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

Abstract

We study variable-speed random walks on Z driven by a family of nearestneighbor time-dependent random conductances {a t (x, x + 1) : x ∈ Z, t ≥ 0} whose law is assumed invariant and ergodic under space-time shifts. We prove a quenched invariance principle for the random walk under the minimal moment conditions on the environment; namely, assuming only that the conductances possess the first positive and negative moments. A novel ingredient is the representation of the parabolic coordinates and the corrector via a dual random walk which is considerably easier to analyze. Dedicated to Jean-Dominique Deuschel1 arXiv:1809.05401v2 [math.PR] 27 Sep 2018 1D DYNAMICAL CONDUCTANCE MODEL 3Similarly, we will call (x, t) K, -good in environment a if (0, 0) is K, -good in the environment τ t,x (a). In light of (7.1) and (7.8),

Symmetric simple exclusion process in dynamic environment: hydrodynamics

“…In particular, we establish this connection in the case of uniformly elliptic dynamic conductances providing, in combination with the results in e.g. [1], [3], [6], plenty of non-trivial examples of dynamic environments in which the assumption of arbitrary starting point invariance principle holds.…”

“…It is worth noticing that in the last ten years there have been several results in this direction, see e.g. [1], [3], [6], [7], [13], though all of them prove a quenched invariance principle for the "initially-anchored-at-the-origin" random walk only, i.e., for a given environment, the diffusively rescaled random walk that starts at time zero in the origin converges in law to a non-degenerate Brownian motion also starting in the origin. Unlike in the case of spatially homogeneous conductances, in our case the laws of the random walks are not translation invariant, therefore the derivation of an invariance principle for random walks centered around arbitrary macroscopic points does not follow at once from the invariance principle for the random walk initialized in the origin.…”

“…However, random walks in dynamic random environment (RWDRE) have been extensively studied in recent years (see e.g. [1], [3], [4], [6], [7], [13], [45] and references therein) 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.…”

The Effect of Obstacle Design Architectures on Randomly Ranging AGVs in a Shared Workspace

“…Conditional probabilities are used in formulating the probability expression and commonly used in complex situations that make the use of a single simple probability calculation inaccurate. SRW is a stochastic process which involves the collection of identical independent random variables where each of those random variables represents the next move [47]. In a rectangular gridded workspace, the AGV has a 25% chance of moving in any of the four possible directions, at any given grid point position other than the boundaries.…”

Quenched invariance principle for random walks on dynamically averaging random conductances

“…Here, the walker moves according to jump rates on the edges of a given base graph which themselves are evolving according to some stochastic process simultaneously with the movement of the walker. We refer the reader to [9] for an excellent overview of the plethora of models studied in the literature and to [1,8] on the topic of invariance principles.…”