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Quantitative Estimates for the Long-Time Behavior of an Ergodic Variant of the Telegraph Process

Part of: Limit theorems Stochastic systems and control Markov processes

Published online by Cambridge University Press:  04 January 2016

Joaquin Fontbona ,
Hélène Guérin  and
Florent Malrieu
Joaquin Fontbona*
Affiliation:
Universidad de Chile
Hélène Guérin*
Affiliation:
Universitè de Rennes 1
Florent Malrieu*
Affiliation:
Universitè de Rennes 1
*
∗ Postal address: CMM-DIM, UMI 2807, UChile-CNRS, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile. Email address: [email protected]
∗∗ Postal address: UMR 6625, CNRS, Institut de Recherche Mathèmatique de Rennes (IRMAR), Universitè de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France.
∗∗ Postal address: UMR 6625, CNRS, Institut de Recherche Mathèmatique de Rennes (IRMAR), Universitè de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France.
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Abstract

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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 nonconstant 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 the velocity and position. Sharpness of the obtained convergence rate is discussed.

Keywords

Piecewise-deterministic Markov processcouplinglong-time behaviortelegraph processchemotaxis model

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 93E15: Stochastic stability 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 977 - 994
Copyright
© Applied Probability Trust 

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