2021/04/14 by Yuanfei Huang, Huang, Yuanfei, Qiao Huang +3 · 3 citations
Economics, Econometrics and Finance · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Complex Systems and Time Series Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2104.06864
openalex publication_date 2021/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.