The minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform mesh, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are 1/2 and 1, respectively. Our main result also reveals the convergence of the stochastic θ-method for SDEs with small noise in terms of large deviations. Numerical experiments are reported to verify the theoretical results.

Publication:

IMA JOURNAL OF NUMERICAL ANALYSIS

http://dx.doi.org/10.1093/imanum/drae038

Author:

Jialin Hong

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Diancong Jin

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China and Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

Corresponding author

jindc@hust.edu.cn

Derui Sheng

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon 999077, Hong Kong