ISSN: 1083-589X ELECTRONIC COMMUNICATIONS

We introduce the time dynamic random conductance model and consider the heat kernel for the random walk on this environment. In the case where conductances are bounded above, an example environment is presented which exhibits heat kernel decay that is asymptotically slower than in the well studied time homogeneous case-being close to O n −1 as opposed to O n −2. The example environment given is a modification of an environment introduced in [4]


Introduction and results
Random walk amongst random conductances has been well studied over recent years.Formally, take a weighted graph on the square lattice G (ω) = Z d , E d , (ω e ) e∈E d with symmetry condition ω xy = ω yx and edge weights independent and identically distributed.The random walk on G (ω) is the Markov process (X n ) n≥0 with transition probabilities where we write x ∼ z if and only if x and z are neighbours in Z d .The associated heat kernel is then
The same paper and [5] prove that these are the best general upper bounds available by presenting examples whose heat kernel displays corresponding lower bounds.
We will consider the dynamic random conductance model.That is, take a time inhomogeneous weighted graph (G t ) t∈K = Z d , E d , ω = (ω e (t)) e∈E d t∈K for K either R or N.For any edge e ∈ E d , the edge weights (ω e (t)) t∈K are taken to be iid Markov processes with unique invariant probability distribution µ.Further, take (ω e (0)) e∈E d to be equal in distribution to µ E d .As in the time homogeneous case, the symmetry condition ω xy (t) = ω yx (t) is assumed for all edges and times.
The natural extension to (1.1) would be the random walk (X n ) n≥0 with transition probabilities This walk presents several technical challenges: if one looks for an invariant measure for the walk considered on the space-time graph then the measure does not generally have a simple form and the walk is not generally reversible.Further, this invariant space-time measure is not temporally consistent -for a given environment, the projection of the invariant measure at time t onto Z d will in general be different from the projection at time s = t.This restricts the tools available with which to analyze the walk.
With this in mind we introduce the variable speed random walk in continuous time.This is the Markov process (X t ) t≥0 with generator at time t The flat measure π (x) = 1 for all x ∈ Z d , is the invariant measure for the variable speed walk with for all t ∈ R.Although this does not imply P (X t = y |X s = x ) = P (X t = x |X s = y ) the fact that L t is self-adjoint can be important.
One would perhaps expect that the environment evolving over time would reduce the time that the walk spends in locally anomalous regions due to the time dynamic removing these regions before the random walk can spend a large quantity of time in them.This heuristic would suggest that the heat kernel behaviour should be no worse than in the time homogeneous case.Intriguingly it is this dynamic -the disappearance of anomalous regions -that leads to a change in the heat kernel.Anomalous heat kernel decay in the static case is due to the walk becoming "trapped" close to the origin so Dynamic random conductance model that when the walk escapes the trap it is much closer to the origin than would normally be expected.However, the random walk must pay a price to enter and exit the trap -a price of O n −1 for both entrance and exit leading to the O n −2 return probabilities stated above.The key idea that we present is that it is possible to choose a dynamic environment where the traps persist for long enough to trap the walk for a good length of time -so that the walk is much closer to the origin than would be expected -but then the trap disappears leaving the walk unimpeded to return to the origin.As the walk only has to pay to enter the trap and not to exit we show that lower heat kernel bounds close to O n −1 can be achieved.
The main result of the paper is the following, to be proved in Section 2.
Theorem 1.2.Take κ > for all i, where X n is the variable speed continuous time random walk on ω.
The statement of the theorem is very similar to Theorem 2.2 of [4].This is deliberate.The environment that we present is a modification of an environment presented in that paper.
Note that Theorem 1.2 implies that the dynamic heat kernel has at least three extrema: when the environment is strongly mixing the walk resembles the walk on the annealed graph and hence has heat kernel bounds of order n −d/2 ; when the environment is highly persistent the walk is close to the walk on the static graph and hence the heat kernel is bounded above by order n −2 ; in between these two cases sit the environments we have outlined with heat kernel lower bounds of order n −1 .
No corresponding upper bound is presented.It is natural to ask whether the walk could also enter a trap for free by being at the correct site when the trap forms.It will become clear that there is a trade-off between persistence of traps and their frequency of occurrence.As we wish the traps to be persistent so that the walk remains trapped for long time periods the traps cannot occur frequently and thus the walk is highly unlikely to be at the trap site when the trap forms.We comment further on this open problem at the end of the paper.Dynamic random environment is not a new topic.There are several papers that prove both annealed and quenched central limit theorems under various conditions, frequently without the symmetry condition ω xy = ω yx (for example [6], [2] and [16]).
