On the Hamiltonian structure of large deviations in stochastic hybrid systems

Paul Bressloff 1 Olivier Faugeras 2, 3
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
3 MATHNEURO - Mathématiques pour les Neurosciences
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We present a new derivation of the classical action underlying a large deviation principle (LDP) for a stochastic hybrid system, which couples a piecewise deterministic dynamical system in R d with a time-homogeneous Markov chain on some discrete space Γ. We assume that the Markov chain on Γ is ergodic, and that the discrete dynamics is much faster than the piecewise deterministic dynamics (separation of timescales). Using the Perron-Frobenius theorem and the calculus-of-variations, we show that the resulting action Hamiltonian is given by the Perron eigenvalue of a |Γ|-dimensional linear equation. The corresponding linear operator depends on the transition rates of the Markov chain and the nonlinear functions of the piecewise deterministic system. We compare the Hamiltonian to one derived using WKB methods, and show that the latter is a reduction of the former. We also indicate how the analysis can be extended to a multi-scale stochastic process, in which the continuous dynamics is described by a piecewise stochastic differential equations (SDE). Finally, we illustrate the theory by considering applications to conductance-based models of membrane voltage fluctuations in the presence of stochastic ion channels.
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Paul Bressloff, Olivier Faugeras. On the Hamiltonian structure of large deviations in stochastic hybrid systems. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2017, 2017, pp.33206. ⟨10.1088/1742-5468/aa64f3⟩. ⟨hal-01072077v3⟩

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