Symmetric simple exclusion process in dynamic environment: hydrodynamics

We consider the symmetric simple exclusion process in $\mathbb Z^d$ with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the single particle invariance principle and the macroscopic behavior of the density field.


INTRODUCTION
Dynamic random environments are natural quantities to be inserted in probabilistic models in order to make them more realistic. But to study such models is challenging, and for a long time only models endowed with a static environment were considered. However, random walks in dynamic random environment (RWDRE) have been extensively studied in recent years [1,2,3,4,5,8,36] and several results on the law of large numbers, invariance principles and heat kernel estimates have been obtained. A natural next step is to consider particle systems in such dynamic environments. There the first question concerns the derivation of hydrodynamic limits. In this article, we answer this question for the nearest neighbor symmetric simple exclusion process.
For interacting particle systems with a form of self-duality and that evolve in a static disorder, the problem of deriving the macroscopic equation governing the hydrodynamic limit has been shown to be strongly connected to the invariance principle for a single random walker in the same environment [12,32]. Indeed, the feature that, if a rescaled test particle converges to a Brownian motion then the interacting particle system has a hydrodynamic limit, appears already in [7,18,27,35]. Our contribution is to carry out this connection between single particle behavior and diffusive hydrodynamic limit in the context of dynamic environment, namely for the nearest-neighbor symmetric simple exclusion process in a dynamic bond disorder, for which we show that a suitable form of self-duality remains valid.
SYMMETRIC SIMPLE EXCLUSION PROCESS. We first introduce the nearest neighbor symmetric simple exclusion process without disorder (SSEP) in Z d with d ≥ 1 [29,38]. In words, SSEP is an interacting particle system consisting of indistinguishable particles which are forbidden to simultaneously occupy the same site, and which jump at a constant rate only to nearest-neighbor unoccupied sites. More precisely, let η ∈ {0, 1} Z d be a configuration of particles, with η(x) denoting the number of particles at site x ∈ Z d . The stochastic process {η t , t ≥ 0} is Markovian and evolves on the state space {0, 1} Z d according to the infinitesimal generator (1. 1) where |x − y| = d i=1 |x i − y i | and ϕ : {0, 1} Z d → R is a bounded cylinder function, i.e. it depends only on a finite number of occupation variables {η(x), x ∈ Z d }. In (1.1) the finite summation is taken over all unordered pairs of nearest-neighboring sites -referred to as bonds -and η x,y is the configuration obtained from η by removing a particle from the occupied site x and placing it at the empty site y. The hydrodynamic limit [7,19,25] of the particle system described by (1.1) is known [7,25] and, roughly speaking, prescribes that the particle density scales to the weak solution of the heat equation.
STATIC ENVIRONMENT. For the SSEP in a static bond disorder in Z d , hydrodynamic limits have been obtained by means of the self-duality property of the particle system. As examples, see [11,32] with d = 1, [12] with d ≥ 1 and [14] on the supercritical percolation cluster with d ≥ 1. The method is rather general as it has been applied also to non-diffusive space-time rescalings, i.e. when the hydrodynamic behavior is not described by a heat equation [15]. On the other side, all hydrodynamic limits obtained via this self-duality technique lack of a proof of relative compactness of the empirical density fields. Indeed, a direct application of the classical Aldous-Rebolledo criterion (see e.g. [25]) fails when following this approach.
Other techniques than self-duality -which also apply to different particle systems -have also been studied in static environments. For instance, in static bond disorder, the method based on the so-called corrected empirical process has been applied to prove hydrodynamics for the the SSEP [23] and for zero-range processes [13,16,22]. The non-gradient method [35,39] (see also [25]) has found many applications to reversible lattice-gas models in a more general static environment, see e.g. [17]. sites, i.e.
We introduce a dynamic bond-disorder on (Z d , E d ). Namely, we assign timedependent positive weights to each bond {x, y} ∈ E d and we define as environment any càdlàg (w.r.t. the time variable t) function where is referred to as the conductance of the bond {x, y} ∈ E d at time t ≥ 0. The environment λ is said to be static if λ t ({x, y}) = λ 0 ({x, y}), for all {x, y} ∈ E d and t ≥ 0.
We will need the following assumption on the environment.

REMARK 2.1. The boundedness of conductances guarantees, via a graphical construction (see Appendix A), that all stochastic processes introduced in Sections 3 and 4 are well-defined.
Given the environment λ as defined in (2.1)-(2. 2), we now introduce as a counterpart to the symmetric simple exclusion process without disorder (1.1) the timeevolution of the symmetric simple exclusion process in the dynamic environment λ (SSEP(λ)) by specifying its time-dependent infinitesimal generator L t . For all t ≥ 0 and every bounded cylinder function ϕ : {0, 1} Z d → R, we have )) . (2. 3) Given any initial configuration η ∈ {0, 1} Z d , the time-dependent infinitesimal generators in (2.3) generate a time-inhomogeneous Markov (Feller) process {η t , t ≥ 0} with sample paths in the Skorokhod space D([0, ∞), {0, 1} Z d ) such that η 0 = η. We postpone to Section 4 the construction of this infinite particle system via a graphical representation.

