Martin boundary of a killed non-centered random walk in a general cone

TL;DR

This paper studies the Martin boundary of a non-centered random walk in a convex cone, using large deviation estimates and ladder processes to identify the boundary structure.

Contribution

It introduces a new approach combining large deviations and ladder processes to analyze the Martin boundary for non-centered random walks in convex cones.

Findings

Identified the Martin boundary for random walks in convex cones with smooth boundaries.

Extended classical results to non-centered walks with general cone geometries.

Provided a framework for analyzing boundary behavior using large deviation techniques.

Abstract

We investigate Martin boundary for a non-centered random walk on ${\mathbb Z}^d$ killed up on the time $\tau_\vartheta$ of the first exit from a convex cone with a vertex at $0$. The approach combines large deviation estimates, the ratio limite theorem and the ladder height process. The results are applied to identify the Martin boundary for a random walk killed upon the first exit from a convex cone having $C^1$ boundary.