It is known that when the diffuse interface thickness $\epsilon$ vanishes, the sharp interface limit of the stochastic reaction-diffusion equation is formally a stochastic geometric flow. To capture and simulate such geometric flow, it is crucial to develop numerical approximations whose error bounds depends on $\frac 1\epsilon$ polynomially. However, due to loss of spectral estimate of the linearized stochastic reaction-diffusion equation, how to get such error bound of numerical approximation has been an open problem. In this paper, we solve this weak error bound problem for stochastic reaction-diffusion equations near sharp interface limit. We first introduce a regularized problem which enjoys the exponential ergodicity. Then we present the regularity analysis of the regularized Kolmogorov and Poisson equations which only depends on $\frac 1{\epsilon}$ polynomially. Furthermore, we establish such weak error bound. This phenomenon could be viewed as a kind of the regularization effect of noise on the numerical approximation of stochastic partial differential equation (SPDE). As a by-product, a central limit theorem of the weak approximation is shown near sharp interface limit. Our method of proof could be extended to a number of other spatial and temporal numerical approximations for semilinear SPDEs.

翻译：众所周知，当扩散界面厚度$\epsilon$趋近于零时，随机反应-扩散方程的尖锐界面极限形式上是随机几何流。为了捕捉和模拟此类几何流，关键在于发展误差界仅依赖于$\frac 1{\epsilon}$多项式量级的数值逼近方法。然而，由于线性化随机反应-扩散方程谱估计性质的缺失，如何获得此类数值逼近的误差界一直是一个悬而未决的问题。本文解决了尖锐界面极限附近随机反应-扩散方程的弱误差界问题。我们首先引入一个具有指数遍历性的正则化问题，随后给出了仅依赖于$\frac 1{\epsilon}$多项式量级的正则化Kolmogorov方程和Poisson方程的正则性分析，并在此基础上建立了弱误差界。这一现象可视为噪声对随机偏微分方程数值逼近的正则化效应。作为副产品，本文还证明了尖锐界面极限附近弱逼近的中心极限定理。本文的证明方法可推广至半线性SPDE的多种空间和时间数值逼近格式。