Stochastic Allen-Cahn equation provides a prototypical class of semilinear SPDEs with non-globally Lipschitz nonlinearities and arises in the modeling of phase transition phenomena under random perturbations. In this talk, I will discuss the weak convergence analysis of a fully discrete approximation for stochastic Allen-Cahn equation driven by multiplicative noise. The numerical scheme combines a drift-implicit Euler method in time with a spectral Galerkin approximation in space.

The main challenges stem from the non-globally Lipschitz drift and the presence of Nemytskii-type multiplicative noise. By establishing suitable regularity estimates for the associated Kolmogorov equations and developing techniques for handling trace terms involving stochastic integrals, we obtain weak convergence rates for the fully discrete scheme. The analysis provides a rigorous framework for weak error estimates of SPDEs with non-globally Lipschitz nonlinearities driven by multiplicative noise.