# Gambler's ruin estimates on finite inner uniform domains

## TL;DR

This paper links gambler's ruin estimates to harmonic measures in finite Markov chains, using Perron-Frobenius eigenfunctions, especially in inner-uniform domains, to derive detailed probabilistic estimates.

## Contribution

It establishes a connection between gambler's ruin estimates and Perron-Frobenius eigenfunctions for Harnack Markov chains in finite inner-uniform domains, providing new analytical tools.

## Key findings

- Gambler's ruin estimates relate to harmonic measures in finite Markov chains.
- In inner-uniform domains, estimates can be explicitly derived using eigenfunctions.
- Understanding the Perron-Frobenius eigenfunction remains a key challenge.

## Abstract

Gambler's ruin estimates can be viewed as harmonic measure estimates for finite Markov chains which are absorbed (or killed) at boundary points. We relate such estimates to properties of the underlying chain and its Doob transform. Precisely, we show that gambler's ruin estimates reduce to a good understanding of the Perron-Frobenius eigenfunction and eigenvalue whenever the underlying chain and its Doob transform are Harnack Markov chains. Finite inner-uniform domains (say, in the square grid $\mathbb Z^n$) provide a large class of examples where these ideas apply and lead to detailed estimates. In general, understanding the behavior of the Perron-Frobenius eigenfunction remains a challenge.