Advances in Applied Probability

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

Joaquin Fontbona, Hélène Guérin, and Florent Malrieu

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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.

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Adv. in Appl. Probab., Volume 44, Number 4 (2012), 977-994.

First available in Project Euclid: 5 December 2012

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Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 93E15: Stochastic stability 60F17: Functional limit theorems; invariance principles

Piecewise-deterministic Markov process coupling long-time behavior telegraph process chemotaxis model


Fontbona, Joaquin; Guérin, Hélène; Malrieu, Florent. Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process. Adv. in Appl. Probab. 44 (2012), no. 4, 977--994. doi:10.1239/aap/1354716586.

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