Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process
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.
Keywords
Cite
@article{arxiv.1006.0982,
title = {Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process},
author = {Joaquin Fontbona and Hélène Guérin and Florent Malrieu},
journal= {arXiv preprint arXiv:1006.0982},
year = {2012}
}
Comments
Definitive version of former paper "Quantitative estimates for the long time behavior of a PDMP describing the movement of bacteria", now accepted in Advances in Applied Probability. Presentation changed. A diffusive scaling limit result is added. Sharpness of the long-time convergence rate is discussed. 20 pages, 3 figures