Hostname: page-component-84b7d79bbc-g7rbq Total loading time: 0 Render date: 2024-07-26T13:42:16.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation∗

Published online by Cambridge University Press:  10 October 2014

A. Genadot  and
M. Thieullen
A. Genadot
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris 6, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France. algenadot@gmail.com; michele.thieullen@upmc.fr
M. Thieullen
Affiliation:
Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris 6, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France. algenadot@gmail.com; michele.thieullen@upmc.fr
Get access

Abstract

In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministic markov process in infinite dimensions. Adv. Appl. Probab. 44 (2012) 749–773], the authors addressed the question of averaging for a slow-fast Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the present paper, we carry on and complete this work by the mathematical analysis of the fluctuations of the slow-fast system around the averaged limit. A central limit theorem is derived and the associated Langevin approximation is considered. The motivation for this work is the study of stochastic conductance based neuron models which describe the propagation of an action potential along a nerve fiber.

Keywords

Piecewise deterministic Markov processaveraging principleneuron model
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 541 - 569
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Austin, T., The emergence of the deterministic hodgkin–huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann. Appl. Probab. 18 (2008) 1279–1325. Google Scholar
R. Azaïs, A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process, preprint arXiv:1211.5579 (2012).
M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic markov processes, preprint arXiv:1204.4143 (2012).
Benaïm, M., Le Borgne, S., Malrieu, F. and Zitt, P.-A., Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab. 17 (2012) 1–14. Google Scholar
N. Berglund and B. Gentz, Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach, vol. 246. Springer Berlin (2006).
Brandejsky, A., De Saporta, B. and Dufour, F., Numerical methods for the exit time of a piecewise-deterministic markov process. Adv. Appl. Probab. 44 (2012) 196–225. Google Scholar
C.-E. Bréhier, Strong and weak order in averaging for spdes. Stochastic Processes Appl. (2012).
Buckwar, E. and Riedler, M.G., An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol. 63 (2001) 1051–1093. Google ScholarPubMed
Cerrai, S. and Freidlin, M., Averaging principle for a class of stochastic reaction–diffusion equations. Probab. Theory Relat. Fields 144 (2009) 137–177. Google Scholar
Costa, O. and Dufour, F., Singular perturbation for the discounted continuous control of piecewise deterministic markov processes. Appl. Math. Optim. 63 (2011) 357–384. Google Scholar
G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Cambridge University Press (2008).
M.H. Davis. Piecewise-deterministic markov processes: A general class of non-diffusion stochastic models. J. Roy. Statist. Soc. Ser. B (Methodological) (1984) 353–388.
M.H. Davis, Markov Models & Optimization, vol. 49. Chapman & Hall/CRC (1993).
M. Doumic, M. Hoffmann, N. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree, preprint arXiv:1210.3240 (2012).
S. Ethier and T. Kurtz, Markov processes. characterization and convergence, vol. 9. John Willey and Sons, New York (1986).
A. Faggionato, D. Gabrielli and M.R. Crivellari, Averaging and large deviation principles for fully-coupled piecewise deterministic markov processes and applications to molecular motors, preprint arXiv:0808.1910 (2008).
Genadot, A. and Thieullen, M., Averaging for a fully coupled piecewise-deterministic markov process in infinite dimensions. Adv. Appl. Probab. 44 (2012) 749–773. Google Scholar
D. Goreac, Viability, invariance and reachability for controlled piecewise deterministic markov processes associated to gene networks, preprint arXiv:1002.2242 (2010).
Hausenblas, E. and Seidler, J., Stochastic convolutions driven by martingales: Maximal inequalities and exponential integrability. Stochastic Anal. Appl. 26 (2007) 98–119. Google Scholar
D. Henry, Geometric theory of semilinear parabolic equations, vol. 840. Springer-Verlag Berlin (1981).
B. Hille, Ionic channels of excitable membranes. Sinauer associates Sunderland, MA (2001).
M. Jacobsen, Point process theory and applications: marked point and piecewise deterministic processes. Birkhauser Boston (2006).
Lopker, A. and Palmowski, Z., On time reversal of piecewise deterministic markov processes. Electron. J. Probab. 18 (2013) 1–29. Google Scholar
M. Métivier, Convergence faible et principe d’invariance pour des martingales à valeurs dans des espaces de sobolev. In Ann. Inst. Henri Poincaré, Probab. Stat., vol. 20. Elsevier (1984) 329–348.
Morris, C. and Lecar, H., Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35 (1981) 193–213. Google ScholarPubMed
K. Pakdaman, M. Thieullen and G. Wainrib, Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic markov processes. Stochastic Proc. Appl. (2012).
G.A. Pavliotis and A.M. Stuart, Multiscale methods: averaging and homogenization, vol. 53. Springer Science (2008).
M.G. Riedler, Spatio-temporal stochastic hybrid models of biological excitable membranes. Ph.D. thesis, Heriot-Watt University (2011).
M.G. Riedler, Almost sure convergence of numerical approximations for piecewise deterministic markov processes. J. Comput. Appl. Math. (2012).
Riedler, M.G. and Buckwar, E., Laws of large numbers and langevin approximations for stochastic neural field equations. J. Math. Neurosci. (JMN) 3 (2013) 1–54. Google ScholarPubMed
M.G. Riedler, M. Thieullen and G. Wainrib, Limit theorems for infinite-dimensional piecewise deterministic markov processes. applications to stochastic excitable membrane models, preprint arXiv:1112.4069 (2011).
Tyran-Kamińska, M., Substochastic semigroups and densities of piecewise deterministic markov processes. J. Math. Anal. Appl. 357 (2009) 385–402. Google Scholar
G. Wainrib, Randomness in neurons: a multiscale probabilistic analysis. Ph.D. thesis, Ecole Polytechnique (2010).
W. Wang, A. Roberts and J. Duan, Large deviations for slow-fast stochastic partial differential equations, preprint arXiv:1001.4826 (2010).
W. Wang and A.J. Roberts, Average and deviation for slow–fast stochastic partial differential equations. J. Differ. Equ. (2012).
G. G. Yin and Q. Zhang, Continuous-time Markov chains and applications, vol. 37. Springer (2013).
G.G. Yin and C. Zhu, Hybrid switching diffusions: properties and applications, vol. 63. Springer (2010).