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

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 for about two decades 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 [Math. Comp. 80 (2011) 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 kinds of spatial and temporal numerical approximations for semilinear SEEs.