r), λ π c (r) + ε ). Note that by Proposition 2.3 λ π c (r) = λc(r). Let n ≥ 0 and fix λ ∈ (λ π c (r), λ π c (r) + ε ) by (37) we know that for every ε > 0 there exists n large enough such that θ π (Bn(x)) > 1 − ε and by Lemma 6.8 we know that for given n and ε there exist s > 0 large enough such that θ πs (Bn(x)) > 1 − ε.

Choose a set {xi : i ∈ N} ⊂ V such that d(xi, xj) > 2n for i = j. Note that by this choice the sets (Bn(xi))i∈N are disjoint. Let us consider the event 

where we used the translation invariance of (C, B) and that A s m,n is independent of the background. Now by (43), (44) and (45) we get that

which yields that limn→∞ θ(λ, r, Bn(x), ∅) = 1, for all λ ∈ (λc(r), λc(r) + ε). Since the map λ → c1(λ, ρ) is strictly decreasing it is possible that there exists λ > λ such that c1(λ , ρ) > κ −1 ρ is no longer satisfied. In this case we can use monotonicity and see that lim n→∞ θ(λ , r, Bn(x), ∅) ≥ lim n→∞ θ(λ, r, Bn(x), ∅) = 1.

In this section we study continuity of the survival probability with respect to the infection rate λ and recovery rate r. We start by determining on which regions of the parameter space the functions λ → θ(λ, r, C, B) and r → θ(λ, r, C, B)

are left or right continuous. Before we proceed we need the following result concerning the limit of a sequence of monotone and continuous functions. Proof. See [Sei21] for a proof of this result.

As a direct consequence of this lemma we can conclude right continuity in the following proposition. (Ct = ∅) is increasing with respect to the infection rate λ, we can use Lemma 6.10 to conclude that λ → θ(λ, r, C, B) is right continuous.

Analogously it follows that r → θ(λ, r, C, B) is left continuous since P (C,B) λ,r (Ct = ∅) is decreasing with respect to the recovery rate r.

The continuity from the respective other side is more difficult to prove. Before we proceed with this we need the following somewhat technical result. Proof. We can assume that π = δ ∅ since otherwise the survival probability is 0 which makes the statement trivial. We omit for most parts of the proof the initial configuration (C, B) since it remains unchanged throughout this proof. Note that since Dn,t is increasing in t, it follows that limt→∞ P(Dn,t) = P(Dn,∞). The idea of this proof is that if a vertex x is infected at time k ∈ N, i.e. x ∈ C k , the probability that all vertices in a radius of n get infected by time k + 1, i.e. C k+1 ⊇ Bn(x), is positive for every fixed n ∈ N. But if we assume that C survives we know that for every t ≥ 0 there exists an x ∈ V such that x ∈ Ct and this will imply P λ,r (Dn,∞) ≥ θ(λ, r) for every n ∈ N. In fact

Recall that F k is the σ-algebra generated from the Poisson point processes Ξ used in the graphical representation until time k. Then we set

We see that P(C k+1 ⊇ Bn(x)|F k ) ≥ ε a.s. on {x ∈ C k }, where we used monotonicity with respect to the initial configurations (see Lemma 4.2). This yields that for any Finally, we are prepared to prove the second continuity property. Recall from (5) that Proof. We assume that C ⊂ V is finite and non-empty. Otherwise the survival probability is 0 or 1 and a constant function is obviously continuous. We only show (i) since (ii) follows analogously, i.e only some minor changes are needed in the proof. We fix r > 0 and assume that {λ : (λ, r) ∈Sc 1 } = ∅. Thus, let (λ, r) ∈Sc 1 , fix some x ∈ V and define τ = τn := inf{t ≥ 0 : ∃x ∈ V s.t. Ct ⊇ Bn(x)}, where n ∈ N. We see that

for any t ≥ 0, where we used again that if Ct = ∅ for t ≥ τ , then this must also be true for all t ≤ τ . Now we use the fact that (C, B) is a Feller process, and see that P(Cs = ∅ ∀s ≥ τ |Fτ ) = P Cτ+s = ∅ ∀s ≥ 0 (Cτ , Bτ ) ,

where we used the strong Markov property. From the definition of τ it is clear that there exists an x ∈ V such that Cτ ⊇ Bn(x). Now we know that P Cτ+s = ∅ ∀s ≥ 0 (Cτ , Bτ ) ≥ P (Bn(x),∅) Cs = ∅ ∀s ≥ 0 , and by translation invariance the right-hand side is independent of x. Thus, we can omit the vertex x and write Bn. So we get that

where we used that {τ < t} = Dn,t. The set Dn,t is defined as in Lemma 6.12. Now let λc(r) < λ < λ < λ, and thus λ , λ ∈ {λ : (λ, r) ∈Sc 1 }. Then we see that

where we used monotonicity, see Lemma 4.2. Letting λ ↑ λ yields

where we used continuity of λ → P λ (Dn,t) which follows by Lemma 4.3. Recall that (C, B) was the initial configuration of the CPERE, using Lemma 6.12 we get that Since we know that λ ∈ {λ : (λ, r) ∈Sc 1 }, by Lemma 6.9 it follows that θ(λ , Bn, ∅) → 1 as n → ∞.

Putting everything together yields θ(λ−, C, B) ≥ θ(λ, C, B). Since we know that this function is increasing in λ, this yields left continuity on the set {λ : (λ, r) ∈Sc 1 }. Right continuity, and therefore continuity follows by Proposition 6.11.

We end this section with the following proof:

Proof of Proposition 2.4. 1. By Proposition 6.11 and Proposition 6.13 it follows that (λ, r) → θ(λ, r, C, B)

is separately continuous on the open setSc 1 ⊂ R 2 , which means that the function is continuous in all variable separately, i.e. λ → θ(λ, r, C, B) and r → θ(λ, r, C, B) are continuous on {λ : (λ, r) ∈Sc 1 } and {r : (λ, r) ∈Sc 1 } respectively. Since the survival probability θ is monotone in the infection rate λ and the recovery rate r it follows that the function is jointly continuous onSc 1 , see [KD69, Proposition 2].

