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

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.