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Motivated by stability questions on piecewise deterministic Markov models ofbacterial chemotaxis, we study the long time behavior of a variant of theclassic telegraph process having a non-constant jump rate that induces a drifttowards the origin. We compute its invariant law and show exponentialergodicity, obtaining a quantitative control of the total variation distance toequilibrium at each instant of time. These results rely on an exact descriptionof the excursions of the process away from the origin and on the explicitconstruction of an original coalescent coupling for both velocity and position.Sharpness of the obtained convergence rate is discussed.

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