Weak Convergence for the Fourth-Order Stochastic Heat Equation with Fractional Noises

In this paper, we study a fourth-order stochastic heat equation with homogeneous Neumann boundary conditions and double-parameter fractional noises. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}$$\end{document} which approximates the noises. Then we provided sufficient conditions ensuring that the real-valued mild solution of the fourth-order stochastic heat equation driven by this family of noises converges in law, in the space of C([0,T]×[0,π])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C([0,T]\times [0,\pi ])$$\end{document} of continuous functions, to the solution of a class of fourth-order stochastic heat equation driven by fractional noises.