ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION

doi:10.1007/s10473-025-0318-0

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 1078-1106.doi: 10.1007/s10473-025-0318-0

ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION

Diancong JIN¹, Derui SHENG^2,†

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China; Hubei Key Laboratory of Engineering Modeling and Scientfic Computing, Huazhong University of Science and Technology, Wuhan 430074, China;
2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon 999077, Hong Kong, China

收稿日期:2023-10-12 修回日期:2024-04-22 出版日期:2025-05-25 发布日期:2025-09-30

ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION

Diancong JIN¹, Derui SHENG^2,†

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China; Hubei Key Laboratory of Engineering Modeling and Scientfic Computing, Huazhong University of Science and Technology, Wuhan 430074, China;
2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon 999077, Hong Kong, China

Received:2023-10-12 Revised:2024-04-22 Online:2025-05-25 Published:2025-09-30
Contact: ^†Derui SHENG, E-mail: sdr@lsec.cc.ac.cn
About author:Diancong JIN, E-mail: jindc@hust.edu.cn
Supported by:
National Natural Science Foundation of China (12201228, 12171047) and the Fundamental Research Funds for the Central Universities (3034011102). National Key R&D Program of China (2020YFA0713701).

摘要/Abstract

摘要： In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn-Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the $\Gamma$-convergence of objective functions. This relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-sided Lipschitz continuous, we derive the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. These play important roles in deriving the $\Gamma$-convergence of objective functions.

关键词: large deviations rate function, finite difference method, convergence analysis, $\Gamma$-convergence, stochastic Cahn-Hilliard equation

Abstract: In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn-Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the $\Gamma$-convergence of objective functions. This relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-sided Lipschitz continuous, we derive the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. These play important roles in deriving the $\Gamma$-convergence of objective functions.

Key words: large deviations rate function, finite difference method, convergence analysis, $\Gamma$-convergence, stochastic Cahn-Hilliard equation

中图分类号:

60F10

Diancong JIN, Derui SHENG. ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION[J]. 数学物理学报(英文版), 2025, 45(3): 1078-1106.

Diancong JIN, Derui SHENG. ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION[J]. Acta mathematica scientia,Series B, 2025, 45(3): 1078-1106.

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ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION

ASYMPTOTICS OF LARGE DEVIATIONS OF FINITE DIFFERENCE METHOD FOR STOCHASTIC CAHN-HILLIARD EQUATION

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