2. If ρ = 0, then it follows that Sc 1 = S(C, B) for all (C, B) with C non-empty but finite. This implies in particular that the critical infection rate λc(r) is independent of the initial configuration (C, B), where we used Theorem 2.1. We know by Proposition 6.13 that the function λ → θ(λ, r, C, B) is left continuous on (λc(r), ∞) and is zero on (0, λc(r)). Now by assumption θ(λc(r), r, C, B) = 0, and thus the function is left continuous on (0, ∞). But we also know by Proposition 6.11 that λ → θ(λ, r, C, B) is right continuous on (0, ∞), and therefore continuous on (0, ∞). Continuity in r follows analogously, and thus it follows that (λ, r) → θ(λ, r, C, B)

is separately continuous on (0, ∞) 2 . We can conclude, analogously as in (i), the joint continuity with [KD69, Proposition 2].

This section is dedicated to proving Theorem 2.5. Recall that c1(λ, ρ) is the solution of (3), κ is the constant from Assumption 1.1 (ii) and ρ denotes the exponential growth of the graph G. Therefore, the main goal is to show that if for given λ, r > 0 there exists a λ ≤ λ such that for (λ , r) the two conditions (6) and (7) are satisfied, i.e.

for all x ∈ V , C ⊂ V and B ⊂ E and lim n→∞ lim sup t→∞ P λ ,r (C

for any x ∈ V , then this implies complete convergence of the CPERE, i.e.

for all C ⊂ V and B ⊂ E. On the other hand if for λ the inequality c1(λ, ρ) > κ −1 ρ and (49) are satisfied, then this already implies that (47) and (48) are satisfied. We begin with the first part, and thus show convergence of the marginals C and B and then conclude that this already implies that the CPERE (C, B) convergences. By Assumption 1.1 (i) we already know that B B t ⇒ π as t → ∞ for all B ⊂ E. Hence it remains to show that the two conditions (47) and (48) imply that the infection process C convergences weakly as t → ∞. We show that for any C ⊂ V and B ⊂ E

as t → ∞ for every C ⊂ V finite, which suffices to conclude weak convergence of the infection process C since the function class {1 {· ∩C =∅} : C ⊂ V finite} is convergence determining. Then we show that the converse holds true as well, which provides that (49) implies (47) and (48). Once we know that the marginals converge we show that this already implies the convergence of the joint distribution, i.e. (Ct, Bt) converges weakly as t → ∞. We first introduce some shorthand notation to keep the formulas somewhat cleaner. For A ⊂ V we set AE := {x, y} ∈ E : x ∈ A ,

is the ball with centre x and radius N with respect to the graph distance of G.

Let ( q B s/2 r ) r≥s/2 denote a process with same dynamics as the background process B, which is coupled with the original background in such a way that it starts at time s/2 with an initial distribution π and is assumed to be independent from (B B r ) r<s/2 , but from s/2 onward it uses the same graphical representation as B B . For a illustration see Figure 7 (b).

Lemma 6.14. Let D ⊂ V be finite and B ⊂ E. Then for every ε > 0 there exists an S > 0 such that for all s ≥ S

Proof. Let x ∈ D. Let c > 0 be chosen such that cκ > ρ, then by Proposition 5.3 we know that P(∃s ≥ 0 : B c −1 t (x) ⊆ Φt ∀t ≥ s) = 1. Let S > 0 be chosen such that D ⊂ B c −1 t for all t ≥ S /2. By continuity of the measure P, for every ε > 0 there exists a S > S > 0 such that P B c −1 t (x) ⊆ Φt, ∀t ≥ S 2 > 1 − ε then this already implies that for all s ≥ S

Let t, s > 0 and recall the dual process ( C A,B,t+s r ) r≤t+s of (C C,B r ) r≤t+s . In the definition of the dual process we fixed the background (B B r ) r≤t+s , reversed the graphical representation with respect to the time axis at the time point t + s and fixed A as the initial set of infected vertices for the dual process.

A,s/2,t+s u ) u≤t+s/2 be a process coupled to C A,B,t+s by using the same time-reversed infection arrows and recovery symbols, but the background at time s/2 (forwards in time) is reset and independently drawn according to the law π, i.e. we use ( q B s/2 r ) r≥s/2 instead of (B B r ) r≥s/2 . Again see Figure 7 for a illustration. For D given via Lemma 6.14 we get that for every ε2 > 0 there exists an S > 0 such that for every s > S

Recall that DE ⊂ E was the set which contains every edge attached to D. Now we see that

But by choosing ε1, ε2 ≤ ε 2 we see that P( C A,B,t+s u = q C A,s/2,t+s u ∀u ≤ t) ≤ ε, which yields the claim. Now we can begin to show the convergence of the first marginal. We will split this in two steps by first proving an upper bound and in the second step we use (47) and (48) to show that this upper bound also acts as a lower bound which provides the desired result. First we observe that for C ⊂ V finite that P (C ,π) (τex > t) → θ π (C ) as t → ∞. Thus, for every ε > 0 there exists a T > 0 such that |P (C ,π) (τex > t) − θ π (C )| < ε for all t > T . So we fix t such that this is satisfied. Note that

and thus it follows that limu→∞ P (C,B) (u < τex < ∞) = 0. Now we can use that {C C,B s = ∅} = {τex > s} to see that for every ε > 0 there exists an S1 > 0 such that

for all s > S1, which implies that Furthermore, we know that for every ε > 0 there exists an S3 > 0 such that

for all s > S3. Note that by construction (C C,B r ) r≤s/2 and ( q C C ,s/2,t+s r ) r≤t+s/2 are independent, and thus

where we used in the second equality that q C C ,s/2,t+s u , q B s/2 t+s−u u≤t+s/2 is again a CPERE with intial distribution δ C ⊗ π. Set κ := 4ε + ε 2 . We obtain at last that for any t > T and s > S := max(S1, S2, S3) (note that S depends on T ) we have P(C C,B s = ∅, C C ,B,t+s t = ∅) ≤ P (C,B) (τex > s/2)P (C ,π) (τex > t) + 2ε ≤ θ(C, B)θ π (C ) + κ, which proves the claim. The next step is to prove a lower bound. For that we need the following stopping time

which is the first time that at least all vertices in A are infected and all edges in H are open.

Lemma 6.17. Let A, C ⊂ V and H, B ⊂ E be non-empty and A and H be finite. Let x ∈ C then

Proof. Suppose that π = δ ∅ . Otherwise P (C,B) (x ∈ Ct i.o.) = 0, and thus the inequality is trivially true. First of all note that

Next we define the stopping times T k = inf{t > T k−1 + 1 : x ∈ Ct}, where T0 = 0. Recall that Ft is the σ-algebra generated from all Poisson point processes used in the graphical representation until time t. Let us assume that x ∈ C, since π = δ ∅ and we know that the background process is translation invariance, we can guarantee that ε = P ({x},∅) (C1 ⊇ A, B1 ⊇ H) > 0. This implies by monotonicity

{T k < ∞}.