MAIN RESULT: HYDRODYNAMICS
In the present section we discuss the hydrodynamic limit of the particle system {η t , t ≥ 0} evolving in the environment λ, described by (2.3). For this we introduce for all N ∈ N the empirical density field {X N t , t ≥ 0} as a process in denotes the class of rapidly decreasing functions on R d and S ′ (R d ) its topological dual. For any test function G ∈ S (R d ), the empirical density evaluated at G reads as So we choose to view the empirical density field as a random tempered distribution rather than as a Radon measure. Indeed, the space S ′ (R d ) has the advantage that it is a good space for tightness criteria (see [31]) and we use the fact that S (R d ) is closed under the action of the Brownian motion semigroup. Let us denote by ·, · the standard scalar product in R d . We denote by {ρ Σ t , t ≥ 0} the unique weak solution to the following Cauchy problem and Σ being a d-dimensional real symmetric positive-definite matrix (see e.g. [10,25]). We recall that for {ρ Σ t , t ≥ 0} being a weak solution of (3.2) means that, for all G ∈ S (R d ) and t ≥ 0, ), the space of tempered distribution-valued continuous functions. Indeed, for all G ∈ S(R d ) and t ≥ 0, we have [10,25] where {S Σ t , t ≥ 0} is the transition semigroup of the Brownian motion in assumption (b) of Theorem 3.2 below, and, by [30,Theorem 1] and ρ • ∈ L ∞ (R d ), Usually (see e.g. [25]), the convergence of the processes {X N t , t ≥ 0} to {π Σ t , t ≥ 0} is derived by starting from the Dynkin martingale associated to the empirical density field, for all G ∈ S (R d ) and t ≥ 0. After obtaining tightness of the sequence {X N t , 0 ≤ t ≤ T } via an application of Aldous-Rebolledo criterion, the rest of the proof is carried out in two steps. First one shows that the martingale term M N t (G) vanishes in probability as N → ∞. Secondly, all the remaining terms in (3.5) can be expressed in terms of the empirical density field only; i.e. one "closes" the equation. Unfortunately, in our case, tightness cannot be derived via usual techniques. Moreover, in presence of (static or dynamic) disorder, "closing" equation (3.5) cannot be directly achieved.
In the static disorder case, in [22,23] the authors solve this problem by introducing an auxiliary observable, called corrected empirical density field. However, here we follow the probabilistic approach initiated in [11,32], which is more natural in our context. Key ingredients of this method are the self-duality property of the particle system and the limiting behavior of a single particle. For this reason, our main result can be stated in terms of properties of suitable random walks, which we define below.

DEFINITION 3.1 (FORWARD AND BACKWARD RANDOM WALKS).
Let {X x s,t , t ≥ s} be the forward random walk starting at x ∈ Z d at time s ≥ 0 and evolving in the environment λ through the time-dependent infinitesimal generator Similarly, for all t ≥ 0, let { X y s,t , s ≤ t} be the backward random walk which starts at y ∈ Z d at time t ≥ 0 and "evolves backwards" in the environment λ through the time-dependent infinitesimal generator We will give in Section 4.1 and Appendix A.1 the construction of both those forward and backward random walks via a graphical representation.
We are now ready to state our main theorem.
where {B Σ t , t ≥ 0} denotes a d-dimensional Brownian motion, starting at the origin and with non-degenerate covariance matrix Σ (Notation: ⇒ stands for convergence in law).
If, for all N ∈ N, the distribution of X N 0 is the distribution on S ′ (R d ) induced by ν N , then, for all T > 0, we have the convergence is the unique weak solution to the Cauchy problem (3.2).

REMARK 3.3 (FORWARD AND BACKWARD INVARIANCE PRINCIPLES).
In case of static environment λ, the laws of forward and backward random walks coincide. In general, this is not true in presence of dynamic environment. However, under Assumption 1, the convergence in (3.9) is equivalent, by keeping the same notation, to For a proof of this claim, see Appendix B.

