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Averaging for slow–fast piecewise deterministic Markov processes with an attractive boundary

Part of: Limit theorems Markov processes Stochastic analysis

Published online by Cambridge University Press:  19 May 2023

Alexandre Génadot
Alexandre Génadot*
Affiliation:
Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IMB, UMR 5251
*
*Postal address: Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IMB, UMR 5251, 33400 Talence, France. Email address: alexandre.genadot@u-bordeaux.fr
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Abstract

In this paper we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMPs) whose dynamic is constrained by the presence of a boundary. On reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMPs.

Keywords

Constrained Markov processmultiscale modellingsingular perturbationnumber of crossings

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60H99: None of the above, but in this section 60J99: None of the above, but in this section
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1439 - 1468
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Azais, R. and Génadot, A. (2019). Estimation of the average number of continuous crossings for non-stationary non-diffusion processes. J. Statist. Planning Infer. 198, 119–138.CrossRefGoogle Scholar
Bao, H., Yu, X., Xu, Q., Wu, H. and Bao, B. (2022). Three-dimensional memristive Morris–Lecar model with magnetic induction effects and its FPGA implementation. Cogn. Neurodynam. Available at https://doi.org/10.1007/s11571-022-09871-6.Google ScholarPubMed
Borovkov, K. and Last, G. (2012). On Rice’s formula for stationary multivariate piecewise smooth processes. J. Appl. Prob. 49, 351–363.CrossRefGoogle Scholar
Burkitt, A. N. (2006). A review of the integrate-and-fire neuron model, I: Homogeneous synaptic input. Biol. Cybernet. 95, 1–19.CrossRefGoogle ScholarPubMed
Costantini, C. and Kurtz, T. (2015). Viscosity methods giving uniqueness for martingale problems. Electron. J. Prob. 20, 1–27.Google Scholar
Costantini, C. and Kurtz, T. G. (2018). Existence and uniqueness of reflecting diffusions in cusps. Electron. J. Prob. 23, 1–21.Google Scholar
Davis, M. H. A. (1993). Markov Models & Optimization (Monographs on Statistics and Applied Probability 49). CRC Press.Google Scholar
Engel, K.-J. and Nagel, R. (2001). One-parameter semigroups for linear evolution equations. Semigroup Forum 63, 278–280.CrossRefGoogle Scholar
Faggionato, A., Gabrielli, D. and Ribezzi Crivellari, M. (2010). Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors. Markov Process. Relat. Fields 16, 497–548.Google Scholar
Génadot, A. (2013). A multiscale study of stochastic spatially-extended conductance-based models for excitable systems. Doctoral thesis, Université Pierre et Marie Curie-Paris VI.Google Scholar
Génadot, . (2019). Averaging for some simple constrained Markov processes. Prob. Math. Statist. 39, 139–158.CrossRefGoogle Scholar
Gerstner, W., Kistler, W. M., Naud, R. and Paninski, L. (2014). Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. Cambridge University Press.CrossRefGoogle Scholar
Gollisch, T. and Meister, M. (2008). Rapid neural coding in the retina with relative spike latencies. Science 319, 1108–1111.CrossRefGoogle ScholarPubMed
Gwizdz, P. and Tyran-Kaminska, M. (2019). Densities for piecewise deterministic Markov processes with boundary. J. Math. Anal. Appl. 479, 384–425.CrossRefGoogle Scholar
Hairer, M. and Li, X.-M. (2020) Averaging dynamics driven by fractional Brownian motion. Ann. Prob. 48, 1826–1860.CrossRefGoogle Scholar
Jakubowski, A. (2007). The Skorokhod space in functional convergence: a short introduction. In International Conference: Skorokhod Space, pp. 11–18.Google Scholar
Kac, M. (1943). On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc. 49, 314–320.CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Krantz, S. and Parks, H. (1999). The Geometry of Domains in Space. Springer.CrossRefGoogle Scholar
Kurtz, T. G. (1990). Martingale problems for constrained Markov processes. In Recent Advances in Stochastic Calculus, ed. J. S. Baras and V. Mirelli, pp. 151–168. Springer.CrossRefGoogle Scholar
Morris, C. and Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213.CrossRefGoogle ScholarPubMed
Pakdaman, K., Thieullen, M. and Wainrib, G. (2012). Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic Markov processes. Stoch. Process. Appl. 122, 2292–2318.CrossRefGoogle Scholar
Pardoux, E. and Veretennikov, A. Y. (2001). On the Poisson equation and diffusion approximation, I. Ann. Prob. 29, 1061–1085.CrossRefGoogle Scholar
Pavliotis, G. and Stuart, A. (2008). Multiscale Methods: Averaging and Homogenization. Springer.Google Scholar
Perthame, B. (2006). Transport Equations in Biology. Springer.Google Scholar
Riedler, M. G. (2013). Almost sure convergence of numerical approximations for piecewise deterministic Markov processes. J. Comput. Appl. Math. 239, 50–71.CrossRefGoogle Scholar
Riedler, M. G. (2011). Spatio-temporal stochastic hybrid models of biological excitable membranes. Doctoral dissertation, Heriot-Watt University.Google Scholar
Tyran-Kaminska, M. (2009). Substochastic semigroups and densities of piecewise deterministic Markov processes. J. Math. Anal. Appl. 357, 385–402.CrossRefGoogle Scholar
Yin, G. G. and Zhang, Q. (2012). Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach. Springer.Google Scholar