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

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.