Oscillating random walk with a moving boundary

Abstract

A discrete-time Markov chain is defined on the real line as follows: When it is to the left (respectively, right) of the “boundary”, the chain performs a random walk jump with distributionU (respectively,V). The “boundary” is a point moving at a constant speed γ. We examine certain long-term properties and their dependence on γ. For example, if bothU andV drift away from the boundary, then the chain will eventually spend all of its time on one side of the boundary; we show that in the integer-valued case, the probability of ending up on the left side, viewed as a function of γ, is typically discontinuous at every rational number in a certain interval and continuous everywhere else. Another result is that ifU andV are integer-valued and drift toward the boundary, then when viewed from the moving boundary, the chain has a unique invariant distribution, which is absolutely continuous whenever γ is irrational.