# Weak convergence of a random walk in a random environment

Research paper by Gregory F. Lawler

Indexed on: 01 Mar '82Published on: 01 Mar '82Published in: Communications in Mathematical Physics

#### Abstract

Let πi(x),i=1,...,d,x∈Zd, satisfy πi(x)≧α>0, and π1(x)+...+πd(x)=1. Define a Markov chain onZd by specifying that a particle atx takes a jump of +1 in theith direction with probability 1/2πi(x) and a jump of −1 in theith direction with probability 1/2πi(x). If the πi(x) are chosen from a stationary, ergodic distribution, then for almost all π the corresponding chain converges weakly to a Brownian motion.