Approximation results for weak solutions of stochastic partial differential equations
Xi Lin
TL;DR
The article addresses the challenge of approximating weak solutions of stochastic partial differential equations (SPDEs) by deterministic systems, extending Wong-Zakai type results from finite-dimensional SDEs to SPDEs with spatial structure. The authors combine a Wong-Zakai discretization of the Wiener process with a time-discretized, test-function-based weak formulation, and perform detailed term-by-term analyses to prove convergence. The main contribution is a rigorous result: for every $T>0$, $\\lim_{\delta\to 0} \mathbb{E}[\sup_{0\le t\le T} \|X(t,w)-X_{\delta}(t,w)\|_{L^2(\mathcal{O})}^4] = 0$, thereby extending classical approximation results to weak SPDEs and enabling applications to stochastic degenerate systems such as cross-diffusion ion transport models. This provides a principled tool for understanding stochastic systems where the solution depends on space, with potential impact on analysis methods and future research directions for relaxing assumptions and leveraging entropy-based approaches.
Abstract
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.