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

By Joaquín Fontbona, H'elene Gu'erin, Florent Malrieu. Advances in Applied Probability, 2010

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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.

Subfields: Biomedical Engineering, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/197bc65369fb9d641664f35581060c9fe5253343