Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T14:47:50.927Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Stochastic systems and control Limit theorems

Published online by Cambridge University Press:  04 January 2016

Joaquin Fontbona ,
Hélène Guérin  and
Florent Malrieu
Joaquin Fontbona*
Affiliation:
Universidad de Chile
Hélène Guérin*
Affiliation:
Universitè de Rennes 1
Florent Malrieu*
Affiliation:
Universitè de Rennes 1
*
∗ Postal address: CMM-DIM, UMI 2807, UChile-CNRS, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile. Email address: fontbona@dim.uchile.cl
∗∗ Postal address: UMR 6625, CNRS, Institut de Recherche Mathèmatique de Rennes (IRMAR), Universitè de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France.
∗∗ Postal address: UMR 6625, CNRS, Institut de Recherche Mathèmatique de Rennes (IRMAR), Universitè de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Piecewise-deterministic Markov processcouplinglong-time behaviortelegraph processchemotaxis model

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes 93E15: Stochastic stability 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 977 - 994
Copyright
© Applied Probability Trust 

References

Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51), 2nd edn. Springer, New York.Google Scholar
Bakry, D., Cattiaux, P. and Guillin, A. (2008). Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254, 727–759.CrossRefGoogle Scholar
Costa, O. L. V. and Dufour, F. (2008). Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optimization 47, 1053–1077.CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Dufour, F. and Costa, O. L. V. (1999). Stability of piecewise-deterministic Markov processes. SIAM J. Control Optimization 37, 1483–1502.Google Scholar
Erban, R. and Othmer, H. G. (2004/05). From individual to collective behavior in bacterial chemotaxis. SIAM J. Appl. Math. 65, 361–391.Google Scholar
Erban, R. and Othmer, H. G. (2005). From signal transduction to spatial pattern formation in E. coli: a paradigm for multiscale modeling in biology. Multiscale Model. Simul. 3, 362–394.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
Herrmann, S. and Vallois, P. (2010). From persistent random walk to the telegraph noise. Stoch. Dynamics 10, 161–196.Google Scholar
Jacobsen, M. (2006). Point Process Theory and Applications. Birkhäuser, Boston, MA.Google Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497–509.CrossRefGoogle Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Lund, R. B., Meyn, S. P. and Tweedie, R. L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Prob. 6, 218–237.Google Scholar
Meyn, S. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548.Google Scholar
Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.Google Scholar
Norris, J. R. (1997). Markov Chains (Camb. Ser. Statist. Prob. Math. 2). Cambridge University Press.Google Scholar
Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion (Fundamental Principles Math. Sci. 293), 2nd edn. Springer, Berlin.Google Scholar
Robert, P. (2003). Stochastic Networks and Queues (Appl. Math. 52). Springer, Berlin.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (1996). Quantitative bounds for convergence rates of continuous time Markov processes. Electron. J. Prob. 1, 21pp.CrossRefGoogle Scholar
Rousset, M. and Samaey, G. (2011). Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction. Preprint. Available at http://arxiv.org/abs/1111.5321v1.Google Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar