Feynman-Kac formula for stochastic hybrid systems
File(s)
Published version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
We derive a Feynman-Kac formula for functionals of a stochastic hybrid system evolving according to a piecewise deterministic Markov process. We first derive a stochastic Liouville equation for the moment generator of the stochastic functional, given a particular realization of the underlying discrete Markov process; the latter generates transitions between different dynamical equations for the continuous process. We then analyze the stochastic Liouville equation using methods recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment generating function, averaged with respect to realizations of the discrete Markov process. The resulting Feynman-Kac formula takes the form of a differential Chapman-Kolmogorov equation. We illustrate the theory by calculating the occupation time for a one-dimensional velocity jump process on the infinite or semi-infinite real line. Finally, we present an alternative derivation of the Feynman-Kac formula based on a recent path-integral formulation of stochastic hybrid systems.
Date Issued
2017-01
Date Acceptance
2017-01-01
Citation
Physical Review E, 2017, 95 (1)
ISSN
2470-0045
Publisher
American Physical Society (APS)
Journal / Book Title
Physical Review E
Volume
95
Issue
1
Copyright Statement
©2017 American Physical Society
Identifier
http://dx.doi.org/10.1103/physreve.95.012138
Publication Status
Published
Article Number
012138
Date Publish Online
2017-01-23