Geometric Ergodicity and Optimal Error Estimates for a Class of Novel Tamed Schemes to Super-linear Stochastic PDEs

2025/02/26 by Zhihui Liu, Jie Shen, Liu, Zhihui +1 · 2 citations
Decision Sciences · Economics, Econometrics and Finance · Physics and Astronomy · #65M60 #FOS: Mathematics #Numerical Analysis (math.NA) #Primary 60H35 #Probabilistic and Robust Engineering Design #Secondary 60H15 #Stochastic processes and financial applications #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2502.19117

openalex publication_date 2025/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a class of novel tamed schemes that can preserve the original Lyapunov functional for super-linear stochastic PDEs (SPDEs), including the stochastic Allen--Cahn equation, driven by multiplicative or additive noise, and provide a rigorous analysis of their long-time unconditional stability. We also show that the corresponding Galerkin-based fully discrete tamed schemes inherit the geometric ergodicity of the SPDEs and establish their convergence towards the SPDEs with optimal strong rates in both the multiplicative and additive noise cases.

Cited by

• Ergodicity and invariant measure approximation of the stochastic Cahn-Hilliard equation via an explicit fully discrete scheme
• A modified tamed scheme for stochastic differential equations with superlinear drifts

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