Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks

Abstract

In this paper we study Markov chains with the state space given by the coordinate axes of $\mathbb R^m$, $m \geq 2$, whose step sizes on each positive half-axis are distributed according to a centered probability distribution with variance $v_i^2 \in (0, \infty)$, $i = 1,\ldots, m$. Under very mild assumptions on the jumps sizes on the negative half-axes, we show that the Donsker scaling limit of such Markov chains is a Walsh Brownian motion whose weights are determined explicitly in terms of stationary distributions of certain embedded Markov chains. This convergence result is applied to integer-valued random walks perturbed on a finite subset of $\mathbb Z$ called a membrane. We show that their Donsker scaling limit is an oscillating skew Brownian motion.