Approximation analysis for weak solutions of stochastic partial differential equations

In probability theory, how to approximate the solution of a stochastic differential equation is an important topic. In Watanabe's classical textbook, by an approximation of the Wiener process, solutions of approximated equations converge to the solution of the stochastic differential equation in probability. In traditional approximation theorems, solutions do not contain the spatial variable. In recent years, stochastic partial differential equations have been playing major roles in probability theory. If the solution is a weak one with the spatial variable, we may not be able to directly apply these classical approximation results. In this work, we try to extend the approximation result to stochastic partial differential equations case. We show that in this case, the approximation result still holds.