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

## TL;DR

This paper establishes conditions under which variable speed random walks in time-dependent, potentially degenerate random environments on integer lattices converge to Brownian motion, using advanced PDE techniques.

## Contribution

It introduces new conditions for diffusive limits of random walks in degenerate, time-dependent environments and employs Moser iteration for the analysis.

## Key findings

- Proves invariance principle for degenerate environments
- Uses Moser iteration to establish sublinearity of the corrector
- Applicable to dynamical percolation and particle systems

## Abstract

We study continuous-time (variable speed) random walks in random environments on $\mathbb{Z}^d$, $d\ge2$, 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 space-time 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-Sj\"ostrand representation of gradient models with certain non-strictly convex potentials.