Martin boundary of random walks in convex cones

Abstract

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

43 pages. This article is based on two papers which have already appeared on arXiv: arXiv:1803.09253v2 (by Kilian Raschel and Pierre Tarrago) and arXiv:1807.07360 (by Jetlir Duraj and Vitali Wachtel)