REMARK 3.4 (NON-DEGENERACY OF Σ).
The simplest example of an environment λ leading to a non-degenerate Σ is obtained by considering a realization of independent flips of conductances between two strictly positive values. In that case, the nondegeneracy of Σ follows from [36,Theorem 3.6]. For other more general contexts when non-degeneracy is proved, see for instance [1,2]. It is possible to obtain a non-degenerate Σ in presence of null conductances. In other words, the invariance principle in assumption (b) of Theorem 3.2 does not a priori exclude this possibility.
, the convergence in (3.10) can also be equivalently rewritten as follows: for all G ∈ S (R d ), P → stands for convergence in probability).
The proof of Theorem 3.2 is divided into two steps. First we prove that the sequence of distributions of by proving tightness in Section 5.2 below. Then, we prove that all limiting measures are supported on weak solutions of the Cauchy problem (3.2). By uniqueness of such a solution (see e.g. [10,25]), Theorem 3.2 will follow.
The characterization of the limiting measures boils down to prove convergence of finite-dimensional distributions: for all n ∈ N, for all 0 ≤ t 1 < . . . < t n ≤ T and for all G 1 , . . . , G n ∈ S (R d ), (3. 12) As joint convergence in probability comes down to checking convergence in probability of the single marginal laws, it suffices to prove (3. 12) for the choice n = 1 only: for all 0 ≤ t ≤ T and G ∈ S (R d ), Before presenting the proof of Theorem 3.2, in Section 4 we provide a mild solution representation for the particle system involving the forward and backward random walks of Definition 3.1. In Section 5, we will then exploit this representation to prove both (3.13) and tightness.

GRAPHICAL CONSTRUCTIONS AND MILD SOLUTION
In Section 4.1 we construct the symmetric simple exclusion process in dynamic environment via a graphical representation. Relying on this construction, we express in Section 4.2 the occupation variables of the symmetric simple exclusion process in dynamic environment as mild solution of a system of Poissonian stochastic differential equations.

GRAPHICAL CONSTRUCTION OF THE PARTICLE SYSTEM
The graphical construction employs, as a source of randomness, a collection of independent Poisson processes, each one attached to a bond of Z d . To take care of both space and time inhomogeneities, their intensities will depend both on the bond and time.
As an intermediate step towards the graphical construction of the particle system, the same Poisson processes provide a graphical construction for all forward and backward random walks introduced in Definition 3.1. We explain this procedure below, leaving a detailed treatment to Appendix A. Finally, we will relate the occupation variables of the particle system to the positions of backward random walks. This must be meant in a pathwise sense, expressing the pathwise duality of the symmetric simple exclusion process in the dynamic environment λ.

POISSON PROCESSES
We consider a family of independent inhomogeneous Poisson processes defined on the probability space (Ω, F, {F t , t ≥ 0}, P), where E denotes expectation w.r.t. P, {F t , t ≥ 0} is the natural filtration for F, and such that N · ({x, y}) has intensity measure λ r ({x, y})dr, that is The associated compensated Poisson processes are a family of locally square integrable martingales w.r.t. {F t , t ≥ 0} of bounded variation, due to Assumption 1.
The associated picture is drawn as follows. On the space Z d × [0, ∞), where Z d represents the sites and [0, ∞) represents time which goes up, for each z ∈ Z d draw a vertical line {z} × [0, ∞). Then for each {x, y} ∈ E d , draw a horizontal two-sided arrow between x and y at each event time of N · ({x, y}).

FORWARD AND BACKWARD RANDOM WALKS
We recover the walks defined in Definition 3.1 as follows.
First, for all ω ∈ Ω, x ∈ Z d and t ≥ s, X x s,t [ω] now denotes the position at time t of the process in Z d that is at x at time s ≥ 0, and that, between times s and t, crosses the bond {z, w} ∈ E d at an event time of N · ({z, w})[ω] whenever at that time the walk is at location either z or w in Z d (i.e. it follows the corresponding arrow in the graphical representation). We prove in Appendix A, thanks to Assumption 1, that the trajectories of those walks are, for P-a.e. realization ω ∈ Ω, well defined for all times and starting positions. Moreover, they are all simultaneously defined on the common probability space (Ω, F, {F t , t ≥ 0}, P}, where {F t , t ≥ 0} denotes the induced natural filtration. In Appendix A, we show that their associated generators are given by (3. 6), so that, indeed, these walks are a version of the processes introduced in Definition 3.1.
We now provide a version of the backward random walks of Definition 3.1. For P-a.e. ω ∈ Ω and y ∈ Z d , we implicitly define backward random walks' trajectories { X y s,t [ω], s ≤ t} by the following identity: In words, X y s,t [ω] denotes the unique position in Z d at which the forward random walk that follows the Poissonian marks ω ∈ Ω and that is at y ∈ Z d at time t ≥ s was at time s ≥ 0. In particular, for P-a.e. ω ∈ Ω and x, y ∈ Z d , we have Again, all these random walks are simultaneously P-a.s. well-defined, and these backward random walks coincide in law with the ones in Definition 3.1 (see Appendix A).

TRANSITION PROBABILITIES
The Poissonian construction and the jump rules explained in the previous subsection ensure that each of the forward and backward random walks is Markovian.
we obtain families of transition probabilities respectively for the forward and backward random walks. In particular, for all x, y ∈ Z d and 0 ≤ s ≤ r ≤ t, we have the Chapman-Kolmogorov equations Then, from (4. 4), we obtain that for all x, y ∈ Z d and t ≥ s. Then, the operators {S s,t , t ≥ s} and { S s,t , s ≤ t}, acting on bounded functions f : correspond to the transition semigroups respectively associated to the forward and backward random walks. Then, as a consequence of (4. 7), we obtain that 10) for all f, g : Z d → R for which the above summations are finite. We refer to Appendix A.2 for further details and properties of these time-inhomogeneous semigroups.

STIRRING PROCESS
The stirring process relates the above introduced random walks with the occupation variables of the symmetric simple exclusion process in the environment λ as follows. Due to the symmetry (2.2) of the environment and the one of the exclusion dynamics, we can rewrite the generator (2. 3) as where η {x,y} stands for the exchange of occupation numbers between sites x and y in configuration η, which takes place even if x, y are both occupied (due to the fact that particles are indistinguishable). This rewriting gives the stirring interpretation of the symmetric simple exclusion process in the environment λ (similar to the stirring interpretation in the case (1.1) without disorder, as described in [7, p. 98] and [29, p. 399]), that we take from now on. This way, the stirring process can be constructed on the same graphical representation as before, and particles evolve as the forward random walks previously introduced.
following the random walk X x 0,.
[ω]. Hence, similarly to [29, p. 399], we can write, for P-a.e. ω ∈ Ω, for any initial configuration η ∈ {0, 1} Z d , for any x ∈ Z d and t ≥ 0, that or, equivalently by using the associated backward random walks and (4. 3), In other words, . ω ∈ Ω. Moreover, from the memoryless property of the inhomogeneous Poisson processes employed in the graphical construction of forward and backward random walks, we recover the Markov property of the process {η t , t ≥ 0} w.r.t. {F η t , t ≥ 0}. What we obtained in (4.11) is the property of pathwise self-duality of the symmetric simple exclusion process with a single dual particle, which thus remains valid also in presence of the dynamic environment λ.

MILD SOLUTION
The above construction provides an alternative way of defining the symmetric simple exclusion process in the environment λ as strong solution of an infinite system of linear stochastic differential equations. This is the content of Proposition 4.1 below. Indeed, the motivation comes from an infinitesimal description of the stirring process, as explained through the following computation.
For all t > 0 and By introducing the compensated Poisson process (4.2) in (4. 12), we obtain Note that the terms in the second sum in the r.h.s. of (4.13) are increments of a martingale as products of predictable terms (w.r.t. the natural filtration of the process {η t , t ≥ 0}) and increments of the compensated Poisson processes. Moreover, like the latter, such martingales are square integrable and of bounded variation. After observing that the first sum on the r.h.s. of (4. 13) corresponds to the definition of the infinitesimal generator in (3.6) at time t of the forward random walk, (4.13) rewrites 14) where A t acts on the x variable and where 15) In the following proposition, whose proof is postponed to Section 6, we prove that the so-called "mild solution" [34, Chapter 9] associated to the system of differential equations (4.14) equals P-a.s. the process obtained via the stirring procedure in (4. 11).
The mild solution is defined as in (4. 16) below, i.e. by formally applying the method of variation of constants to (4. 14). Recall that { S s,t , s ≤ t} and { p s,t (y, x), x, y ∈ Z d , s ≤ t} are, respectively, the semigroup and transition probabilities of the backward random walks of Definition 3.1. .11), η 0 = η and dM r is given in (4. 15).
Then, for P-a.e. ω ∈ Ω, In [34] systems of equations of type (4. 14) are studied in the context of Hilbert spaces. There it is proved that for a large class of linear SDE's the so-called mild solutions coincide with weak solutions.

