Abstract
We investigate harmonic functions and the convergence of the sequence of ratios \(({\mathbb {P}}_x(\tau _\vartheta> n)/{\mathbb {P}}_e(\tau _\vartheta > n))\) for a random walk on a countable group killed upon the time \(\tau _\vartheta \) of the first exit from some semigroup 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 multi-dimensional random walks (X(t)) killed upon the times of first exit from a convex cone. Our approach combines large deviation estimates and an extension of Choquet–Deny theory.
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Ignatiouk-Robert, I. Harmonic Functions of Random Walks in a Semigroup via Ladder Heights. J Theor Probab 34, 34–80 (2021). https://doi.org/10.1007/s10959-019-00974-1
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DOI: https://doi.org/10.1007/s10959-019-00974-1