Quenched invariance principle for long range random walks in balanced random environments

TL;DR

This paper proves a quenched invariance principle for long-range random walks in balanced environments, showing convergence to stable processes or Brownian motion depending on the parameter, using martingale techniques.

Contribution

It introduces a probabilistic method to establish the invariance principle for long-range walks in non-i.i.d. environments, extending results to non-balanced cases for certain parameters.

Findings

Proves invariance principle for $eta$-stable limits with $eta eq 2$.

Establishes convergence to Brownian motion when $eta=2$.

Works for non-balanced environments when $eta<1$.

Abstract

We establish via a probabilistic approach the quenched invariance principle for a class of long range random walks in independent (but not necessarily identically distributed) balanced random environments, with the transition probability from $x$ to $y$ on average being comparable to $|x-y|^{-(d+\alpha)}$ with $\alpha\in (0,2]$. We use the martingale property to estimate exit time from balls and establish tightness of the scaled processes, and apply the uniqueness of the martingale problem to identify the limiting process. When $\alpha\in (0,1)$, our approach works even for non-balanced cases. When $\alpha=2$, under a diffusive with the logarithmic perturbation scaling, we show that the limit of scaled processes is a Brownian motion.