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.
Mots clés : piecewise deterministic Markov process, averaging principle, neuron model
@article{PS_2014__18__541_0,
author = {Genadot, A. and Thieullen, M.},
title = {Multiscale {Piecewise} {Deterministic} {Markov} {Process} in infinite dimension: central limit theorem and {Langevin} approximation},
journal = {ESAIM: Probability and Statistics},
pages = {541--569},
publisher = {EDP-Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2013051},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2013051/}
}
TY - JOUR AU - Genadot, A. AU - Thieullen, M. TI - Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation JO - ESAIM: Probability and Statistics PY - 2014 SP - 541 EP - 569 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2013051/ DO - 10.1051/ps/2013051 LA - en ID - PS_2014__18__541_0 ER -
%0 Journal Article %A Genadot, A. %A Thieullen, M. %T Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation %J ESAIM: Probability and Statistics %D 2014 %P 541-569 %V 18 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2013051/ %R 10.1051/ps/2013051 %G en %F PS_2014__18__541_0
Genadot, A.; Thieullen, M. Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 541-569. doi : 10.1051/ps/2013051. http://www.numdam.org/articles/10.1051/ps/2013051/
[1] , 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. | MR | Zbl
[2] , A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process, preprint arXiv:1211.5579 (2012).
[3] , , and , Qualitative properties of certain piecewise deterministic markov processes, preprint arXiv:1204.4143 (2012). | MR
[4] , , and , Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab. 17 (2012) 1-14. | MR
[5] and , Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach, vol. 246. Springer Berlin (2006). | MR | Zbl
[6] , and , Numerical methods for the exit time of a piecewise-deterministic markov process. Adv. Appl. Probab. 44 (2012) 196-225. | MR | Zbl
[7] , Strong and weak order in averaging for spdes. Stochastic Processes Appl. (2012). | MR | Zbl
[8] and , An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol. 63 (2001) 1051-1093. | MR | Zbl
[9] and , Averaging principle for a class of stochastic reaction-diffusion equations. Probab. Theory Relat. Fields 144 (2009) 137-177. | MR | Zbl
[10] and , Singular perturbation for the discounted continuous control of piecewise deterministic markov processes. Appl. Math. Optim. 63 (2011) 357-384. | MR | Zbl
[11] and , Stochastic equations in infinite dimensions. Cambridge University Press (2008). | MR | Zbl
[12] . Piecewise-deterministic markov processes: A general class of non-diffusion stochastic models. J. Roy. Statist. Soc. Ser. B (Methodological) (1984) 353-388. | MR | Zbl
[13] , Markov Models & Optimization, vol. 49. Chapman & Hall/CRC (1993). | MR | Zbl
[14] , , and , Statistical estimation of a growth-fragmentation model observed on a genealogical tree, preprint arXiv:1210.3240 (2012).
[15] and , Markov processes. characterization and convergence, vol. 9. John Willey and Sons, New York (1986). | MR | Zbl
[16] , and , Averaging and large deviation principles for fully-coupled piecewise deterministic markov processes and applications to molecular motors, preprint arXiv:0808.1910 (2008). | MR | Zbl
[17] and , Averaging for a fully coupled piecewise-deterministic markov process in infinite dimensions. Adv. Appl. Probab. 44 (2012) 749-773. | MR | Zbl
[18] , Viability, invariance and reachability for controlled piecewise deterministic markov processes associated to gene networks, preprint arXiv:1002.2242 (2010). | Numdam | MR
[19] and , Stochastic convolutions driven by martingales: Maximal inequalities and exponential integrability. Stochastic Anal. Appl. 26 (2007) 98-119. | MR | Zbl
[20] , Geometric theory of semilinear parabolic equations, vol. 840. Springer-Verlag Berlin (1981). | MR | Zbl
[21] , Ionic channels of excitable membranes. Sinauer associates Sunderland, MA (2001).
[22] , Point process theory and applications: marked point and piecewise deterministic processes. Birkhauser Boston (2006). | MR | Zbl
[23] and , On time reversal of piecewise deterministic markov processes. Electron. J. Probab. 18 (2013) 1-29. | MR | Zbl
[24] , 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. | Numdam | MR | Zbl
[25] and , Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35 (1981) 193-213.
[26] , and , Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic markov processes. Stochastic Proc. Appl. (2012). | MR | Zbl
[27] and , Multiscale methods: averaging and homogenization, vol. 53. Springer Science (2008). | MR | Zbl
[28] , Spatio-temporal stochastic hybrid models of biological excitable membranes. Ph.D. thesis, Heriot-Watt University (2011).
[29] , Almost sure convergence of numerical approximations for piecewise deterministic markov processes. J. Comput. Appl. Math. (2012). | MR | Zbl
[30] and , Laws of large numbers and langevin approximations for stochastic neural field equations. J. Math. Neurosci. (JMN) 3 (2013) 1-54. | MR | Zbl
[31] , and , Limit theorems for infinite-dimensional piecewise deterministic markov processes. applications to stochastic excitable membrane models, preprint arXiv:1112.4069 (2011). | MR | Zbl
[32] , Substochastic semigroups and densities of piecewise deterministic markov processes. J. Math. Anal. Appl. 357 (2009) 385-402. | MR | Zbl
[33] , Randomness in neurons: a multiscale probabilistic analysis. Ph.D. thesis, Ecole Polytechnique (2010).
[34] , and , Large deviations for slow-fast stochastic partial differential equations, preprint arXiv:1001.4826 (2010). | Zbl
[35] and , Average and deviation for slow-fast stochastic partial differential equations. J. Differ. Equ. (2012). | Zbl
[36] and , Continuous-time Markov chains and applications, vol. 37. Springer (2013). | MR | Zbl
[37] and , Hybrid switching diffusions: properties and applications, vol. 63. Springer (2010). | MR | Zbl
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