Quantitative ergodicity for some switched dynamical systems

TL;DR

This paper establishes explicit exponential bounds on the convergence to equilibrium for a class of switched Markov processes, with applications to neurobiological models, advancing understanding of their long-term behavior.

Contribution

It provides the first quantitative bounds on ergodicity for certain switched dynamical systems with state-dependent jump rates.

Findings

Explicit exponential convergence bounds in Wasserstein distance.

Application to a stochastic neurobiological model.

Conditions ensuring stability and ergodicity.

Abstract

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd \times E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology.