Quantitative speeds of convergence for exposure to food contaminants
Bouguet, Florian1
1 UMR 6625 CNRS Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 482-501.

In this paper, we consider a class of piecewise-deterministic Markov processes modeling the quantity of a given food contaminant in the body. On the one hand, the amount of contaminant increases with random food intakes and, on the other hand, decreases thanks to the release rate of the body. Our aim is to provide quantitative speeds of convergence to equilibrium for the total variation and Wasserstein distances via coupling methods.

Reçu le :
DOI : 10.1051/ps/2015002
Classification : 60J25, 60K15, 60B10
Mots-clés : Piecewise deterministic Markov processes, coupling, renewal Markov processes, convergence to equilibrium, exponential ergodicity, dietary contamination
Bouguet, Florian 1

1 UMR 6625 CNRS Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France 
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     zbl = {1343.60110},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2015002/}
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Bouguet, Florian. Quantitative speeds of convergence for exposure to food contaminants. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 482-501. doi : 10.1051/ps/2015002. http://archive.numdam.org/articles/10.1051/ps/2015002/

S. Asmussen, Applied Probability and Queues. Vol. 51 of Appl. Math. Stoch. Model. Appl. Prob. 2nd edition. Springer-Verlag, New York (2003). | Zbl

J.-B. Bardet, A. Christen, A. Guillin, F. Malrieu and P.-A. Zitt, Total variation estimates for the TCP process. Electron. J. Probab. 18 (2013) 1–21. | Zbl

M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab. 17 (2012) 14. | Zbl

P. Bertail, S. Clémençon and J. Tressou, A storage model with random release rate for modeling exposure to food contaminants. Math. Biosci. Eng. 5 (2008) 35–60. | Zbl

J.-L. Bon, Fiabilité des systèmes: Méthodes mathématiques. Masson (1995).

D. Chafaï, F. Malrieu and K. Paroux, On the long time behavior of the TCP window size process. Stochastic Process. Appl. 120 (2010) 1518–1534. | Zbl

W. Feller, An introduction to probability theory and its applications, 2nd edition. In Vol. II. John Wiley & Sons Inc., New York (1971). | Zbl

M. Hairer, Convergence of Markov processes. Available at http://www.hairer.org/notes/Convergence.pdf (2010).

M. Jacobsen, Point Process Theory and Applications. Marked Point and Piecewise Deterministic Processes. Probab. Appl. Birkhäuser Boston Inc., Boston, MA (2006). | Zbl

T. Lindvall, On coupling of renewal processes with use of failure rates. Stoch. Process. Appl. 22 (1986) 1–15. | Zbl

T. Lindvall, Lectures on the Coupling Method. Wiley Series in Probab. Math. Stat. A Wiley-Interscience Publication. Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York (1992). | Zbl

S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability. Commun. Control Eng. Series. Springer-Verlag London Ltd., London (1993). | Zbl

P. Monmarché, Hypocoercive relaxation to equilibrium for some kinetic models. Kinet. Relat. Models 7 (2014) 341–360. | Zbl

P. Monmarché, On ℋ 1 and entropic convergence for contractive PDMP. Preprint arXiv:1404.4220 (2014).

S. Resnick, Adventures in stochastic processes. Birkhäuser Boston Inc., Boston, MA (1992). | Zbl

C. Villani, Optimal transport, Old and new. Vol. 338 of Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (2009). | Zbl

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