PROOF OF THEOREM 3.2
The key ingredient to prove Theorem 3.2 is the decomposition of the occupation variables provided in Proposition 4.1. Let G ∈ S (R d ) and consider the empirical density fields X N t (G) as in (3.1). By using first (3.1), then Proposition 4.1 and, finally, identity (4.10), we obtain Observe that, for any fixed initial configuration η ∈ {0, 1} Z d , the first term in the r.h.s. of (5.1) is deterministic whereas the second term has mean zero and contains all stochasticity derived from the stirring construction. Indeed, we have the term inside expectation being an integral of a deterministic function w.r.t. martingales, whereas, starting with (3.1), then using (4.11) and (4. 10), we obtain Thus, the decomposition (5.1) can be written as where the first term is the expectation of the empirical density field and the second is "noise", i.e. the (stochastic) deviation from the mean. Therefore, when deriving the hydrodynamic limit -basically a WLLN -the proof of (3. 13) reduces to proving that the "noise" vanishes in probability and that the expectation converges to the correct deterministic limit corresponding to the macroscopic equation. More precisely, we have to show that, for any δ > 0, We prove (5.3) and (5.4) in the following subsection. In Section 5.2 we exploit the decomposition (5.1) together with the tightness criterion given in Appendix C to prove relative compactness of the empirical density fields.
is a consequence of Chebyshev's inequality and the following lemma.
PROOF. By (4.15), we can rearrange as follows: Recall that the compensated Poisson processes {N · ({x, y}), {x, y} ∈ E d } are of bounded variation in view of Assumption 1 and, moreover, they are independent over bonds. Thus by Itō's isometry for jump processes and the independence over the bonds of the Poisson processes in (4.1), we obtain where in the second-to-last identity we used Kolmogorov backward equation (A. 6) for the forward transition semigroup. After integration, we further write In what follows, the invariance principle of forward and backward random walks, i.e. assumption (b) of Theorem 3.2, will play a crucial role. We exploit conditions, given in terms of convergence of semigroups, that are equivalent to the invariance principle. In the time-homogeneous context, the correspondence between weak convergence of Feller processes and convergence of Feller semigroups is due to Trotter and Kurtz [9,24,26]. For the sake of completeness, in the next theorem we point out how this correspondence translates in the time-inhomogeneous setting.  Having an invariance principle for both the forward and the backward random walks in the environment λ allows to replace the uniform convergence (w.r.t. x ∈ Z d ) in (5. 6) with convergence in mean (w.r.t. the counting measure). The more precise statement is the content of the following proposition. We state only the forward case, the backward one being analogous.

PROPOSITION 5.3. Keep the same notation as in Theorem 5.2. Assume that assumption (b) in Theorem 3.2 holds true. Then, for all
PROOF. The proof consists of proving a compact containment condition uniformly over time and space. More precisely, we want to show that, for all ε > 0, we can find a compact subset K ε ⊂ R d for which we have lim sup The bound (5.8) is a consequence of the uniform bound for the tails of {S Σ t G, t ≥ 0} over finite time intervals. Indeed, there exist a compact subset J ⊂ R d and a constant C > 0 such that This follows from the fact that G ∈ S (R d ), S Σ t acts as convolution with a nondegenerate Gaussian kernel and the use of Fourier transformation. Then, it suffices to choose K ε ⊃ J such that We turn now to (5.9). Let H ε ⊂ R d be a compact subset such that, for all N ∈ N, it holds As a consequence, for all N ∈ N, we have the following upper bound: where |H ε | denotes the Lebesgue measure of H ε and in the second inequality we used x N / ∈Kε p sN 2 ,tN 2 (y, x) ≤ 1 and (5. 10). After observing that, for all y ∈ Z d , Then, up to choose K ε ⊂ R d large enough so that H ε ⊂ K ε and the bound (5.9) holds. The bounds (5.8) and (5.9) together with (5.6) lead to (5.7).
We apply Proposition 5.3 and hypothesis (a) of Theorem 3.2 to prove (5.4) and conclude the characterization of the finite-dimensional distributions of the limiting density field.
Let {ρ Σ t , t ≥ 0} be the unique weak solution of the Cauchy problem as given in Hence, for any family of probability measures {ν N , N ∈ N} associated to the density profile ρ • (see (3.8) for the definition), we obtain 12) for all t ≥ 0 and all δ > 0. In turn, (5.4) comes as a consequence of (5.12) and the following lemma.
Then we obtain (5.13) via Proposition 5.3 together with the Markov's inequality.

TIGHTNESS
In this section we prove tightness of the sequence of density fields {X N · , N ∈ N} in the Skorokhod space D([0, T ], S ′ (R d )). Note that tightness of the distributions {X N · , N ∈ N} is implied by tightness of the single density fields evaluated at all functions G ∈ S (R d ) (see [31]). Hence, it suffices to discuss tightness of the sequence The criterion we use is given in Appendix C. Note that we cannot use Aldous-Rebolledo criterion (see e.g. [25]), which relies ultimately on Doob's maximal martingale inequality. Indeed, instead of decomposing the empirical density fields into a predictable term and a martingale term, we employed the mild solution representation (4. 16) for which maximal inequalities for martingales do not apply. We postpone to Appendix C any precise statements and anticipate that in our case the proof boils down to prove the following.
As a consequence, {X N · , N ∈ N} is a tight sequence in D([0, T ], S ′ (R d )). PROOF. Statement (a) is a direct consequence of (3.13) proved in Section 5.1. We prove equicontinuity. For all N ∈ N, 0 ≤ h ≤ T and 0 ≤ t ≤ T , writing X N t+h (G) via (5.1), then using for these terms (4.10), Chapman-Kolmogorov equation for {S s,t , t ≥ s} (see Proposition A.2(f)), and (4. 16), we get the decomposition Thus, we obtain 15) and we estimate separately the two terms in (5.14) and (5. 15). We start with the term in (5.14), that we write A. The bound η t (x) ≤ 1 yields and the probability on the r.h.s. vanishes as N → ∞. This can be seen as follows: (α) by Proposition 5.3 we can deduce that there exists a sufficiently large N ε ∈ N such that, for all N ≥ N ε , we have from which the conclusion follows.
To bound the term in (5. 15), that we write B, we combine Chebyshev's inequality and the argument in the proof of Lemma 5.1 and we get Now, let N ε ∈ N and h ε > 0 be the values obtained from conditions (5. 16) and (5. 17).
By the strong continuity of the transition semigroup {S s,t , t ≥ s} (see Proposition A.2(g)), for all N ≥ N ε we have ψ N ε (h) → 0 as h → 0, giving the first part of (i). We obtain (ii) from (5.18) and our choices for N ε and h ε . For the last items (iii) and (iv), use that the forward transition semigroup is a contraction (see Proposition A.2(d) ), and note that Hence, we conclude the proof by setting the first term on the r.h.s. equal to ψ ε (h) and the second term equal to φ N .

