## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1010-1032.

### Exponential mixing properties for time inhomogeneous diffusion processes with killing

Pierre Del Moral and Denis Villemonais

#### 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 evolution operator; 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1010-1032.

**Dates**

Received: June 2015

Revised: March 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980887

**Digital Object Identifier**

doi:10.3150/16-BEJ845

**Mathematical Reviews number (MathSciNet)**

MR3706785

**Zentralblatt MATH identifier**

06778356

**Keywords**

process with absorption time-inhomogeneous diffusion process uniform mixing property

#### Citation

Del Moral, Pierre; Villemonais, Denis. Exponential mixing properties for time inhomogeneous diffusion processes with killing. Bernoulli 24 (2018), no. 2, 1010--1032. doi:10.3150/16-BEJ845. https://projecteuclid.org/euclid.bj/1505980887