Harmonic functions of random walks in a semigroup via ladder heights

TL;DR

This paper extends classical renewal theory to multidimensional random walks in semigroups, introducing ladder height processes and renewal functions to analyze harmonic functions and convergence of ratio sequences.

Contribution

It introduces a multidimensional ladder height process and renewal function, extending renewal theory and Choquet-Deny theory to semigroup-based random walks.

Findings

Extended renewal theory to multidimensional settings.

Analyzed harmonic functions for random walks in semigroups.

Provided convergence results for ratio sequences of exit probabilities.

Abstract

We investigate harmonic functions and the convergence of the sequence of ratios $(P_x(\tau_\vartheta {>} n)/P_e(\tau_\vartheta {>} n))$ for a random walk on a countable group killed up on the time $\tau_\vartheta$ of the first exit from some semi-group with an identity element $e$. Several results of classical renewal theory for one dimensional random walk killed at the first exit from the positive half-line are extended to a multi-dimensional setting. For this purpose, an analogue of the ladder height process and the corresponding renewal function $V$ are introduced. The results are applied to multidimensional random walks killed upon the times of first exit from a convex cone. Our approach combines large deviation estimates and an extension of Choquet-Deny theory.