On the Hamiltonian structure of large deviations in stochastic hybrid systems

Abstract

We develop the connection between large deviation theory and more applied approaches to stochastic hybrid systems by highlighting a common underlying Hamiltonian structure. A stochastic hybrid system involves the coupling between a piecewise deterministic dynamical system in $R^d$ and a time-homogeneous Markov chain on some discrete space $\Gamma$. We assume that the Markov chain on $\Gamma$ is ergodic, and that the discrete dynamics is much faster than the piecewise deterministic dynamics (separation of time-scales). Using the Perron-Frobenius theorem and the calculus-of-variations, we evaluate the rate function of a large deviation principle in terms of a classical action, whose Hamiltonian is given by the Perron eigenvalue of a $|\Gamma |$-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. The resulting Hamiltonian is identical to one derived using path-integrals and WKB methods. We illustrate the theory by considering the example of stochastic ion channels.

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• The large deviation approach to statistical mechanics
• Spectral theory of metastability and extinction in birth-death systems
• Stochastic Representations of Ion Channel Kinetics and Exact Stochastic Simulation of Neuronal Dynamics
• Bistable switching asymptotics for the self regulating gene
• Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors

Cited by in corpus (2)

• Large Deviations of Irreversible Processes
• Path integrals for stochastic hybrid reaction-diffusion processes