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

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 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, in some special cases, the most probable transition paths can be determined by first order deterministic differential equations with only a initial condition. Then we show that for general nonlinear stochastic systems with small noise, 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.

An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumps

“…It may be because the noise in the non-Gaussian Lévy case is with exponentially light jumps. And when the noise intensity tends to zero, the deterministic velocity field has the main impact on the shape of the most likely transition path as the Gaussian case, for related work, see [41].…”

Data-driven method to learn the most probable transition pathway and stochastic differential equations

“…These transition paths can be considered as the simulation of the original stochastic differential equations (2.1). The most probable transition pathway of stochastic differential equations (2.1) and (2.10) coincides [25]. In next section, we will use the expectation of transition paths as the most probable transition pathway observation data to extract the drift function.…”