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

Joaquin Fontbona (1) , Hélène Guérin (2) , Florent Malrieu (2)

1 CMM - Center for Mathematical Modeling
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: long-time behavior; chemotaxis model; coupling; telegraph process; Piecewise deterministic Markov process

Domaines: Probabilités [math.PR]

Joaquin Fontbona, Hélène Guérin, Florent Malrieu. Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process. Advances in Applied Probability, 2012, 44 (4), pp.977-994. ⟨10.1239/aap/1354716586⟩. ⟨hal-00763424⟩