Since A k ∈ G k+1 for all k ∈ N we can analogously as in Lemma 6.12 apply the extension of the Borel-Cantelli Lemma [Dur19, Theorem 4.3.4] and we get that for every C ⊂ V finite.

Proof. First assume that for λ the inequality c1(λ, ρ) > κ −1 ρ is satisfied. Let A ⊂ V and H ⊂ E both be finite sets. We can assume that π = δ ∅ , since if π = δ ∅ , then θ π (C ) = 0 for all C ⊂ V finite, and thus the right hand side is zero. Recall from (52) that the first time that at least all vertices in A are infected and all edges in H are open is denoted by τA,H (C, B) . Furthermore, set σ N A := τ A,A N E and τA := τ A,∅ . Now we see that

where we used that (C, B) is a strong Markov process. One major issue is that in comparison to the classical case our duality is weaker in the sense that C C ,A N E ,t+u+r is not a CPERE, and therefore our process is not self dual. But now we show that the difference is not big if we choose t + u large enough. Recall that Φt was the set of all vertices x such that all edges attached to x are contained in the permanently coupled region at time t. By Proposition 5.4 we know that P(∃s > 0 : C A t ⊂ Φt ∀t ≥ s) = 1, and thus for every ε > 0 there exists an S > 0 such that

As an application of Lemma 6.5 we find an N = N (S) ∈ N such that

Furthermore, by Lemma 6.7 there exists an M > N such that

where B ⊂ E is chosen arbitrarily and ε is independent of the choice of B. Thus, we can conclude for a given A ⊂ V that for every ε > 0 there exists an S = S(ε) > 0, N = N (S) ∈ N and M > N such that

Note that ε depends on A. On this event the process C A,A M E does not differ from C A,A M E ∪B for any B ⊂ E, since on this event the infection paths have either not yet left A N and the edges in A N E will have the same state open or closed with the two chosen initial configuration or the infection paths stay in Φ, the area where every edge attached to an infected vertex has already been coupled. Thus, we get

Furthermore, by monotonicity (see Lemma 4.2) it follows that

Using this and the fact that if the background is started stationary the CPERE is self dual by Proposition 6.1 we get with (55) that

Then, analogously to (55) by considering τD with D ⊂ V finite instead of σ N A we can find a similar lower bound for the last probability such that

For A ⊂ V and B ⊂ E finite we know by Lemma 6.17 that

and thus by letting s, t, u → ∞ we see that

Now for an arbitrary x ∈ V we choose A = D = Bn(x) and use (48) which means that for all δ > 0 there exists n0 ∈ N such that lim inft→∞ P (Bn(x),∅) (Bn(x) ∩ Ct = ∅) > 1 − δ for all n > n0. Note that ε depends on Bn(x), which means we first need to choose n0 and then the parameter accordingly such that (56) holds for ε = δ and such that lim inf t→∞ P (C,B) (Ct ∩ C = ∅) ≥ θ(C, B)θ π (C ) − 2δ.

Since this holds for all δ > 0, the claim follows. Now assume that c1(λ, ρ) ≤ κ −1 ρ holds but there exists a λ ≤ λ such that c1(λ , ρ) > κ −1 ρ and for λ and r the conditions (47) and (48) are satisfied. Then we just showed that lim inf t→∞ P (C,B) λ ,r (Ct ∩ C = ∅) ≥ θ(λ , r, C, B)θ π (λ , r, C ).

Since (48) is satisfied this implies in particular a positive survival probability. Thus, (λ, r) ∈ Sc 1 because c1(λ , ρ) > κ −1 ρ and θ(λ , r, {x}, ∅) > 0 for any x ∈ V . Since λ ≥ λ by monotonicity it follows that P (Ct ∩ C = ∅) ≥ θ(λ, r, C, B)θ π (λ, r, C ). by letting λ ↑ λ.

We showed one direction of the equivalence. Next we show the converse direction.

Proposition 6.19. Let (λ, r) ∈ Sc 1 . Suppose (50) holds and assume that ν λ,r = δ ∅ ⊗ π, then (47) and (48) are satisfied.

Proof. Note that ν λ,r = δ ∅ ⊗ π can only occur if π = δ ∅ . Choose C = C = Bn and B = ∅, then by (50) it follows that limt→∞ P (Bn,∅) (Bn ∩ Ct = ∅) = θ(Bn, ∅)θ π (Bn). Using Lemma 6.9 yields that the right hand side converges to 1 as n → ∞. This proves (48) . Now all what is left to show is (47). We see that

and thus by continuity of the law P we get that

where we again used (50). Now using the fact that P ({y},∅) (x ∈ C1) > 0 for all x, y ∈ V it follows analogously as in the proof of Lemma 6.12, that the event {x ∈ C C,B Furthermore, if we choose C = Bn and let n → ∞, then Lemma 6.9 yields that

for all x ∈ V and C ⊂ V . Since the reversed inequality "≤" obviously holds as well, this provides (47).

Since we have shown that the conditions (47) and (48) are equivalent to the fact that the two marginal processes converge, the only thing left to show is that convergence of the marginals already implies convergences of the joint distribution. Proof. Let A, C ⊂ V and B, H ⊂ E be chosen arbitrary with A ⊂ V and H ⊂ E finite. We consider these sets as fixed. We again exploit the duality relation we derived in Proposition 6.1, which states that Now we fix an arbitrary ε > 0. Then by a combination of Proposition 6.16 and (50) we get that there exists a S1 > 0 and T > 0 such that

for all s > S1 and t > T . By using the duality relation (58) together with (59) we can conclude that

for all s > S1 and t > T . Furthermore, there exists an S2 = S2(C, B, ε) > 0 such that

for s ≥ S2, which can be shown analogously to (51). In the last step we conclude that there exists an S3 = S3(t, A, H, ε) > 0 such that for s ≥ S3

which follows as a combination of Lemma 6.14 and Lemma 6.15. Finally by putting everything together and using the triangle inequality we get that for every t > T there exists an S > 0 such that This means that if we first let s → ∞ and then t → ∞, the two probabilities converge to the same limit. So it suffices to show that

as s, t → ∞. Recall that we already concluded above that (C C,B r ) r<s/2 is independent of ( q C A,s/2,t+s r ) r≤t+s/2 and it is also independent of ( q B s/2 r ) r≥s/2 . Thus, we get that

Next we use that ( q C A,s/2,t+s r , q B s/2 t+s−r ) r≤t is again a CPERE with initial distribution δA ⊗ π, and thus by duality

which converges to the desired limit since we have already shown that (δV ⊗ π)T (t) ⇒ ν as t → ∞ in Lemma 6.2. The claim follows, since P (C,B) (τ > s/2) → θ(C, B) as s → ∞.

