Reflecting random walks in curvilinear wedges

Abstract

We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form x 2 = a + x β + 1 and x 2 = −a − x β − 1 , with x 1 ≥ 0. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle α + or α − to the relevant inwards-pointing normal vector. Here we focus on the case where α + and α − are equal but opposite, which includes the case of normal reflection. For 0 ≤ β + , β − < 1, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.