The weak averaging principle of stochastic functional partial differential equations with H$\ddot{\text{o}}$lder continuous coefficients and infinite delay

TL;DR

The paper addresses the weak averaging problem for stochastic functional partial differential equations with infinite delay and Hölder continuous coefficients, extending averaging theory beyond Lipschitz settings. It develops a generalized coupling framework together with Galerkin projections to prove weak well-posedness for finite- and infinite-dimensional systems and then establishes convergence in law of the fast-slow SFPDEs to an averaged equation on a fixed horizon $[0,T]$ using time discretization in the spirit of Khasminskii. The main result asserts that for any bounded continuous functional on $C([0,T],U_1)$, $ ext{E}ig[ ext{L}(u^ ext{ε})ig|_{[0,T]}ig] o ext{E}ig[ ext{L}(u^*)ig|_{[0,T]}ig]$ as $ ext{ε} o 0$, provided the initial data converge in the $h$-norm. Applications to stochastic generalized porous media and stochastic reaction-diffusion equations illustrate the method and extend averaging theory to SPDEs with infinite delay and non-Lipschitz coefficients.

Abstract

In this paper, we establish the weak averaging principle for stochastic functional partial differential equations (in short, SFPDEs) with H$\ddot{\text{o}}$lder continuous coefficients and infinite delay by a new generalized coupling approach. Firstly, we rigorously establish the existence and uniqueness of weak solutions for a specific class of finite-dimensional systems by the generalized coupling approach. Then we extend these results to their infinite-dimensional counterparts using the variational approach and Galerkin projection technique. Subsequently, we establish the averaging principle for SFPDEs with infinite delay in the weak sense, i.e., we prove that the solution of the original system converges in law to that of the averaged system on a finite interval $[0,T]$ as the small parameter $\varepsilon\to 0$. To illustrate our findings, we present two applications: stochastic generalized porous media equations and stochastic reaction-diffusion equations.