These papers generally show that if the walk is uniformly elliptic and the environment is well mixing in time then the rescaled process converges to non-degenerate Brownian motion.In terms of heat kernel estimates, there is not a huge amount in the literature.The Appendix of [12] proves full off-diagonal results in the case where weights are bounded away from zero and infinity.The paper [10] proves that if is independent of t and the environment satisfies both a uniform ellipticity condition and uniform Sobolev inequality then standard off-diagonal upper bounds and on diagonal lower bounds hold.One can in fact extend this result in the spirit of [14] to prove standard on-diagonal upper bounds under assumptions of asymptotic isoperimetric dimension and ergodicity of the spatial environment over time [9], although this is not proved here.There are also recent results in the case where space is taken to be finite, these can be found in [17] and [18].

Heat kernel lower bounds
We take a space-time environment that only switches at discrete time points.This choice of discrete environment does lead to a somewhat peculiar hybrid pair as the walk is in continuous time.The choice does, however, make the combinatorics easier and removes issues relating to exceptional times.Take weights ω e to be supported on {2 −n : n ≥ 0} and let the transition probabilities for the Markov chain (ω e (n)) n∈Z be: where p n ∈ (0, 1) and n≥0 s n = 1.Then ω e (n) has an invariant distribution if and only if the Markov chain is positive recurrent.Hence, the invariant distribution, µ, exists if and only if Assume this to be the case, then for n ≥ 1 the invariant distribution satisfies We define our space-time environment to be ω = (ω e (n)) with ω being iid in space with P ω e (0) = 2 −k = µ 2 −k .Assume for the moment that s n and p n are chosen such that µ (1) > p c (d) and hence for every t ∈ R, the bonds of unit conductance percolate in G t .
The traps that we consider are of the form shown in Figure 1.They consist of the following.At time zero there is a strong spatial path (made of bonds of unit conductance) connecting the origin to a vertex x. x is connected to y by a weak bond of strength 2 −n and all other bonds are of lesser conductance.The trap, without necessarily the path to the origin, remains in place until some time T n (to be chosen later) at which point ω xy switches to unit conductance.At time T n there again exists a strong spatial path to the origin.
If such a trap exists, we obtain a lower bound on P ω (0,0) (X Tn+1 = 0) by conditioning on the walk moving directly to x within one unit of time, jumping from x to y in one unit of time, not jumping from y until time T n and then proceeding directly back to the origin within one unit of time: where l 1 n and l 2 n are the graph distances between the origin and x at times 0 and T n respectively.In the final line the 2 −n is the cost to the walk of crossing the bond ω xy (1), the exponential terms come from the lower bounds on the probability of the walk moving a large distance in a short time proven in [15] (and given in more detail in the proof of Theorem 1.2 below), and we have assumed that T n ≤ 1 2d 2 n and hence the walk does not jump from y between time 2 and T n with probability bounded independently of n.If we can take l i n = O (log n) and T n = O (2 n ) then we would see for some function f that decays slower than 2 −n .
In order to prove the theorem we require some combinatorial estimates on the strong paths that will connect the traps to the origin.For t ∈ R write x ↔ y at time t if there exists a nearest neighbour path from x to y consisting purely of bonds of unit conductance in the graph G t .As µ (1) > p c (d) there exists a unique infinite cluster at each t ∈ R.Call this C ∞ (t).The following proposition encapsulates the results required.
d and define the events There exist constants c 1 , M 1 > 0 such that for m ≥ M 1  : ω e (0) ≤ 2 −M , the set of all light edges -we will throw these edges away as we cannot control their mixing properties.
Take M 1 sufficiently large so that for all m ≥ M 1 P ω e (m) = 1 ω e (0) = 2 −n > p c (d) + 2ε for all n < M .M 1 exists since the Markov chain on edge weights is irreducible, aperiodic, positive recurrent, n is bounded and p c (d) + 2ε < µ (1) through the choice of ε.
