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 [Article] [arXiv] [MR]



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=axβ1, with x10. 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 0β+,β<1, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.