OSCILLATING RANDOM-WALK WITH A MOVING BOUNDARY

A discrete-time Markov chain is defined on the real line as follows: W hen it is to the left (respectively, right) of the ''boundary'', the c hain performs a random walk jump with distribution U (respectively, V) . The ''boundary'' is a point moving at a constant speed gamma. We exa mine certain long-term properties and their dependence on gamma. For e xample, if both U and V drift away from the boundary, then the chain w ill eventually spend all of its time on one side of the boundary; we s how that in the integer-valued case, the probability of ending up on t he left side, viewed as a function of gamma, is typically discontinuou s at every rational number in a certain interval and continuous everyw here else. Another result is that if U and V are integer-valued and dr ift toward the boundary, then when viewed from the moving boundary, th e chain has a unique invariant distribution, which is absolutely conti nuous whenever gamma is irrational.