Note that analogously as before (57) is equivalent to complete convergence, i.e for every initial configuration C ⊂ V and B ⊂ E

since the function class {1 { · ∩A =∅, · ∩H =∅} : C ⊂ V, H ⊂ E finite} is convergence determining. Now we can conclude the main result of this chapter.

Proof of Theorem 2.5. The theorem follows as a combination of the four Propositions 6.16, 6.18, 6.19 and 6.20. To be precise Propositions 6.16 and 6.18 yield that (47) 

In the previous sections we considered the CPERE in a fairly general setting. In this section we focus on the dynamical percolation as the background process, which was introduced in Example 1.2 (i). Recall that B is a Feller process with transitions

One reason to study this special case is that we can compare a CPERE (C, B) with spin rates of the background given through q(·, ·) and a CPDP with adequately chosen rates for the background. Recall from (10) that αmin := min Proof of Proposition 2.10. Recall that in Section 4.1 we constructed a finite range spin system with range R and with spin rate q(·, ·) via the maps up e,F and downe,F . Furthermore, we chose the rates to be rup e,F = q(e, F ) and r down e,F = q(e, F ∪ {e}) for any x ∈ E and any F ⊂ N L e (R). Then we obtained a spin system B with state space P(E) and spin rate q. Note that by choice of the rates it is not difficult so see that for the rates defined in (10)

Now we adjust the construction as follows. We use the same maps but choose the rates r up e,F = αmin and r down e,F = βmax for any e ∈ E and any F ⊂ N L e (R). Then it is not difficult to see that the resulting spin system B has the spin rate Next we show Proposition 2.6 which states that in the case of the CPDP even if the chosen infection rate λ does not satisfy the growth condition c1(λ, ρ) > κ −1 ρ the interior of the survival region does not depend on the initial configuration. To be precise we will show a slightly stronger but more technical result which implies Proposition 2.6. Let us define the parameter set S π := {(λ, r, α, β) : θ π (λ, r, α, β, {x}) > 0}.

Note that since the background is started stationary this set does not depend on the choice of x ∈ V . This follows by translation invariance since this yields that θ π ({x}) = θ π ({y}) for all x, y ∈ V , and thus if there exists x ∈ V such that θ π ({x}) > 0 then for any non-empty and finite C ⊂ V it follows that θ π (C) > 0.

Proposition 7.1. Let C ⊂ V be finite and non-empty and (C, B) be a CPDP.

(i) If there exists an ε > 0 such that (λ, r, α + δ, β − δ) ∈ S π for all δ ∈ (−ε, ε) then (λ, r, α, β) ∈ S({x}, ∅).

Proof. This proof is a modification of the proof of Proposition 5.5, where we use again a comparison of C with the two processes C s and C s . Let s > 0 be given, then recall that the lower bound C s ignores all transmission arrows in [0, s] and the upper bound C s ignores all recovery symbols as well as the background since infection arrows are used in any case, whether or not the background is open or closed. Let us begin with showing (i). By assumption we know that there exists an ε > 0 such that (λ, r, α + δ, β − δ) ∈ S π for all δ ∈ (−ε , ε ). Note that we omit the λ and r as variables in the following since they are constant throughout the whole proof. Now we fix an 0 < ε < ε such that θ(α − ε, β + ε, {x}) > 0. Furthermore, we see that

Thus, for B = ∅ and the time s := 1 α+β log α ε it follows that

Since the events {e1 ∈ B ∅ s } and {e2 ∈ B ∅ s } are independent for any e1, e2 ∈ E we see that

Recall from the proof of Proposition 5.5 that C s is a process which is constructed analogously to C s with the difference that on [0, s] also no recovery symbols have an effect. Therefore, C s is just a delayed CPERE. By construction it is clear that it is only possible for 

By construction it follows that (C s t ) t≤s and (B B t ) t≤s are independent. Now we can thin the open events and draw independently new closing events respectively with rate ε on (s, ∞). Since Bs ∼ π α−ε,β+ε a coupling via the graphical representation yields

Thus, we can use that C 

where the last inequality follows by assumption. This proves (i). Next, we proceed with showing (ii). Similarly to before we know that an ε > 0 exists such that (λ, r, α + δ, β − δ) ∈ (0, ∞) 4 \S π for all δ ∈ (−ε , ε ). Thus, we find again an ε > ε > 0 such that θ π (α + ε, β − ε, {x}) = 0. Analogously to before we see that

Thus, fixing ε and choosing B = E we see for the time s :

Since the events {e1 ∈ B E s } and {e2 ∈ B E s } are independent for any e1, e2 ∈ E it follows that B E s ∼ π α+ε,β−ε . By construction of C we see that

We can again thin out the closed events and draw independently new open events with a rate ε.

Since B E s ∼ π α+ε,β−ε and C {x} t = C

{x},E,s t for all t ∈ [0, s], where C is the coupled contact process that ignores the background and recoveries, it follows that

where we used that by assumption θ π (α + ε, β − ε, C) = 0 for all finite C and | C {x} s | < ∞ a.s. Now Proposition 2.6, i.e.S(C, E) =S({x}, ∅) for any finite and non-empty C ⊂ V , follows as a corollary.

Proof of Proposition 2.6. By definition we know that S({x}, ∅) ⊂ S π . Furthermore, if (λ, r, α, β) ∈ S π then in particular there must exists an ε > 0 such that (λ, r, α + δ, β − δ) ∈ S π for all δ ∈ (−ε, ε). Thus, we know by Proposition 7.1 (i) that (λ, r, α, β) ∈ S({x}, ∅) must hold. But this implies that S({x}, ∅) =S π . Similarly, we know that S π ⊂ S({x}, E), and therefore we also know that

Both of these sets are open, and thus if (λ, r, α, β) ∈ (0, ∞) 4 \S π , then there must exist an ε > 0 such that (λ, r, α + δ, β − δ) ∈ (0, ∞) 4 \S π for all δ ∈ (−ε, ε). Hence, by Proposition 7.1 (ii) it follows that (λ, r, α, β) ∈ (0, ∞) 4 \S({x}, E) which then yields (0, ∞) 4 \S({x}, E) = (0, ∞) 4 \S π which impliesS π =S(x, E). Since the definition of S π does not depend on x the claim follows.

