Quantitative ergodicity for some switched dynamical systems

Michel Benaïm (Université de Neuchâtel)
Stéphane Le Borgne (Université de Rennes I)
Florent Malrieu (Université de Rennes 1)
Pierre-André Zitt (Université de Bourgogne)

Abstract


We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space $\mathbb{R}^d\times E$ where $E$ is a finite set. The continous component evolves according to a smooth vector field that it switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances.

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Pages: 1-14

Publication Date: December 3, 2012

DOI: 10.1214/ECP.v17-1932

References

  • Alsmeyer, Gerold; Iksanov, Alex; Rösler, Uwe. On distributional properties of perpetuities. J. Theoret. Probab. 22 (2009), no. 3, 666--682. MR2530108
  • Y. Bakhtin and T. Hurth. Invariant densities for dynamical systems with random switching, Preprint available on arXiv, 2012.
  • Bardet, Jean-Baptiste; Guérin, Hélène; Malrieu, Florent. Long time behavior of diffusions with Markov switching. ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010), 151--170. MR2653702
  • J.B. Bardet, A. Christen, A. Guillin, A. Malrieu, and P.-A. Zitt, Total variation estimates for the TCP process, Preprint available on arXiv, 2012.
  • M. Benaïm, S. Le~Borgne, F. Malrieu, and P.-A. Zitt. On the ergodicity of some piecewise deterministic markov processes, preprint, 2012.
  • M. Benaïm, S. Le~Borgne, F. Malrieu, and P.-A. Zitt. On the stability of planar randomly switched systems, preprint, 2012.
  • Boxma, Onno; Kaspi, Haya; Kella, Offer; Perry, David. On/off storage systems with state-dependent input, output, and switching rates. Probab. Engrg. Inform. Sci. 19 (2005), no. 1, 1--14. MR2104547
  • E. Buckwar and M. G. Riedler. An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution, J. Math. Biol. 63 (2011), no. 6, 1051--1093.
  • Caputo, Pietro; Dai Pra, Paolo; Posta, Gustavo. Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 3, 734--753. MR2548501
  • Chafaï, Djalil; Malrieu, Florent; Paroux, Katy. On the long time behavior of the TCP window size process. Stochastic Process. Appl. 120 (2010), no. 8, 1518--1534. MR2653264
  • Costa, O. L. V. Stationary distributions for piecewise-deterministic Markov processes. J. Appl. Probab. 27 (1990), no. 1, 60--73. MR1039184
  • Costa, O. L. V.; Dufour, F. Ergodic properties and ergodic decompositions of continuous-time Markov processes. J. Appl. Probab. 43 (2006), no. 3, 767--781. MR2274799
  • Costa, O. L. V.; Dufour, F. Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optim. 47 (2008), no. 2, 1053--1077. MR2385873
  • Davis, M. H. A. Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. With discussion. J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353--388. MR0790622
  • Davis, M. H. A. Markov models and optimization. Monographs on Statistics and Applied Probability, 49. Chapman & Hall, London, 1993. xiv+295 pp. ISBN: 0-412-31410-X MR1283589
  • de Saporta, Benoîte; Yao, Jian-Feng. Tail of a linear diffusion with Markov switching. Ann. Appl. Probab. 15 (2005), no. 1B, 992--1018. MR2114998
  • Diaconis, Persi; Freedman, David. Iterated random functions. SIAM Rev. 41 (1999), no. 1, 45--76. MR1669737
  • Dufour, François; Costa, Oswaldo L. V. Stability of piecewise-deterministic Markov processes. SIAM J. Control Optim. 37 (1999), no. 5, 1483--1502 (electronic). MR1710229
  • Dumas, Vincent; Guillemin, Fabrice; Robert, Philippe. A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. in Appl. Probab. 34 (2002), no. 1, 85--111. MR1895332
  • J. Fontbona, H. Guérin, and F. Malrieu. Quantitative estimates for the long time behavior of a PDMP describing the movement of bacteria, arXiv, 2010.
  • Goldie, Charles M.; Grübel, Rudolf. Perpetuities with thin tails. Adv. in Appl. Probab. 28 (1996), no. 2, 463--480. MR1387886
  • Graham, Carl; Robert, Philippe. Interacting multi-class transmissions in large stochastic networks. Ann. Appl. Probab. 19 (2009), no. 6, 2334--2361. MR2588247
  • Graham, Carl; Robert, Philippe. Self-adaptive congestion control for multi-class intermittent connections in a communication network, arXiv, 2010.
  • Guyon, Xavier; Iovleff, Serge; Yao, Jian-Feng. Linear diffusion with stationary switching regime. ESAIM Probab. Stat. 8 (2004), 25--35 (electronic). MR2085603
  • Hitczenko, Paweł; Wesołowski, Jacek. Perpetuities with thin tails revisited. Ann. Appl. Probab. 19 (2009), no. 6, 2080--2101. MR2588240
  • Kesten, Harry. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), 207--248. MR0440724
  • Pakdaman, K.; Thieullen, M.; Wainrib, G. Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. in Appl. Probab. 42 (2010), no. 3, 761--794. MR2779558
  • Rachev, Svetlozar T. Probability metrics and the stability of stochastic models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1991. xiv+494 pp. ISBN: 0-471-92877-1 MR1105086
  • O. Radulescu, A. Muller, and A. Crudu, Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moléculaire, Technique et Science Informatiques 26 (2007), no. 3-4, 443--469.
  • Saloff-Coste, Laurent. Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996), 301--413, Lecture Notes in Math., 1665, Springer, Berlin, 1997. MR1490046
  • Vervaat, Wim. On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 (1979), no. 4, 750--783. MR0544194
  • Villani, Cédric. Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003. xvi+370 pp. ISBN: 0-8218-3312-X MR1964483
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