Free Access
Volume 18, 2014
Page(s) 541 - 569
Published online 10 October 2014
  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. [CrossRef] [Google Scholar]
  2. R. Azaïs, A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process, preprint arXiv:1211.5579 (2012). [Google Scholar]
  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). [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  5. N. Berglund and B. Gentz, Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach, vol. 246. Springer Berlin (2006). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  7. C.-E. Bréhier, Strong and weak order in averaging for spdes. Stochastic Processes Appl. (2012). [Google Scholar]
  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. [Google Scholar]
  9. S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction–diffusion equations. Probab. Theory Relat. Fields 144 (2009) 137–177. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  11. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Cambridge University Press (2008). [Google Scholar]
  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. [Google Scholar]
  13. M.H. Davis, Markov Models & Optimization, vol. 49. Chapman & Hall/CRC (1993). [Google Scholar]
  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). [Google Scholar]
  15. S. Ethier and T. Kurtz, Markov processes. characterization and convergence, vol. 9. John Willey and Sons, New York (1986). [Google Scholar]
  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). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  18. D. Goreac, Viability, invariance and reachability for controlled piecewise deterministic markov processes associated to gene networks, preprint arXiv:1002.2242 (2010). [Google Scholar]
  19. E. Hausenblas and J. Seidler, Stochastic convolutions driven by martingales: Maximal inequalities and exponential integrability. Stochastic Anal. Appl. 26 (2007) 98–119. [CrossRef] [Google Scholar]
  20. D. Henry, Geometric theory of semilinear parabolic equations, vol. 840. Springer-Verlag Berlin (1981). [Google Scholar]
  21. B. Hille, Ionic channels of excitable membranes. Sinauer associates Sunderland, MA (2001). [Google Scholar]
  22. M. Jacobsen, Point process theory and applications: marked point and piecewise deterministic processes. Birkhauser Boston (2006). [Google Scholar]
  23. A. Lopker and Z. Palmowski, On time reversal of piecewise deterministic markov processes. Electron. J. Probab. 18 (2013) 1–29. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  25. C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35 (1981) 193–213. [CrossRef] [PubMed] [Google Scholar]
  26. K. Pakdaman, M. Thieullen and G. Wainrib, Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic markov processes. Stochastic Proc. Appl. (2012). [Google Scholar]
  27. G.A. Pavliotis and A.M. Stuart, Multiscale methods: averaging and homogenization, vol. 53. Springer Science (2008). [Google Scholar]
  28. M.G. Riedler, Spatio-temporal stochastic hybrid models of biological excitable membranes. Ph.D. thesis, Heriot-Watt University (2011). [Google Scholar]
  29. M.G. Riedler, Almost sure convergence of numerical approximations for piecewise deterministic markov processes. J. Comput. Appl. Math. (2012). [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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). [Google Scholar]
  32. M. Tyran-Kamińska, Substochastic semigroups and densities of piecewise deterministic markov processes. J. Math. Anal. Appl. 357 (2009) 385–402. [CrossRef] [Google Scholar]
  33. G. Wainrib, Randomness in neurons: a multiscale probabilistic analysis. Ph.D. thesis, Ecole Polytechnique (2010). [Google Scholar]
  34. W. Wang, A. Roberts and J. Duan, Large deviations for slow-fast stochastic partial differential equations, preprint arXiv:1001.4826 (2010). [Google Scholar]
  35. W. Wang and A.J. Roberts, Average and deviation for slow–fast stochastic partial differential equations. J. Differ. Equ. (2012). [Google Scholar]
  36. G. G. Yin and Q. Zhang, Continuous-time Markov chains and applications, vol. 37. Springer (2013). [Google Scholar]
  37. G.G. Yin and C. Zhu, Hybrid switching diffusions: properties and applications, vol. 63. Springer (2010). [Google Scholar]

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