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

摘要

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 vertical bar x - y vertical bar(-(d +) (alpha)) with alpha is an element of (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 is an element of (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.