PROOF OF PROPOSITION 4.1
We have to show in this section that the infinite summation on the r.h.s. of (4. 16) is absolutely convergent and that it equals η t (x) [ω]. The proof relies on the construction of active islands (introduced in Appendix A.1) and on a finer control on their radius, which allows to obtain exponential bounds on the transition probabilities of the random walks. As a consequence, we prove identity (4. 16) for all initial conditions η ∈ {0, 1} Z d and all times t ≥ 0. The plan is the following. First we show that, when restricting to a finite summation, formula (4. 16) indeed holds for P-a.e. ω ∈ Ω. Then, based only on a percolation result on the radius of active islands in sufficiently small time intervals [20] and the uniform boundedness assumption of the conductances (Assumption 1), we obtain an exponential upper bound for the heat kernel. In conclusion, we prove that, for P-a.e. ω ∈ Ω, the infinite summation in (4. 16) is absolutely convergent, hence a rearrangement of the order of the summation, which does not change its value, gives us the result. [ω] in terms of the stirring process, poses no problem. In the following lemma, due to the finiteness of active islands, we can give a precise meaning to (4. 16) when restricting the summation only to particle positions within the same active island. < h c (d, a). Then, for P-a.e. ω ∈ Ω and any configuration η s ∈ {0, 1} Z d , we have c (d, a). By recalling the definition of dM r in (4. 15) and the following backward master equation (obtained by using (A.6))

PROOF. For notational convenience, let us set s = 0 and t < h
we rearrange the l.h.s. in (6.1) to obtain: which further simplifies as In turn, there will be exactly one term in the following sum which cancels the corresponding k-th term in (6.4). Hence, after reordering these finite summations, (6.3) reduces to the following The observation that p t,t (x, y) = 1l{x = y} concludes the proof.

RADIUS OF ACTIVE ISLANDS AND ABSOLUTE CONVERGENCE.
We start by presenting a key estimate, direct consequence of [20,Theorem 3.4] and Assumption 1, on the radius of active islands: for all n ∈ N and x ∈ Z d . In words, the probability that the active island in (s, t] containing x ∈ Z d contains at least one site at distance n from x ∈ Z d is smaller than e −χ(t−s)n , for all n ∈ N. The function χ : (0, h c (d, a)) → (0, ∞) can be chosen to be non-increasing.
For all x ∈ Z d , t ≥ 0 and η ∈ {0, 1} Z d , we need to give a precise meaning to the infinite sum in (4. 16) for P-a.e. realization ω ∈ Ω. More precisely, we need to ensure that this infinite sum is absolutely convergent, allowing us to reorder the summation so as to sum over finite active islands (over space and time) first, and then, to apply Lemma 6.1. This is the content of the following lemma.

PROOF.
For item (a), all terms being non-negative, we can reorder the summation on the l.h.s. in (6.6) to obtain Hence, by (6.11) and (6.12), (6.10) is bounded above by By iterating this procedure for a finite number of steps, we obtain the following upper bound for (6.10): If we bound the last summation in parenthesis as follows (see also (6.11)) ≤ P ∃ w ∈ Z d : |w − x| = m 1 and w ∈ G [0,t 1 ] (x) ≤ e −χ(t 1 )m 1 , (6. 13) we obtain For the second term, we observe that, for all m ∈ N and y ∈ Z d with |y − x| = m, by independence of the Poisson processes over the edges and Assumption 1, we have for some constant c > 0 independent of m. As a consequence, we obtain ∞ m=0 y:|y−x|=m Hence, by a Borel-Cantelli argument, we can conclude that, for P-a.e. ω ∈ Ω, there exists a constant c[ω] > 0 for which 14) holds for all m ∈ N and y ∈ Z d with |y − x| = m. Therefore, for P-a.e. ω ∈ Ω, by (6.14), processes to a bond percolation model in Z d [20]. Using the latter, we can construct the families of random walks by piecing together paths defined on sufficiently small time intervals which cover the whole positive real line.
REMARK A.1. The uniform boundedness assumption could in principle be relaxed as long as results from bond percolation models transfer to our inhomogeneous setting. For examples of weaker assumptions on the conductances when d = 1, see e.g. [11,Lemma 2.1] or [4].
Moreover, P-a.s. each G I k (x) contains at most finitely-many Poissonian marks (and no marks at the times kh, for all k ∈ N). As a consequence, for P-a.e. ω ∈ Ω, if we choose s, t ∈ I k for some k ∈ N with

