Qualitative properties of certain piecewise deterministic Markov processes

Benaïm, Michel; Le Borgne, Stéphane; Malrieu, Florent; Zitt, Pierre-André

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1040-1075.

Nous étudions une classe de processus de Markov déterministes par morceaux, sur espace d’états ${ℝ}^d\times E$ où $E$ est un ensemble fini. La composante continue du processus évolue suivant le flot d’un champ de vecteur, qui change lorsque la composante discrète saute. Les taux de saut peuvent dépendre des deux composantes. Sous l’hypothèse que le processus reste dans un ensemble compact, nous détaillons une construction possible et caractérisons son support en termes de solution d’une inclusion différentielle. Nous étudions ensuite le comportement en temps long, en faisant apparaître un certain ensemble de points accessibles, qui se trouve être fortement lié au support des mesures invariantes. Sous des conditions de type Hörmander sur les crochets de Lie entre les champs de vecteurs, nous montrons qu’il existe une unique mesure invariante vers laquelle le processus converge en variation totale. Nous donnons enfin des exemples où la condition d’unicité n’est pas vérifiée, et où le nombre de mesures invariantes dépend des taux de saut entre les flots.

We study a class of piecewise deterministic Markov processes with state space ${ℝ}^d\times E$ where $E$ is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hörmander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.

[1] J. P. Aubin and A. Cellina. Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, New York, 1984. | MR | Zbl

[2] Y. Bakhtin and T. Hurth. Invariant densities for dynamical systems with random switching. Nonlinearity 25(10) (2012) 2937–2952. | MR | Zbl

[3] M. Benaïm, J. Hofbauer and S. Sorin. Stochastic approximations and differential inclusions. SIAM J. Control Optim. 44 (2005) 328–348. | DOI | MR | Zbl

[4] M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt. Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab. 17, no. 56, 14 (2012). | MR | Zbl

[5] M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt. On the stability of planar randomly switched systems. Ann. Appl. Probab. 24(1) (2014) 292–311. | MR | Zbl

[6] O. Boxma, H. Kaspi, O. Kella and D. Perry. On/off storage systems with state-dependent inpout, outpout and swithching rates. Probab. Engrg. Inform. Sci. 19 (2005) 1–14. | DOI | MR | Zbl

[7] E. Buckwar and M. G. Riedler. An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol. 63(6) (2011) 1051–1093. | MR | Zbl

[8] B. Cloez and M. Hairer. Exponential ergodicity for Markov processes with random switching. Bernoulli 21 (2015) 505–536. | DOI | MR | Zbl

[9] O. L. V. Costa and F. Dufour. Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optim. 47(2) (2008) 1053–1077. | MR | Zbl

[10] M. H. A. Davis. Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Stat. Soc. Ser. B Stat. Methodol. 46(3) (1984) 353–388. With discussion. | MR | Zbl

[11] M. H. A. Davis. Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. Chapman & Hall, London, 1993. | MR | Zbl

[12] P. Diaconis and D. Freedman. Iterated random functions. SIAM Rev. 41(1) (1999) 45–76. | MR | Zbl

[13] M. Duflo. Random Iterative Models. Springer, Paris, 2000. | MR | Zbl

[14] V. Dumas, F. Guillemin and Ph. Robert. A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. in Appl. Probab. 34(1) (2002) 85–111. | MR | Zbl

[15] R. Durrett. Stochastic Calculus: A Practical Introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. | MR | Zbl

[16] H. Furstenberg. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961) 573–601. | DOI | MR | Zbl

[17] C. Graham and P. Robert. Interacting multi-class transmissions in large stochastic networks. Ann. Appl. Probab. 19(6) (2009) 2334–2361. | MR | Zbl

[18] C. Graham and P. Robert. Self-adaptive congestion control for multiclass intermittent connections in a communication network. Queueing Syst. 69(3–4) (2011) 237–257. | MR | Zbl

[19] M. Jacobsen. Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Probability and Its Applications. Birkhäuser Boston, Boston, MA, 2006. | MR | Zbl

[20] R. Karmakar and I. Bose. Graded and binary responses in stochastic gene expression. Phys. Biol. 197(1) (2004) 197–214.

[21] J. M. Lee. Introduction to Smooth Manifolds, 2nd edition. Graduate Texts in Mathematics 218. Springer, New York, 2013. | MR | Zbl

[22] P. A. W. Lewis and G. S. Shedler. Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. 26(3) (1979) 403–413. | MR | Zbl

[23] T. Lindvall. Lectures on the Coupling Method. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York, 1992. | MR | Zbl

[24] R. Mañé. Ergodic Theory and Differentiable Dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 8. Springer, Berlin, 1987. Translated from the Portuguese by Silvio Levy. | MR | Zbl

[25] K. Pakdaman, M. Thieullen and G. Wainrib. Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. in Appl. Probab. 42(3) (2010) 761–794. | MR | Zbl

[26] N. S. Papageorgiu. Existence theorems for differential inclusions with nonconvex right-hand side. Int. J. Math. Sci. 9(3) (1986) 459–469. | EuDML | MR | Zbl

[27] O. Radulescu, A. Muller and A. Crudu. Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moléculaire. Tech. Sci. Inform. 26(3–4) (2007) 443–469.

[28] S. M. Ross. Simulation, 2nd edition. Statistical Modeling and Decision Science. Academic Press, San Diego, CA, 1997. | MR | Zbl

[29] G. G. Yin and C. Zhu. Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability 63. Springer, New York, 2010. | MR | Zbl