Weak approximation for stochastic reaction-diffusion equation near sharp interface limit

Abstract: 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.