The Most Probable Transition Paths of Stochastic Dynamical Systems: Equivalent Description and Characterization

Authors:
Preprint
Publisher website
Google Scholar
Abstract
This work is devoted to show an equivalent description for the most probable transition paths of stochastic dynamical systems with Brownian noise, based on the theory of Markovian bridges. The equivalence is proved by showing the relationships between Markovian bridge measures and the Onsgaer-Machlup action functional. This cannot be done by the existing methods because Markovian bridge measures are no longer quasi translation invariant. We develop a new method to handle this problem. The most probable transition path for a stochastic dynamical system is the minimizer of the Onsager-Machlup action functional, and thus determined by the Euler-Lagrange equation (a second order differential equation with initial-terminal conditions) via a variational principle. After showing that the Onsager-Machlup action functional can be derived from a Markovian bridge process, we first demonstrate that, for some special cases (one is a class of linear stochastic systems, another one is the class of general stochastic systems with small noise), the most probable transition paths can be determined by first order deterministic differential equations with only initial conditions. Though for general such equations do not have analytical representations, the most probable transition paths can be well approximated by solving a first order differential equation or an integro differential equation on a certain time interval. Finally, we illustrate our results with several examples.

This publication has 26 references indexed in Scilit: