Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have, loosely speaking, been investigated since 2003 and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions. In the recent article (Conus et al. in Ann Appl Probab 29(2):653–716, 2019) this weak convergence problem has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome this weak convergence problem in the case of a class of time-discrete Euler-type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild Itô-type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.

具有平滑和规则非线性的半线性随机演化方程（SEE）的时间离散数值逼近的强收敛速度在文献中已广为人知。松散地说，此类SEE的时间离散数值逼近的弱收敛速度自2003年以来就进行了研究，并且距离人们的理解还很遥远：大致而言，对于抛物型SEE的时间离散数值逼近，基本上没有明显的弱收敛速度。具有非线性扩散系数函数。在最近的文章中（Conus等人，在Ann Appl Probab 29（2）：653-716，2019中），在具有非线性扩散系数函数的半线性SEE的空间光谱Galerkin逼近的情况下，已经解决了该弱收敛问题。在本文中，我们克服了在一类时间离散的Euler型逼近方法（包括指数和线性隐式Euler逼近作为特殊情况）的情况下的弱收敛问题，尤其是，对于以下情况，我们建立了基本尖锐的弱收敛速度具有非线性扩散系数函数的半线性SEE的线性-隐式Euler近似。我们方法的关键要素是应用温和的Itô型公式以及使用时离散数值逼近过程的适当半线性积分。我们为具有非线性扩散系数函数的半线性SEE的线性-隐式Euler逼近建立了本质上较弱的收敛速度。我们方法的关键要素是应用温和的Itô型公式以及使用时离散数值逼近过程的适当半线性积分。我们为具有非线性扩散系数函数的半线性SEE的线性-隐式Euler逼近建立了本质上很弱的收敛速度。我们方法的关键要素是应用温和的Itô型公式以及使用时离散数值逼近过程的适当半线性积分。