Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE
For researchers studying numerical approximation of invariant measures for SPDEs, this work provides rigorous convergence rates and measure approximation results for a class of non-globally Lipschitz problems, which is an incremental advance over existing methods.
This paper establishes the sharp weak convergence rate of a full discretization (spectral Galerkin + implicit Euler) for parabolic SPDEs with non-globally Lipschitz coefficients, and proves that the invariant measure can be approximated by the discretization. Numerical experiments confirm the theoretical results.
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.