A.2 FELLER TRANSITION SEMIGROUPS AND GENERATORS
We study properties of the transition semigroups {S s,t , t ≥ s} and { S s,t , s ≤ t} introduced in (4. 8) and (4. 9) and their associated infinitesimal generators solving the associated Kolmogorov forward and backward equations as in (A. 5) and (A. 6), which turn out to be {A t , t ≥ 0} and {A t − , t ≥ 0} defined in (3. 6) and (3.7). Indeed, for all where the following limits λ s ± ({x, y}) = lim h↓0 λ s±h ({x, y}) exist and λ s + ({x, y}) = λ s ({x, y}) as the conductances are assumed to be càdlàg.
In what follows, for a differentiable function φ : Moreover, C 0 (R d ) denotes the Banach space of real-valued continuous functions on R d vanishing at infinity endowed with the sup norm · ∞ . By C 0 (Z d ), resp. S (Z d ), we denote the space of functions obtained as restrictions to Z d of functions in C 0 (R d ), resp. S (R d ).
The proofs of next propositions, which follow from Assumption 1, are left to the reader. For notational convenience, we extend the definitions of conductances, transition semigroups and generators to negative times.
(c) Kolmogorov forward and backward equations. and FELLER PROPERTY. We now consider the space-time processes [41,Section 8.5.5] 8) associated to forward and backward random walks, respectively. These processes are time-homogeneous Markov processes on the state space Z d ×(−∞, ∞) with infinitesimal generators B and B given by 10) for all f ∈ S (Z d × (−∞, ∞)) [6], [37,Chapter III.2], [41,Section 8.5.5]. Hence, by passing to this formulation, Propositions A.2 and A.3 guarantee that the forward and backward random walks are Feller processes.

B FORWARD AND BACKWARD INVARIANCE PRINCIPLE
As announced in Remark 3.3, we prove that an invariance principle for the forward random walks (3.9) holds if and only if an analogous result holds for the backward random walks (3.11). For this, next to the two equivalent formulations (A) and (B) of the invariance principle for the forward random walks in Theorem 5.2, we add a third one below: and A Σ = 1 2 ∇(Σ · ∇) is the infinitesimal generator of the Brownian motion {B Σ t , t ≥ 0} with covariance matrix Σ. The proof of the equivalence of (A), (B) and (C) can be found in [24,Theorem 19.25], [9] after considering the generator (A.9) of the associated space-time process. The analogous condition for the backward random walks reads as follows: where the notation is as in (C).
As a consequence, if holds for all G ∈ S (R d ), then, by triangle inequality, (C) and ( C) are equivalent. In turn, the invariance principles in Theorem 3.2(b) would also be equivalent. We end this section by showing that in our context (C) and ( C) are always equivalent, even without relying on (B.1).

As a consequence, (C) holds if and only if ( C) holds.
PROOF. We start with the proof of (B.2). Let ℓ ∈ N. By Assumption 1, we obtain where c f is a constant depending only on f ∈ S (Z d ). For any ε > 0, we choose ℓ > 0 large enough and, consequently, δ > 0 small enough so that This proves (B.2). Now assume (C). Then, for all N ∈ N, we have The uniform bound (B. 3) and (C) give ( C). The converse implication is obtained analogously.

