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

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 e...

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Bibliographic Details
Published in:Advances in applied probability 2012-12, Vol.44 (4), p.977-994
Main Authors: Fontbona, Joaquin, Guérin, Hélène, Malrieu, Florent
Format: Article
Language:English
Subjects:
60F17
60J25
60J75
93E15
Brownian motion
Chemotaxis
chemotaxis model
coupling
Dynamical systems
Ergodic theory
Estimating techniques
General Applied Probability
Laplace transformation
long-time behavior
Markov analysis
Markov processes
Mathematical models
Mathematics
Perceptron convergence procedure
Piecewise-deterministic Markov process
Probability
Random variables
Studies
Telegraph
telegraph process
Telegraph service
Temporal logic
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Summary: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.
ISSN:0001-8678
1475-6064
DOI:10.1239/aap/1354716586