For i ∈ {1, 2}, take ωi e e∈Z d to be independent bond percolation realizations with Define ω1 e (m) e∈Z d , ω2 e (m) e∈Z d to be independent environments with law ωi e (m) = The laws of ω (0) and ω (m) conditioned on E M , are stochastically dominated by the conditioned laws of ω1 (m) and ω2 (m) respectively.Hence, on a suitably extended probability space, the event C0 (k, x) where the final line follows from the definition of E M and standard percolation estimates.
The second claim is straight forward to verify via standard percolation arguments.
Consider first the event D i (n) for i ∈ {0, n}.In either case this corresponds to static percolation and hence Theorem 8.65 of [13] gives upper bounds on P (D i (n) c ) that are independent of i and summable over n.Now, and hence is also summable over n.The Borel-Cantelli Lemma ensures that the event D 0 (n) ∩ D n (n) happens only finitely often with probability one.
For a fixed point x ∈ Z d , let y = x + e 1 and A n (x) be the event that: • In the spatial environment at time zero, G 0 , x is connected to the boundary of the spatial box of side c 2 (log l n ) 2 centred at x by a path of unit conductors.
• ω • x is connected to the boundary of the spatial box of side c 2 (log l n ) 2 centred at x by a path of unit conductors in the spatial environment G Tn .
Proposition 2.1 holds with the events C i modified to ensure that the paths connecting x to ∂B x [k] avoid using the edges emanating from y.Call these modified events Ci . Then by the choice of p n .Taking G n to be a grid of sites in The events {A n (x) : x ∈ G n } are independent, so using 1 − x ≤ e −x for x ∈ [0, 1], , hence by Borel-Cantelli, the intersection occurs for only finitely many n.By Proposition 2.1, every connected component of diameter at least (log and all large enough n.Now, the origin at time zero will not necessarily be connected to this largest component.Take z to be the closest point to the origin that lies on the infinite component at time zero.On the event A n (x) for n large, take l 1 n to be the shortest path from the origin to x in G 0 that goes from 0 to z and then follows a strong path to x.
Take n i to be a subsequence such that there is a strong path from x to 0 at time T ni of length l 2 n with l 2 n bounded by c 9 l n .Such a subsequence (and constant c 9 ) exist by [1].The results of [1] also imply that l 1 n ≤ c 9 l n for all large enough n.We take a subsequence as we then avoid the complication of the origin being surrounded by many weak bonds as such a situation would make it difficult for the walk to return to the origin.It is shown in [15] that for a one dimensional walk the following lower bound holds: there exist constants c i such that for any x, y ∈ Z d and d (x, y) ≥ t ≥ 1 We wish to bound the probability that the walk travels fully along a strong path of length l i n in a unit of time.As we can bound the probability that the walk deviates from this one dimensional path from below by e −c8l 1 n , we can condition so that the walk only sees the one dimensional path and hence Similarly for the initial strong path from z to x, with a constant dependent on the local environment around the origin replacing c 6 .
Page 7/11 taking ε small concludes the proof.
With this example of ω supported on {2 −n : n ∈ N} in mind we can demonstrate three distinct behaviours for the heat kernel for the dynamic random walk.Let G t (ω) be the environment at time t for ω ∈ Ω.For m > 0 define the dynamic graph G m t := G tm , that is the graph speeded up so that edges flip m times per unit of time.Let X m t be the space-time random walk on G m t started at (0, 0).Theorem 1.2 proves that if m = 1 we have a lower bound close to O t −1 .As we let m tend to zero and infinity then we have two further distinct behaviours.Proposition 2.2.For almost all ω ∈ Ω there exist a constant C (ω) such that for all t > 0 we have the limits: e −(log t) κ t 2 for d ≥ 5 Proof.The first line is due to [4], as m → 0 corresponds to the static case.As m gets large the probability that the walk crosses an edge in time t tends towards the probability that the random walk crosses an edge of conductance E (ω e ) in time t.Hence the limit as m → ∞ corresponds to the annealed random walk.This is the simple random walk on Z d with speed 2dE (ω e ) and hence exhibits standard on diagonal heat kernel behaviour.
We have been slightly disingenuous when suggesting that Theorem 1.2 shows behaviour more anomalous than presented in [4] as we are considering a different model -the variable speed walk as opposed to the constant speed walk investigated in [4].In the constant speed case the above trap is not a trap at all as the transition rates are normalized by y∼x ω xy so that the walk always moves at unit speed.There are thus two obvious questions: what is the behaviour of the variable speed walk in the static case and what is the behaviour of the constant speed walk in the dynamic case?
