Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE

Abstract

Approximating the invariant measure and the expectation of the functionals for parabolic stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients is an active research area and is far from being well understood. In this article, we study such problem in terms of a full discretization based on the spectral Galerkin method and the temporal implicit Euler scheme. By deriving the a priori estimates and regularity estimates of the numerical solution via a variational approach and Malliavin calculus, we establish the sharp weak convergence rate of the full discretization. When the SPDE admits a unique $V$-uniformly ergodic invariant measure, we prove that the invariant measure can be approximated by the full discretization. The key ingredients lie on the time-independent weak convergence analysis and time-independent regularity estimates of the corresponding Kolmogorov equation. Finally, numerical experiments confirm the theoretical findings.

47 pages

References in corpus (5)

• Strong Approximation of Stochastic Allen-Cahn Equation with White Noise
• Analysis of Some Splitting Schemes for the Stochastic Allen-Cahn Equation
• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• Numerical analysis of a full discretization for stochastic Cahn--Hilliard equation driven by additive noise
• Strong and weak convergence rates of finite element method for stochastic partial differential equation with non-sided Lipschitz coefficient

Cited by in corpus (6)

• Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise
• Numerical analysis of a full discretization for stochastic Cahn--Hilliard equation driven by additive noise
• Error estimates of semi-discrete and fully discrete finite element methods for the Cahn-Hilliard-Cook equation
• Strong and weak convergence rates of finite element method for stochastic partial differential equation with non-sided Lipschitz coefficient
• Finite-Volume approximation of the invariant measure of a viscous stochastic scalar conservation law
• Approximation of Invariant Measures for Stochastic Differential Equations with Piecewise Continuous Arguments via Backward Euler Method