In this subsection we focus solely on the CPDP on the d-dimensional lattice with nearest neighbour structure, i.e. V = Z d and E = {{x, y} ⊂ Z d : ||x − y||1 = 1}, where || · ||1 denotes the 1-norm. Our next goal is to adapt a block construction which was initially developed by Bezuidenhout and Grimmett [BG90] for the classical contact process. This construction will allow us to show that the CPDP dies out at criticality and that complete convergence is satisfied. In essence, the construction allows us to couple the CPDP with a oriented percolation on a macroscopic grid in such a way that the percolation model survives if and only if the infection process of the CPDP survives.

The block construction of [BG90] is quite sophisticated and technical. In order to adapt the original version to our case only minor changes are needed. One of the main differences is the additional background process B. But it is not difficult to incorporate this additional feature. As in the original construction we just need to be careful to restart the process in an appropriate configuration at the beginning of a block. Thus, we only give a broad description and sketch the proofs of the major results. For a more detailed description of the procedure for the CPDP based on exposition in [Lig13, Part I.2] we refer to [Sei21] . We should also mention that we are not the first to adapt these techniques to a variation of the contact process. In a setting similar to ours this has already been done, for example by Reminik [Rem08] and Steif and Warfheimer [SW08] for a contact process with varying recovery rates and by Deshayes [Des14] for a contact process with ageing.

The idea of the construction is to formulate finite space-time conditions which are equivalent to survival of the process. Thus, for an arbitrary but fixed L ∈ N we first introduce a truncated version This process can again be defined via a graphical representation by only considering recovery symbols on vertices x ∈ VL and infection events which emanate from a vertex x ∈ (−L, L) d ∩ Z d . Therefore, we also only need to consider flip events influencing edges in EL. On the other hand, the event A2 states that we find a spatially shifted version of the box [−n, n] d at the "right" boundary (in direction of the first coordinate) of the bigger space-time box. In broad terms one could say that these events guarantee that throughout this big space-time box the infection survives at least as "strongly" as it started. We illustrate the cross section of these events (in the direction of the first coordinate axis) in Figure 8 . The main difference to the original construction is that we also need to pay attention to the state of the background process. We consider it to be in the most unfavorable state from the perspective of the infection, which is the empty configuration ∅. Figure 8 : Illustration of the events in (64) and (65) The finite space-time condition which we impose states that we can choose the parameters n, L and T in such a way that these events happen with "high" probability. Proof. We only sketch this proof and refer for details to [Sei21, Theorem 6.1.7]. The proof is basically split in two parts. First one shows that with high probability we have sufficiently many infected vertices at the top face and at lateral faces of the space-time box S(L, T ). Lemma 6.9 is fundamental for this proof. This lemma states that for a given 0 < δ < 1 there exists an n such that

In words this means that we can push the extinction probability below any given bound by choosing n large enough. Next we use that for any n, N ≥ 1 , which means that if a contact process survives, then it does so with infinitely many vertices being infected. This fact can be used to construct two strictly increasing sequences (T k ) k≥0 and (L k ) k≥0 such that T k , L k ↑ ∞ and

for all k ≥ 0 and with [Sei21, Lemma 6.1.5] one can conclude that for some M ≥ 1 also

for all k ≥ 0. Note that we chose the constant 2 d N since by positive correlation one can conclude that 

Let us focus on the first part of Condition 7.2. We now know by (68) that with high probability we have more than N infected vertices at the top of the space-time box. Now the key idea is that if N is large enough at least one of theses vertices x will infect the spatially shifted box x + [−n, n] d after a time step of length 1. Of course these events are highly correlated. Our way of dealing with this is to choose N large enough such that we still find a sufficiently large number of vertices which are additionally far enough apart such that we can disregard this correlation: Given δ and n choose an N such that

Furthermore, we then choose N large enough such that for any configuration A ⊂ Z d with |A| > N there exists a D ⊂ A with |D| > N and ||x − y||∞ ≥ 2n + 1 for all x, y ∈ D with x = y. In words, N needs to be large enough such that any set A which contains N or more vertices will contain at least N vertices that are all a distance of 2n + 1 apart. Let us now consider for such a subset D and T > 0 the event 

Note that W T D only uses the graphical representation between T and T + 1, and thus it is independent of FT , i.e. P W T D |FT = P W T D . It is not difficult to see that

where D(A) is a subset of A containing at least N elements, which are all spaced a distance 2n + 1 apart and also that

Thus, (68) and (71)-(73) yield together that

Choosing δ accordingly yields the first part of Condition (7.2).

The second inequality can be concluded in a similar fashion since (69) and translation invariance yield that we get with high probability more than M infected space-time points (x, t) on an arbitrary but fixed intersection of a lateral face of S(L, T ) and an arbitrary orthant. Recall that S+(L, T ) was the intersection of the lateral face in direction of the first coordinate x1 and the first orthant.

Similarly to before we can choose M large enough such that for any finite set F ⊂ Z d × R+ which contains more than M space-time points such that if the spatial coordinate of two points agree then their time coordinates are at least a distance of 1 apart, we find a subset which contains at least M many space-time points which are all 2n + 1 apart in spatial distance. Again with high probability at least one point (x, t) infects via an ∅−infection path the spatially shifted space-time box x + [0, 2n] × [−n, n] d−1 × {t + 1}. By a similar strategy as before we can deal with the dependency and get

Again choose δ accordingly such that we get the second inequality of Condition (7.2).

Now it remains to show that Condition 7.2 implies survival of the CPDP. The strategy is to use Condition 7.2 to define so-called "good blocks" and with that to construct an oriented percolation model on a macroscopic lattice which is coupled to the CPDP in the sense that if the percolation model survives it implies that also the CPDP survives.

The first step is to combine (64) and (65) into one, since this is more convenient for the construction. We consider the event

Similarly to before we illustrate in Figure 9 the cross section in direction of the first coordinate of the event in (74). This event states that we start with a space box [−n, n] d of infected vertices in the Figure 9 : Illustration of the event A 3 defined in (74). The blue space-time box shows the area where the infected space box of length 2n will be contained.

worst possible background configuration, then we find again such an infected space box at some later time shifted at least by L + n and at most by 2L + n to the right along the first spatial coordinate.

