Exponential mixing properties for time inhomogeneous diffusion processes with killing

TL;DR

This paper proves exponential mixing properties for time-inhomogeneous diffusion processes with killing in bounded domains, providing uniform estimates and a novel analysis of their particle interpretations in both soft and hard obstacle settings.

Contribution

It establishes the first uniform exponential mixing results for non-absorbed diffusions with killing in time-inhomogeneous settings, including particle system estimates.

Findings

Proved exponential strong mixing for conditioned diffusion processes.

Provided uniform estimates for particle system behavior.

Extended results to both soft and hard obstacle scenarios.

Abstract

We consider an elliptic and time-inhomogeneous diffusion process with time-periodic coefficients evolving in a bounded domain of $\mathbb{R}^d$ with a smooth boundary. The process is killed when it hits the boundary of the domain (hard killing) or after an exponential time (soft killing) associated with some bounded rate function. The branching particle interpretation of the non absorbed diffusion again behaves as a set of interacting particles evolving in an absorbing medium. Between absorption times, the particles evolve independently one from each other according to the diffusion semigroup; when a particle is absorbed, another selected particle splits into two offsprings. This article is concerned with the stability properties of these non absorbed processes. Under some classical ellipticity properties on the diffusion process and some mild regularity properties of the hard obstacle boundaries, we prove an uniform exponential strong mixing property of the process conditioned to not be killed. We also provide uniform estimates w.r.t. the time horizon for the interacting particle interpretation of these non-absorbed processes, yielding what seems to be the first result of this type for this class of diffusion processes evolving in soft and hard obstacles, both in homogeneous and non-homogeneous time settings.