# Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

## TL;DR

This paper develops analytic-geometric techniques to provide detailed quantitative estimates on the behavior of finite Markov chains before absorption, focusing on chains like random walks on lattice subsets with boundary absorption.

## Contribution

It introduces a novel analytic-geometric framework using inequalities and domain concepts to analyze quasi-stationary behavior of finite Markov chains.

## Key findings

- Quantitative estimates for chain behavior before absorption
- Application to random walks on lattice subsets
- Use of Poincaré, Nash, and Harnack inequalities

## Abstract

For a relatively large class of well-behaved absorbing (or killed) finite Markov chains, we give detailed quantitative estimates regarding the behavior of the chain before it is absorbed (or killed). Typical examples are random walks on box-like finite subsets of the square lattice $\mathbb Z^d$ absorbed (or killed) at the boundary. The analysis is based on Poincar\'e, Nash, and Harnack inequalities, moderate growth, and on the notions of John and inner-uniform domains.