Reflecting random walks in curvilinear wedges
Mikhail V. Menshikov, Aleksandar Mijatović, and Andrew R. Wade
In: Vares M.E., Fernández R., Fontes L.R., Newman C.M. (eds) In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius.
Progress in Probability, vol 77. Birkhäuser.
DOI: 10.1007/978-3-030-60754-8_26
Abstract
We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form x2=a+xβ+1 and x2=−a−xβ−1, with x1≥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.