Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients

摘要

Strong convergence rates for time-discrete numerical approximations ofsemilinear stochastic evolution equations (SEEs) with smooth and regularnonlinearities are well understood in the literature. Weak convergence ratesfor time-discrete numerical approximations of such SEEs have been investigatedsince about 12 years and are far away from being well understood: roughlyspeaking, no essentially sharp weak convergence rates are known fortime-discrete numerical approximations of parabolic SEEs with nonlineardiffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weakapproximation of stochastic partial differential equations: the nonlinear case,Math. Comp. 80 (2011), no. 273, 89-117] for details. In the recent article [D.Conus, A. Jentzen & R. Kurniawan, Weak convergence rates of spectral Galerkinapproximations for SPDEs with nonlinear diffusion coefficients,arXiv:1408.1108] the weak convergence problem emerged from Debussche's articlehas been solved in the case of spatial spectral Galerkin approximations forsemilinear SEEs with nonlinear diffusion coefficient functions. In this articlewe overcome the problem emerged from Debussche's article in the case of a classof time-discrete Euler-type approximation methods (including exponential andlinear-implicit Euler approximations as special cases) and, in particular, weestablish essentially sharp weak convergence rates for linear-implicit Eulerapproximations of semilinear SEEs with nonlinear diffusion coefficientfunctions. Key ingredients of our approach are applications of a mild It^otype formula and the use of suitable semilinear integrated counterparts of thetime-discrete numerical approximation processes.