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May 2018 Exponential mixing properties for time inhomogeneous diffusion processes with killing
Pierre Del Moral, Denis Villemonais
Bernoulli 24(2): 1010-1032 (May 2018). DOI: 10.3150/16-BEJ845


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.


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Pierre Del Moral. Denis Villemonais. "Exponential mixing properties for time inhomogeneous diffusion processes with killing." Bernoulli 24 (2) 1010 - 1032, May 2018.


Received: 1 June 2015; Revised: 1 March 2016; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778356
MathSciNet: MR3706785
Digital Object Identifier: 10.3150/16-BEJ845

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability


Vol.24 • No. 2 • May 2018
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