Reflecting random walks in curvilinear wedges

We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form $x_2=a^{+}{x}_1^{{\beta }^{+}}$ and $x_2=-a^{-}{x}_1^{{\beta }^{-}}$, with $x_1\ge 0$. 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 ${\alpha }^{+}$ or ${\alpha }^{-}$ to the relevant inwards-pointing normal vector. Here we focus on the case where ${\alpha }^{+}$ and ${\alpha }^{-}$ are equal but opposite, which includes the case of normal reflection. For $0\le {\beta }^{+},{\beta }^{-}<1$, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.