Long time behavior of telegraph processes under convex potentials

TL;DR

This paper investigates the long-term dynamics of position-dependent telegraph processes with gradient-like drifts, deriving their invariant measures and providing quantitative convergence estimates using probabilistic coupling methods.

Contribution

It extends existing methods to analyze complex telegraph processes with variable jump-rates and gradient drifts, offering new insights into their invariant laws and convergence behavior.

Findings

Derived explicit invariant laws for the processes.

Provided quantitative estimates of convergence to equilibrium.

Extended coupling techniques to more complex models.

Abstract

We study the long-time behavior of variants of the telegraph process with position-dependent jump-rates, which result in a monotone gradient-like drift toward the origin. We compute their invariant laws and obtain, via probabilistic couplings arguments, some quantitative estimates of the total variation distance to equilibrium. Our techniques extend ideas previously developed for a simplified piecewise deterministic Markov model of bacterial chemotaxis.