The Most Probable Transition Paths of Stochastic Dynamical Systems: A Sufficient and Necessary Characterization

TL;DR

This paper establishes a complete characterization of the most probable transition paths in stochastic systems with Brownian noise, showing they are governed by a first-order differential equation, simplifying analysis and computation.

Contribution

It provides a necessary and sufficient condition for transition paths, linking the Onsager-Machlup functional to a first-order ODE, extending previous Euler-Lagrange based descriptions.

Findings

Most probable paths are determined by a first-order ODE under certain conditions.

Analytical derivation for linear systems and Hongler's model confirms the approach.

Numerical experiments demonstrate the effectiveness of the first-order characterization.

Abstract

The most probable transition paths of a stochastic dynamical system are the global minimizers of the Onsager-Machlup action functional and can be described by a necessary but not sufficient condition, the Euler-Lagrange equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devoted to showing a sufficient and necessary characterization for the most probable transition paths of stochastic dynamical systems with Brownian noise. We prove that, under appropriate conditions, the most probable transition paths are completely determined by a first-order ordinary differential equation. The equivalence is established by showing that the Onsager-Machlup action functional of the original system can be derived from the corresponding Markovian bridge process. For linear stochastic systems and the nonlinear Hongler's model, the first-order differential equations determining the most probable transition paths are shown analytically to imply the Euler-Lagrange equations of the Onsager-Machlup functional. For general nonlinear systems, the determining first-order differential equations can be approximated, in a short time or for the small noise case. Some numerical experiments are presented to illustrate our results.