Let us emphasize again that it is important to consider the background B to be started in the empty configuration ∅ since one of the key ideas is, roughly speaking, that we will stack appropriately shifted versions of A3 so that at the transition from one block to the next we need to ensure that the background process is again restarted in the same configuration. Proof. We briefly describe the broad idea, illustrated also in Figure 10 , and refer to [Sei21, Proposition 6.2.1] for the proof details. Broadly speaking, if for the CPDP the event A2 occurs then there exists a point (x, τ ) with 1 ≤ τ ≤ T + 1 and x ∈ {L + n} × [0, L) d−1 such that

Then we restart the CPDP at time τ in the state (x + [−n, n] d , ∅). If for this restarted process an appropriately spatially shifted version of the event A1 occurs then it can be shown that this implies that for the original CPDP the event A3 must have occurred. Now, due to Condition 7.2 both of these events happen with high probability. Figure 10 : Here it is illustrated how A 3 is constructed by first using A 2 and then A 1 , where we restart the process with only a copy of [−n, n] d being infected and the background in the state ∅. Now we use the event A3 and Proposition 7.4, to successively define what in our case good blocks are and show that they occur with sufficiently high probability. Let us set

where j, k ∈ Z and a, b > 0. Proof. The idea here is that one can apply Proposition 7.4 multiple times since we can stack appropriately spatially shifted and if necessary reflected versions of the event A3 repeatedly to move the centre (x, s) of the initially fully infected hypercube x + [−n, n] d , where x ∈ [−(2L + n), 2L + n] d , in five to ten steps to a new centre (y, t) with y ∈ [2L + n, 3(2L + n)] × [−(2L + n), (2L + n)] d−1 . Note that in every step we use the strong Markov property to restart the background process in the empty configuration.

We visualized the procedure in Figure 11 . In Figure 11 (a) we illustrated how we are able, with an appropriate spatial shift, to move the center of the hypercube in direction of a given coordinate axis such that it is contained in [2L + n, 3(2L + n)]. Furthermore, in Figure 11 (b) it is illustrated how we ensure that via reflection over the remaining coordinate axes we keep the remaining coordinates in [−(2L + n), (2L + n)] d−1 . See [Sei21, Proposition 6.2.2] for the detailed proof. Now we define good block events so that we can construct a suitable coupled oriented percolation model on the "macroscopic" lattice {(j, k) ∈ Z × N0 : j + k even}. We identify the points (j, k) with the space-time boxes (b) All other coordinates: Assume x i = 0, while using successively (74) reflect along the coordinate plane if i-th coordinate changes its sign. Note that after achieving x 1 ∈ [a, 3a] we apply this strategy to the first coordinate as well until t ∈ [5b, 6b] See the solid boxes in Figure 12 for an illustration. At last we now formulate the events which we call "good blocks" B ± = B ± (j, k, (x, s)), where j, k ∈ Z and (x, s) ∈ S j,k . We set (75)

For these events, similarly to Proposition 7.5, the following lemma holds.

Lemma 7.6. Suppose Condition 7.2 holds. Then for every ε > 0 there are choices of n, a, b with n < a such that if (x, s) ∈ S j,k then P(B ± ) > 1 − ε where (j, k) ∈ Z × N0.

Proof. This is a direct consequence of Proposition 7.5 in the sense that we can again stack spatially shifted versions of the events formulated in Proposition 7.5 in an appropriate manner to achieve this result. Note that for B − the statement of Proposition 7.5 for a reflected version over the first coordinate axis of the considered event is needed. This also holds true, i.e. we reflect the whole construction in the direction of the first coordinate at (2ja, 0, . . . , 0) ∈ Z d such that at the end (y, t) ∈ D j−1,k+1 . For more details see [Sei21, Lemma 6.2.4].

Note that the boxes B ± only depend on a finite sector of the graphical representation and only overlap with the adjacent boxes (see Figure 12 ). At first this last step seems a bit redundant, since we could very well work with the events defined in Proposition 7.5, but with this additional step we made the dependency between the respective events clearer. Now we are ready to prove the main theorem of this section.

Theorem 7.7. Suppose Condition 7.2 holds. Then for every q < 1 there are choices of n, a, b such that if the initial configurations W0 ⊂ 2Z and C0 = C satisfy j ∈ W0 ⇒ C ⊃ x + [−n, n] d for some x ∈ [a(12j − 1), a(12j + 1)] × [−a, a] d−1

then {(Ct, Bt) : t ≥ 0} can be coupled with an oriented site percolation (W k ) k≥0 with parameter q such that j ∈ W k ⇒ Ct ⊃ x + [−n, n] d for some (x, t) ∈ S j,k (77)

In particular this implies that the CPDP survives. Figure 12 : Here we see a visualization of the space-time boxes B ± (defined in (75)), where the solid line visualizes the box B + (j, k, ·). We also see that B + (j, k, ·) only overlaps with B − (j, k, ·) and B − (j + 2, k, ·), where the dotted lines visualizes B − (j + 2, k, ·) and the dashed B + (j − 2, k, ·).

Proof. The idea is that we construct our percolation model recursively with the help of Lemma 7.6. Thus, let for an arbitrary ε > 0 the numbers n, a, b be chosen as in the Lemma 7.6. Note that since the events we use are not independent we need to use a comparison of independent and locally dependent Bernoulli random variables to obtain an independent oriented site percolation in a second step. We now construct random variables (Xj(k), Yj(k)) with k ≥ 0 and j ∈ Z: The variables Xj(k) will either be 1 if there exists an (x, s) ∈ S j,k such that (x, s) + ([−n, n] d × {0}) is infected and otherwise 0. Additionally if such a point exists we set Yj(k) = (x, s) and if not Yj(k) = †, where † is a designated state such that the state space of these random variables is {0, 1} × Z d × [0, ∞) ∪ { †}.

Without loss of generality we assume that W0 = {0}. By assumption (76) there exists an x0 such that (x0, 0) ∈ S0,0 and x0 + [−n, n] d is initially infected. We set (X0(0), Y0(0)) = (1, (x0, 0)) and (Xj(0), Yj(k)) = (0, †) for all j = 0. Now with respect to k we recursively construct these random variables. Suppose that (Xj(k), Yj(k))j∈Z are defined for all k ≤ m, then we proceed with the step m → m + 1.

We are finally ready to show that survival is impossible at criticality for the CPDP on Z d .

Proof of Theorem 2.7. We first note that the good blocks B ± defined in (75) only depend on a bounded section of the graphical representation, so by Lemma 4.3 and Lemma 7.10 we get that P λ,r,α,β (B ± ) is continuous seen as a function of any of the four parameters. Let us take as usual the infection parameter λ as an example. By Proposition 7.5 we know that for every ε > 0 we can find a, b, n such that P λ (B ± ) > 1 − ε Then because of continuity there must exist a λ < λ such that P λ (B ± ) > 1 − ε as well and hence by Theorem 7.7 it follows that the CPDP also survives with infection rate λ . This proves the claim.

Recall that we call a function f : R d ⊂ U → R separately continuous if it is continuous in each coordinate separately. In comparison to that one calls f jointly continuous if it is continuous with respect to the Euclidean topology on R d .

Proposition 7.11. Let C ⊂ V with C finite and non-empty and B ⊂ E.

(i) The survival probability θ(λ, r, α, β, C, B) is separately right continuous seen as a function in (λ, r, α, β) on (0, ∞) 4 .

(ii) The survival probability θ(λ, r, α, β, C, B) is separately left continuous seen as a function in (λ, r, α, β) onS.

Proof. We already shown right and left continuity in λ and r on the respective parameter sets in Proposition 6.11 and 6.13. Right and left continuity in α and β can be shown with the same approach.

We have seen that Corollary 2.7 states that the infection process C cannot survive at criticality. As a consequence of this fact we can conclude that the survival probability is jointly continuous with respect to its parameters (λ, r, α, β).

Proof of Proposition 2.8. Proposition 7.11 shows that the survival probability is separately left continuous seen as a function in the four parameters (λ, r, α, β) onS and is separately right continuous on (0, ∞) 4 . Now let us again prove continuity for λ → θ(λ, r, α, β, C, B). The proof is analogous for the remaining three parameter. By Proposition 7.11 it is clear that the function is everywhere continuous expect at criticality. Now obviously in case of λ the left limit at criticality exists, since we come from the subcritical parameter region where the survival probability is constant 0. By Theorem 2.7 we know that the CPDP a.s. goes extinct at criticality, which means that the survival probability is 0. But with that we have shown that the left limit and the right limit at the critical value are the same since λ → θ(λ, r, α, β, C, B) is right continuous on (0, ∞) by Proposition 7.11 and thus, the function is continuous. Now we know that the survival probability is separately continuous seen as a function of the four parameters. But the function is also monotone in each coordinate, so we can use [KD69, Proposition 2] , which states that if a function is continuous and monotone in each coordinate, then it is jointly continuous.

In this section we show complete convergence of the CPDP on Z d and thus Theorem 2.12. We start by showing that the second condition (7) holds. Proof. By translation invariance it suffices to prove the claim for x = 0. For d ≥ 2 the claim follows analogously as in the second part of the proof of [Lig13, Theorem 2.27]. Hence, we only need to consider d = 1.

Again by Theorem 7.7 for every 0 < q < 1 there exist n, a, b such that an oriented site percolation (W k ) k≥0 with parameter q exists, which satisfies (76) and (77). Now let us consider the set Dm = (−15am − 2, 15am + 1). By construction of the oriented site percolation in the proof of Theorem 7.7 (see Figure 12 for a visualization) it follows that for m > 0, 

since the infection is always contained in the blocks B ± . Now by [DS87, Theorem 2] it follow that the right-hand side in (78) converges to 1 as m → ∞.

Now it is left to prove that (6) holds true. We split the proof of this condition in two parts. First we show with Theorem 7.7 that a positive survival probability already implies that the probability that a single vertex is infinitely often infected is positive as well.

Proposition 7.13. Let (C, B) be a CPERE on Z d and (λ, r, α, β) ∈ S, then P (C,B) λ,r,α,β (x ∈ Ct i.o.) > 0 for all x ∈ V and all non-empty C ⊂ V and B ⊂ E.

Proof. We briefly summarize the proof strategy which is again analogous to that of the classical contact process with the difference that one needs to choose the background appropriately when the strong Markov property is used to restart the process. For a detailed proof see [Sei21, Proposition 6.3.6].

First, one observes that it suffices to show P (0,∅) λ,r,α,β (0 ∈ Ct i.o.) > 0. Then the rough idea is that by Theorem 7.7 we can choose an oriented site percolation (W k ) k≥0 which satisfies (76) and (77) with parameter 0 < q < 1 large enough such that P {0} (0 ∈ W 2k i.o.) > 0, i.e. 0 is hit infinitely often. That the latter is true for large enough q is a well-known fact for oriented percolation. But this means that by (76) and (77) there is a positive probability for the event C [−n,n] d ,∅ t ⊃ x + [−n, n] d ∩ Z d for some (x, t) ∈ S 0,k for infinitely many k. This means that with positive probability we have infinitely many times t k such that at this time a vertex x k exist such that the box x + [−n, n] d ∩ Z d is fully infected. Furthermore we know that there exists a uniform bound on the distance of all x k to 0 since all (t k , x k ) are contained in S 0,k . Thus, using a generalized version of Borel-Cantelli's Lemma we obtain the claim.

Next we show (for a general CPERE on Z d ) that if we have a positive probability that a single vertex is infinitely often infected we can already conclude that (6) holds.

Proposition 7.14. Let (C, B) be a CPERE on Z d . Suppose that P Proof. Again there are only minor adjustments needed for the proof compared to how this is proven for the classical contact process, for the details we refer to [Sei21, Proposition 6.3.6].

Finally we are able prove that complete convergence holds for the CPDP on the whole parameter set (0, ∞) 4 .

Proof of Theorem 2.9. Suppose that (λ, r, α, β) ∈ S. Then by Proposition 7.12, Proposition 7.13 and Proposition 7.14 we know that (6) and (7) are satisfied, and thus by Theorem 2.5 it follows that On the other hand if (λ, r, α, β) ∈ S c , then by Proposition 6.3 it follows that ν = δ ∅ ⊗ π. Thus, it follows that (C V,E t , B E t ) ⇒ δ ∅ ⊗ π as t → ∞. By monotonicity shown in Lemma 4.2 we then know that (C C,B t , B B t ) ⇒ δ ∅ ⊗ π as t → ∞ for all C ⊂ V and B ⊂ E, which proves the claim.

We conclude this section by showing that for a general CPERE on Z d complete convergence holds on a suitable subset of its survival region, namely on the survival region of a suitably chosen CPDP, which lies "below" the CPERE. We use the subscript DP in order to distinguish between a CPERE and a CPDP.

