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

A. Genadot; M. Thieullen

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 541-569
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Genadot, A., and Thieullen, M.. "Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation." ESAIM: Probability and Statistics 18 (2014): 541-569. <http://eudml.org/doc/274343>.

@article{Genadot2014,
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.},
author = {Genadot, A., Thieullen, M.},
journal = {ESAIM: Probability and Statistics},
keywords = {piecewise deterministic Markov process; averaging principle; neuron model},
language = {eng},
pages = {541-569},
publisher = {EDP-Sciences},
title = {Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation},
url = {http://eudml.org/doc/274343},
volume = {18},
year = {2014},
}

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
PB - EDP-Sciences
VL - 18
SP - 541
EP - 569
AB - 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.
LA - eng
KW - piecewise deterministic Markov process; averaging principle; neuron model
UR - http://eudml.org/doc/274343
ER -

References

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  1. [1] T. Austin, 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. Zbl1157.60066MR2434172
  2. [2] R. Azaïs, A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process, preprint arXiv:1211.5579 (2012). 
  3. [3] M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic markov processes, preprint arXiv:1204.4143 (2012). Zbl1325.60123MR3005729
  4. [4] M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab.17 (2012) 1–14. Zbl06346849MR3005729
  5. [5] N. Berglund and B. Gentz, Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach, vol. 246. Springer Berlin (2006). Zbl1098.37049MR2197663
  6. [6] A. Brandejsky, B. De Saporta and F. Dufour, Numerical methods for the exit time of a piecewise-deterministic markov process. Adv. Appl. Probab.44 (2012) 196–225. Zbl1244.60073MR2951552
  7. [7] C.-E. Bréhier, Strong and weak order in averaging for spdes. Stochastic Processes Appl. (2012). Zbl1266.60112MR2926167
  8. [8] E. Buckwar and M.G. Riedler, An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution. J. Math. Biol.63 (2001) 1051–1093. Zbl1284.92038MR2855804
  9. [9] S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction–diffusion equations. Probab. Theory Relat. Fields144 (2009) 137–177. Zbl1176.60049MR2480788
  10. [10] O. Costa and F. Dufour, Singular perturbation for the discounted continuous control of piecewise deterministic markov processes. Appl. Math. Optim.63 (2011) 357–384. Zbl1222.60056MR2784836
  11. [11] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Cambridge University Press (2008). Zbl1140.60034MR1207136
  12. [12] 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. Zbl0565.60070MR790622
  13. [13] M.H. Davis, Markov Models & Optimization, vol. 49. Chapman & Hall/CRC (1993). Zbl0780.60002MR1283589
  14. [14] 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). 
  15. [15] S. Ethier and T. Kurtz, Markov processes. characterization and convergence, vol. 9. John Willey and Sons, New York (1986). Zbl1089.60005MR838085
  16. [16] 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). Zbl1266.60048MR2759771
  17. [17] A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministic markov process in infinite dimensions. Adv. Appl. Probab.44 (2012) 749–773. Zbl1276.60023MR3024608
  18. [18] D. Goreac, Viability, invariance and reachability for controlled piecewise deterministic markov processes associated to gene networks, preprint arXiv:1002.2242 (2010). Zbl1319.49040MR2954632
  19. [19] E. Hausenblas and J. Seidler, Stochastic convolutions driven by martingales: Maximal inequalities and exponential integrability. Stochastic Anal. Appl.26 (2007) 98–119. Zbl1153.60035MR2378512
  20. [20] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840. Springer-Verlag Berlin (1981). Zbl0456.35001MR610244
  21. [21] B. Hille, Ionic channels of excitable membranes. Sinauer associates Sunderland, MA (2001). 
  22. [22] M. Jacobsen, Point process theory and applications: marked point and piecewise deterministic processes. Birkhauser Boston (2006). Zbl1093.60002MR2189574
  23. [23] A. Lopker and Z. Palmowski, On time reversal of piecewise deterministic markov processes. Electron. J. Probab.18 (2013) 1–29. Zbl1294.60099MR3035741
  24. [24] 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. Zbl0549.60041MR771893
  25. [25] C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber. Biophys. J.35 (1981) 193–213. 
  26. [26] K. Pakdaman, M. Thieullen and G. Wainrib, Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic markov processes. Stochastic Proc. Appl. (2012). Zbl1261.60031MR2922629
  27. [27] G.A. Pavliotis and A.M. Stuart, Multiscale methods: averaging and homogenization, vol. 53. Springer Science (2008). Zbl1160.35006MR2382139
  28. [28] M.G. Riedler, Spatio-temporal stochastic hybrid models of biological excitable membranes. Ph.D. thesis, Heriot-Watt University (2011). 
  29. [29] M.G. Riedler, Almost sure convergence of numerical approximations for piecewise deterministic markov processes. J. Comput. Appl. Math. (2012). Zbl1255.65028MR2991958
  30. [30] M.G. Riedler and E. Buckwar, Laws of large numbers and langevin approximations for stochastic neural field equations. J. Math. Neurosci. (JMN) 3 (2013) 1–54. Zbl1295.60030MR3033899
  31. [31] 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). Zbl1255.60126MR2955047
  32. [32] M. Tyran-Kamińska, Substochastic semigroups and densities of piecewise deterministic markov processes. J. Math. Anal. Appl.357 (2009) 385–402. Zbl1179.60050MR2557653
  33. [33] G. Wainrib, Randomness in neurons: a multiscale probabilistic analysis. Ph.D. thesis, Ecole Polytechnique (2010). 
  34. [34] W. Wang, A. Roberts and J. Duan, Large deviations for slow-fast stochastic partial differential equations, preprint arXiv:1001.4826 (2010). Zbl1251.35201
  35. [35] W. Wang and A.J. Roberts, Average and deviation for slow–fast stochastic partial differential equations. J. Differ. Equ. (2012). Zbl1251.35201
  36. [36] G. G. Yin and Q. Zhang, Continuous-time Markov chains and applications, vol. 37. Springer (2013). Zbl1277.60127MR2985157
  37. [37] G.G. Yin and C. Zhu, Hybrid switching diffusions: properties and applications, vol. 63. Springer (2010). Zbl1279.60007MR2559912

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