Exponential ergodicity of a degenerate age-size piecewise deterministic process

TL;DR

This paper investigates the long-term behavior of a degenerate, measure-valued stochastic process with age and size variables, establishing exponential ergodicity through a novel approach involving Doob h-transforms and Harris' Theorem.

Contribution

It introduces a new method to prove exponential ergodicity for a degenerate, non-conservative process with deterministic dependencies, addressing challenges of degeneracy and non-compactness.

Findings

Proved exponential ergodicity of the process.

Developed a method to construct explicit trajectories for state space exploration.

Applied the method to a bacterial growth fragmentation model.

Abstract

We study the long-time behaviour of the first-moment semigroup of a non conservative piecewise deterministic measure-valued stochastic process with support on R 2 + driven by a deterministic flow between random jump times, with a transition kernel which has a degenerate form. Using a Doob h-transform where the function h is taken as an eigenfunction of the associated generator, we can bring ourselves back to the study of a conservative process whose exponential ergodicity is proven via Harris' Theorem. Particular attention is given to the proof of Doeblin minoration condition. The main difficulty is the degeneracy of one of the two variables, and the deterministic dependency between the two variables, which make it no trivial to uniformly bound the expected value of the trajectories with respect to a non-degenerate measure in a two-dimensional space, which is particularly hard in a non-compact setting. Here, we propose a general method to construct explicit trajectories which explore the space state with positive probability and witch permit to prove a petite-set condition for the compact sets of the state space. An application to an age-structured growthfragmentation process modelling bacterial growth is also shown.