C TIGHTNESS CRITERION
We present a tightness criterion for processes in the Skorokhod space D([0, T ], R). This criterion relies on the notion of uniform conditional stochastic continuity of a process [40,Appendix A]. The study of this property allows to extract information on the modulus of continuity of the trajectories. By following closely the argument in [40, Appendix A], we get a quantitative estimate for the modulus of continuity leading to a sufficient condition for tightness. To the best of our knowledge, this strategy has not been remarked before with this purpose, therefore we provide below a detailed proof.
As a first step, we specify the topological setting. max{w z ((s, r)), w z ((r, t))} .
Roughly speaking, for all δ > 0, the D-modulus of continuity w ′ z (δ) "allows" for one jump in intervals of size at least δ. Moreover, one can show [40]  (T1) For all 0 ≤ t ≤ T and ε > 0, there exists a value m t,ε > 0 such that In Theorem C.4, we will present a condition alternative to (T2) on the uniform control of the D-modulus of continuity. First we need Theorem C.3, which is a slight modification of [40,Theorem A.6]. Indeed, the proof of Theorem C.3 follows closely the one of [40,Theorem A.6]. Only in the last part, the two proofs differ yielding a different upper bound (C. 2). For the sake of completeness, though, we include the whole proof at the end of this section.
Then the process {Z t , t ≥ 0} can be realized in the Skorokhod space D([0, T ], R) and the bound holds for all ε > 0, 0 ≤ h ≤ h ε 2 and k large enough. (ii) For all 0 ≤ h ≤ h ε and 0 ≤ t ≤ T , we have Then, condition (T2) in Theorem C.2 holds for the processes {Z N · , N ∈ N}.
PROOF. Due to (i) and (ii) we can apply Theorem C.3 to get an estimate for P N (w ′ Z N (h) > ε) of the form (C. 2) with ψ N ε 2 . By using, in addition, item (iv), we obtain the bound , which is valid for all ε > 0, 0 ≤ h ≤ h ε , N ≥ N ε and k large enough. Now observe that, by (iii), we have lim sup .
(C. 3) We are left to show that the r.h.s. of the previous inequality vanishes as h → 0. We use the fact that ψ ε (h) → 0 as h → 0. First observe that, for any arbitrary small σ > 0, there exists k σ large enough such that We can then choose j σ ≤ h ε 2 so that to control also the first term on the r.h.s. of (C. 3). Namely, we can pick a value j σ such that we have for all 0 ≤ h ≤ j σ . Choosing σ sufficiently small leads to condition (T2).
PROOF OF THEOREM C.3. We follow here [40,Theorem A.6]. At first, we prove a bound of the type (C.2) for a discrete-time process that we then extend to a continuoustime process by means of stopping times. Consider Y = {Y 0 , . . . , Y n } and let (O, G, P) be a probability space for which Y : (O, G, P) → R n+1 is Borel-measurable. Moreover, let {G m , 0 ≤ m ≤ n} be the natural filtration associated to Y, i.e. G m = σ{Y ℓ , 0 ≤ ℓ ≤ m}. In this setting we prove that, if Y 0 = 0 P-a.s. and for all ε > 0 there exists a q ∈ [0, 1) for which sup 0≤m≤n P (|Y n − Y m | > ε | G m ) ≤ q , P-a.s. , (C. 4) then we have (C. 5) We start with the simple observation that, since Y 0 = 0 P-a.s. P sup 0≤m≤n |Y m | > 2ε = P sup 0<m<n |Y m | > 2ε, |Y n | < ε + P sup 0<m≤n |Y m | > 2ε, |Y n | ≥ ε (C. 6) and we estimate the two terms on the r.h.s. of (C. 6) separately. On the one hand, if we consider the event where the last inequality follows from a rephrasing in terms of conditional probabilities and (C. 4). Moreover, observe that P(sup 0<m<n |Y m | > 2ε) = n−1 m=1 P(H m ). On the other hand, we obtain P sup 0<m≤n |Y m | > 2ε, |Y n | ≥ ε ≤ P (|Y n | ≥ ε) ≤ q.
To pass to the continuous-time process {Z t , t ≥ 0} we employ stopping times. Fix ε > 0 and define τ 1 as the first time |Z t − Z 0 | exceeds ε, τ 1 + τ 2 as the first time |Z t − Z τ 1 | does and so on, up to reach time T and with the convention that, if τ 1 + . . . + τ n > T , we discard it by setting it equal to T + 1. As a consequence of these definitions, if we set σ n = τ 1 + . . . + τ n , we have: for all n ∈ N, |Z σn+h ′ − Z σn | > ε F σn = P (τ n+1 ≤ h | F σn ) . (C. 9) By (C. 1), which holds true also when considering the stopping time σ-field, as the bound is uniform in time, and the discrete-time argument used to derive (C. 5) from (C. 4), we obtain, for all h > 0 and n ∈ N, .
(C. 10) Recall Definition C.1 of the D-modulus of continuity. For any choice of k ∈ N, the probability P (w ′ Z (h) > 2ε) can be bounded from above by (C. 12) The probability in (C. 11) vanishes. Indeed, if the events {σ k > T } and {min{τ 1 , . . . , τ k } > h} occur, then necessarily in any subinterval of size h of [0, T ] there can be at most one σ ℓ , for some 0 ≤ ℓ ≤ k, making the event {w ′ Z (h) > 2ε} impossible. Now we estimate each term in (C. 12). For the second one, by (C. 9) and (C. 10), we get .
(C. 13) The third term can be controlled thanks to (C.10); indeed, it yields . (C. 14) It is slightly more involved to control the first term in (C. 12). We have .
where in the last inequality we have used (C. 9) and (C. 10). Hence, whenever δ · k > T , we obtain .