Vladislav Vysotskiy: Stationary switching random walks

Abstract: A switching random walk, commonly known under the misnomer “oscillating random walk”, is a real-valued Markov chain whose increment distribution is determined by the sign of its current position. We explicitly identify an invariant measure of this chain and study its uniqueness, up to a constant factor, among locally finite invariant measures on a suitably chosen state space. We then provide sufficient conditions for the topological recurrence of the switching random walk, and prove its topological irreducibility. As a corollary, we recover the corresponding uniqueness, recurrence, and irreducibility results for reflected random walks.