Limit theory for random walks in degenerate time-dependent random environments

Abstract

We study continuous-time (variable speed) random walks in random environments on Z d , d ≥ 2, where, at time t, the walk at x jumps across edge (x, y) at time-dependent rate a t (x, y). The rates, which we assume stationary and ergodic with respect to spacetime shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer-Sjöstrand representation of gradient models with certain non-strictly convex potentials.

Quenched invariance principle for random walks among random degenerate conductances

“…It is left to consider the case s = 1. In [15] it is proven that (24) and (25) by a simple extension argument. Indeed, functions defined on a box B(n) can easily extended by successive reflections (see e.g.…”

Quenched central limit theorem for random walks in doubly stochastic random environment

“…Throughout the paper conditions (1), (8) and (9) are assumed. (Recall that (7) is formally implied by (9), so we don't state it as a separate condition.) Propositions 2 and 3 are valid under these conditions.…”

“…Two of these, (KT35) and Proposition 4(ii) of [17] are of particular interest, since we shall use them. Note that the H −1 -condition (9) actually formally implies the no-drift condition (7). It is proved in Proposition 4 (ii) of [17] that the H −1 -condition (9) is equivalent to the existence of a stationary and square integrable stream-tensor-field whose curl (or, rotation) is exactly the source-less (divergence-free) flow v. More explicitly, there exist h k,l ∈ H , k, l ∈ E , with symmetries…”

An invariance principle for one-dimensional random walks among dynamical random conductances

“…Our second remark concerns the situation when we actually allow the conductances to vanish over sets of times of positive Lebesgue measure. This has been addressed by Biskup and Rodriguez [9], albeit only in d ≥ 2, by requiring sufficiently high (namely, 4d + ) moments of the quantity…”

“…Here we note that, unlike the case of static environments, in dynamical environments different ways to assign speed -i.e., normalize the generator -cannot be related by a time change of the underlying process. At this point, all the existing studies of invariance principles in these cases (namely, the aforementioned references [2,9]) are restricted to the variable speed case. It is thus of interest to see whether the present approach can be extended to include other versions, most notably discrete-time, as well.…”