On quantitative convergence to quasi-stationarity

TL;DR

This paper investigates the convergence behavior of finite Markov processes conditioned on non-absorption, providing explicit estimates and linking to classical spectral gap results, especially in reversible cases.

Contribution

It introduces a method using Doob transforms and eigenvector ratios to analyze convergence to quasi-stationarity, extending known results for ergodic processes.

Findings

Explicit bounds on convergence to quasi-stationarity.

Recovery of the spectral gap as the optimal exponential rate in reversible cases.

Illustrative examples demonstrating the bounds.

Abstract

The quantitative long time behavior of absorbing, finite, irreducible Markov processes is considered. Via Doob transforms, it is shown that only the knowledge of the ratio of the values of the underlying first Dirichlet eigenvector is necessary to come back to the well-investigated situation of the convergence to equilibrium of ergodic finite Markov processes. This leads to explicit estimates on the convergence to quasi-stationarity, in particular via functional inequalities. When the process is reversible, the optimal exponential rate consisting of the spectral gap between the two first Dirichlet eigenvalues is recovered. Several simple examples are provided to illustrate the bounds obtained.