Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Abstract

Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak convergence problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Itô type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak convergence problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kind of spatial, temporal, and noise numerical approximations for semilinear SEEs.

References in corpus (7)

• An exponential integrator scheme for time discretization of nonlinear stochastic wave equation
• Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE
• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions
• Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values
• Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces

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• Weak error analysis for semilinear stochastic Volterra equations with additive noise
• Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise
• Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces
• Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values
• Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations
• Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces
• Weak error estimates for trajectories of SPDEs for Spectral Galerkin discretization
• Optimal Adaptive Ridgelet Schemes for Linear Transport Equations
• Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient
• On the differentiability of solutions of stochastic evolution equations with respect to their initial values
• Weak error analysis via functional Itô calculus
• Malliavin regularity and weak approximation of semilinear SPDE with Lévy noise