Reflected random walks and unstable Martin boundary

TL;DR

This paper studies a family of two-dimensional reflected random walks in the positive quadrant, revealing an unusual instability in their Martin boundary depending on whether a parameter is rational or not, with implications for potential theory.

Contribution

It introduces a new class of reflected random walks and demonstrates a rare instability phenomenon in their Martin boundary structure, combining probabilistic and analytic methods.

Findings

Martin boundary is countable for rational parameters

Martin boundary is uncountable for non-rational parameters

Provides precise Green function estimates for complex boundary behaviors

Abstract

We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp.\ non-rational), then the Martin boundary is countable, homeomorphic to $\mathbb Z\cup\{\pm\infty\}$ (resp.\ uncountable, homeomorphic to $\mathbb R\cup\{\pm\infty\}$). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.