We begin with the first question.The trapping demonstrated above will lead to lower bounds close to O n −2 in the static case (the n −2 is due to the walk now having to pay a price of O n −1 to exit the trap as well as to enter).The upper bounds follow from slight modifications to the arguments of [4], with summations replaced by integrals.
The answer to the second question is presented in Proposition 2.3 below where lower bounds close to O n −1 are again proven.The example that displays these lower bounds is a very similar example of space-time trapping in the constant speed caseagain the environment changes at discrete time points with the walk being a continuous time Markov process.
Proposition 2.3.Take d ≥ 3.For any α > 0 and κ > 1 d , there exist non-static random space-time environments of the above form such that for almost every ω ∈ Ω there exists C (ω) > 0 and an increasing sequence (n i ) i≥0 such that lim i→∞ n i = ∞ and for all i Proof.The proof is very similar to above.We will simply outline the type of traps that lead to this behaviour.
Figure 2 demonstrates the types of trap we consider around a point x.Take y = x+e 1 and z = y + 2e 1 .We initially take the bond between y and z to be of weight 1, with the bond between x and y being of weight 2 −n and all other bonds emanating from y and z being of weight ω e ≤ 2 −n .As time evolves all the weak bonds remain at their initial value.The strong bond will weaken but will never be weaker than 2 −cn for some constant c that we can take to be as small as we like.At time T n the bond between x and y switches to unit weight.We condition on there existing strong paths between 0 and x in the spatial environments G 0 and G Tn .
Take r n to be the length of the space-time path from 0 to x at time zero and r n to be the length from x to 0 time T n .As in equation (2.2) above, if T n = O (2 cn ) we have The details are similar to the proof of Theorem 1.2.
Note that the traps introduced in the proofs of Theorem 1.2 and Proposition 2.3 would also be traps if one considered the discrete time random walks on these environments.However, as lower bounds of the form (2.3) can never hold in discrete time, one would have to prove an exponential lower bound on the probability of moving straight from the origin to the trap and then back from the trap to the origin.One method for doing this would be to prove the existence of strong space-time paths from the origin to the trap and back again -that is, paths in space-time consisting purely of edges of unit weight that if followed would take the walk directly from the origin to the trap and vice-versa.The requisite combinatorics are beyond this paper.
We conclude by remarking on whether or not this lower bound is sharp.If the trapping strategy detailed above is the dominant strategy, then an upper bound of the same order will hold.The strategy employed above is similar to the strategy that maximizes sup y P t (0, y) in the time homogeneous case -as the dominating strategy only pays to enter a trap and not to leave.Such a link between the time homogenous and time inhomogeneous case is appealing.However, one must also consider the possibility that this is not the dominating strategy in the time inhomogeneous case.
It is possible for the walk to escape from a trap for free, is it also possible for the walk to be at a trap site when the trap forms and hence enter the trap for free too?As we assume the existence of an invariant distribution for the Markov chain on edge weights, there is a trade-off between persistence of a trap and how often a trap can form.In the particular example discussed above, this is crystallized in equation (2.1).
In particular this ensures that if we wish a trap to persist for time O (n) then the trap is unlikely to form logarithmically close to the origin with respect to the space-time distance.This somewhat compromises the lower bound calculations of equation (2.2), as crudely bounding the probability that the walk moves from the origin to the space-time point where the trap forms by the negative exponential of the corresponding distance will lead to lower bounds that are smaller than those presented in Theorem 1.2 due to the distance being large.
Although we have found no strategy to obtaining a more anomalous lower bound than presented, we have no proof that such a bound fails to exist.

Proof.
For the first claim we use the mixing properties of the Markov chain on edge weights.We have to be careful as if B x [k] contains edges with very light conductance then these edges can take a long time to mix.To avoid this problem we delete all light edges and percolate on what remains of the box.Let ε := µ (1) − p c (d) 4 and take M such that ∞ n=M µ 2 −n < ε.

Figure 1 :
Figure 1: A space time trap

Figure 2 :
Figure 2: A space time trap for the constant speed walk 1d and d ≥ 3.There exists a law on environments P with iid edge marginals that evolve in an ergodic Markov fashion with edge weights bounded by one such that for almost all ω ∈ Ω there exists a constant C (ω) > 0 and a sequence (n i (ω)) with lim i→∞ n i = ∞ such that