Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions

TL;DR

This paper establishes criteria for exponential convergence to quasi-stationary distributions in Markov processes with absorption, with applications to multi-dimensional diffusions in bounded domains or manifolds, enhancing understanding of their long-term behavior.

Contribution

It introduces new criteria based on transition kernel estimates and gradient bounds for ensuring exponential convergence to quasi-stationary distributions.

Findings

Criteria based on transition kernel estimates for convergence

Gradient estimates on semigroup ensure convergence

Applications to multi-dimensional diffusions in bounded domains

Abstract

We consider general Markov processes with absorption and provide criteria ensuring the exponential convergence in total variation of the distribution of the process conditioned not to be absorbed. The first one is based on two-sided estimates on the transition kernel of the process and the second one on gradient estimates on its semigroup. We apply these criteria to multi-dimensional diffusion processes in bounded domains of $R^d$ or in compact Riemannian manifolds with boundary, with absorption at the boundary.