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Multiscale Piecewise Deterministic Markov Process in infinitedimension: 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
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Abstract

In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministicmarkov process in infinite dimensions. Adv. Appl. Probab. 44(2012) 749–773], the authors addressed the question of averaging for a slow-fastPiecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the presentpaper, we carry on and complete this work by the mathematical analysis of the fluctuationsof the slow-fast system around the averaged limit. A central limit theorem is derived andthe associated Langevin approximation is considered. The motivation for this work is thestudy of stochastic conductance based neuron models which describe the propagation of anaction 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

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