Work

Numerical Unique Ergodicity of Monotone SDEs Driven by Nondegenerate Multiplicative Noise

Year: 2025

Type: article

Abstract: We first establish the unique ergodicity of the stochastic theta method (STM) with $$\theta \in [1/2, 1]$$ for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and $$\theta $$ and deriving a minorization condition for the STM. We then generalize the arguments to the Galerkin-based full discretizations for a class of monotone SPDEs driven by infinite-dimensional nondegenerate multiplicative trace-class noise. Applying these results to the stochastic Allen–Cahn equation indicates that its Galerkin-based full discretizations are uniquely ergodic for any interface thickness. Numerical experiments verify our theoretical results.

Source: Journal of Scientific Computing

DOI: 10.1007/s10915-025-02902-4

Authors: Zhihui Liu, Zhihui Liu, Zhizhou Liu, Zhizhou Liu

Institutions: Hong Kong University of Science and Technology, Southern University of Science and Technology

Language: English

Topic: Numerical methods for differential equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290

Funders: National Natural Science Foundation of China, Basic and Applied Basic Research Foundation of Guangdong Province