Proof of Theorem 2.12. Let (C, B) be a CPERE whose background process has spin rate q(·, ·). Recall from (10) the rates αmin := min F ⊂N L e q(e, F ) and βmax := max F ⊂N L e q(e, F ∪ {e}). By Proposition 2.10 there exists a CPDP (C, B) with rates αmin, βmax and the same initial configuration as (C, B), i.e. C0 = C 0 and B0 = B 0 , such that C t ⊂ Ct and B t ⊂ Bt for all t ≥ 0. This implies that P(x ∈ C t i.o.) ≤ P(x ∈ Ct i.o.)

If Xj−1(m) = 0 and Xj+1(m) = 0 then we set (Xj(m + 1), Yj(m + 1)) = (0, †)

Again the events B + (j − 1, m, Yj−1(m)) and B − (j + 1, m, Yj+1(m)) only guarantee existence of a point (y, t) ∈ Sj,m+1 such that (z, r) + ([−n, n] d × {0}) is completely infected, but there might exist more than one. We let Yj(m + 1) be the smallest space-time point (y, t) in the sense that we take the earliest with respect to time and, if that does not yield a unique point

Yj(k))j∈Z with k ≤ m. By the choice of n, a, b made at the By assumption θDP(λ, r, αmin, βmax, {0}, ∅) > 0, and thus Proposition 7.13 and (79) imply that P (C,B) λ,r (x ∈ Ct i.o.) > 0 for any finite and non-empty set C ⊂ Z d and any B ⊂ E. Furthermore, by Proposition 7.14 it follows that the first condition (6) holds. Due to the fact that C t ⊂ Ct and B t ⊂ Bt for all t ≥ 0 and Proposition 7.12 one also obtains that (7) is satisfied. Since we assumed that (i)-(iii) of Assumption 1.1 are satisfied Theorem 2.5 now implies that if θDP(λ, r, αmin, βmax, {0}, ∅) > 0, then

We only need to assume that the graph G is distance transitive, which is a stronger version of transitivity, see Remark 2.13, since this property together with translation invariance provides us with the following symmetry P ({x},∅) (0 ∈ Ct for some t ≥ 0) =

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The critical contact process in arandomly evolving environment dies out

A course in interacting particle systems

Upper bound of critical values for contact processes on open clusters of bond percolation

The complete convergence theorem holds for contact processes in a random environment on Z d × Z +

beginning of the proof we see that P(Xj(m + 1) = 1|Gm) > 1 − ε on {Xj(m) = 1 or Xj−1(m) = 1}.We use now a standard result concerning k-dependent Bernoulli random variables. Since B ± only overlap with their adjacent boxes, by construction the (Xj(m + 1))j∈Z are conditional on Gm a 1-dependent family of Bernoulli variables. By [Lig13, Theorem B26] we find a family of independent Bernoulli variables such that we can define an oriented site percolation (W k ) k≥0 with parameter q := (1 − ε −3 ) 2 which satisfies (76) and (77). Since ε was arbitrary this finishes the proof.

In this section we can finally reap the benefits of all the work we have done so far in Subsection 7.1 for the CPDP on Z d . First we prove that at criticality, survival is not possible and as direct consequence we gain continuity of the survival probability. Then, we use Theorem 7.7 to show that for the CPDP the two conditions (6) and (7) are satisfied if the survival probability is positive such that by Theorem 2.5 it follows that complete convergence for the CPDP holds.

Here, we show extinction at criticality (Theorem 2.7) and continuity of the survival probability (Proposition 2.8). Recall that we defined in (9) the survival region asIn case of the CPDP we have two additional parameters α and β for which we can easily deduce similar monotonicity and continuity properties as for the infection and recovery rate λ and r in Section 4.3.Lemma 7.8 (Monotonicity with respect to the background). Let (C, B) be a CPDP with parameters λ, r, α, β. Let α ≥ α, then there exists a CPDP ( C, B) with parameter λ, r, α, β and the same initial configuration such that Ct ⊆ Ct and Bt ⊆ Bt for all t ≥ 0. In words C is monotone increasing in α. On the other hand C is monotone decreasing in β.Proof. This follows with an analogous coupling as in the proof of Lemma 4.2. Since if we consider α ≥ α then we can define ( C, B) via Ξ λ,r and add additional independent opening events with rate ( α − α). Thus we get a CPDP ( C, B) which is coupled to (C, B) such that Ct ⊂ Ct and Bt ⊂ Bt for all t ≥ 0. The monotonicity in β follows analogously.Remark 7.9. Obviously π α,β π α,β if α ≤ α. Thus, if we consider the stationary case, i.e. C0 = C ⊂ V and B0 ∼ π α,β , then there exists an CPDP ( C, B) with parameter λ, r, α, β and C0 = C ⊂ V and B0 ∼ π α,β such that Ct ⊆ Ct and Bt ⊆ Bt for all t ≥ 0. This follows by first coupling the initial state of the background with Strassen's Theorem such that B0 ⊆ B0 and then using Lemma 7.8.Lemma 7.10 (Continuity for finite times and finite initial infections). Let (C, B) be a CPDP withλ,r,α,β (Cs) s≤t ∈ A and β → P (C,B) λ,r,α,β (Cs) s≤t ∈ A are continuous. (ii) If B0 ∼ π α,β , then α → P (C,π α,β ) λ,r,α,β (Cs) s≤t ∈ A and β → P (C,π α,β ) λ,r,α,β (Cs) s≤t ∈ A are continuous.Proof.(i) The proof for α and β is similar to the proof of Lemma 4.3. To show the continuity in α we can again construct a CPDP ( C, B) with ratesα and β withα > α such that these processes are coupled in the way that C0 = C0, B0 = B0, Ct ⊆ Ct and Bt ⊆ Bt for all t ≥ 0. Then, it again suffices to show that as α → α it follows that P(Cs = Cs for some s ≤ t) → 0 to conclude right continuity. Left continuity follows via an analogous procedure. The continuity in β follows analogously.(ii) Here the difference to (i) is that the the invariant law π is the initial state of the background and thus we additionally need to ensure that the initial state of the process B also converges. This is not difficult to show since all edges are independent.(x ∈ Ct i.o.) = θ(λ, r, C, B). Therefore, as in the proof of Theorem 2.12 we can compare a CPERE on a distance transitive graph G with a CPDP on the 1-dimensional integer lattice to conclude complete convergence for a certain choice of parameters, as described in Remark 2.13.