Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process

TL;DR

This paper analyzes a modified telegraph process with a drift towards the origin, providing explicit invariant measures, proving exponential ergodicity, and quantifying convergence rates using novel coupling techniques.

Contribution

It introduces a new variant of the telegraph process with a non-constant jump rate, and develops explicit methods to analyze its long-term behavior and convergence to equilibrium.

Findings

Explicit invariant law derived

Proven exponential ergodicity with quantitative bounds

Constructed a novel coalescent coupling for analysis

Abstract

Motivated by stability questions on piecewise deterministic Markov models of bacterial chemotaxis, we study the long time behavior of a variant of the classic telegraph process having a non-constant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both velocity and position. Sharpness of the obtained convergence rate is discussed.