The most probable transition paths of stochastic dynamical systems: a sufficient and necessary characterisation

Abstract

The most probable transition paths (MPTPs) of a stochastic dynamical system are the global minimisers of the Onsager–Machlup action functional and can be described by a necessary but not sufficient condition, the Euler–Lagrange (EL) equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devoted to showing a sufficient and necessary characterisation for the MPTPs of stochastic dynamical systems with Brownian noise. We prove that, under appropriate conditions, the MPTPs 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 MPTPs are shown analytically to imply the EL 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.