The spatial average of solutions to SPDEs is asymptotically independent of the solution

Abstract

Let $\left(u(t,x),t\ge 0,x\in {\mathbb{R}}^d\right)$ be the solution to the stochastic heat or wave equation driven by a Gaussian noise which is white in time and white or correlated with respect to the spatial variable. We consider the spatial average of the solution $F_R(t)=\frac{1}{{\sigma }_R}{\int }_{|x|\le R}\left(u(t,x)-1\right)dx$, where ${\sigma }_R^2=\mathbf{E}{\left({\int }_{|x|\le R}\left(u(t,x)-1\right)dx\right)}^2$. It is known that, when $R$ goes to infinity, $F_R(t)$ converges in law to a standard Gaussian random variable $Z$. We show that the spatial average $F_R(t)$ is actually asymptotic independent by the solution itself, at any time and at any point in space, meaning that the random vector $(F_R(t),u(t,x_0))$ converges in distribution, as $R\to \infty$, to $(Z,u(t,x_0))$, where $Z$ is a standard normal random variable independent of $u(t,x_0)$. By using the Stein-Malliavin calculus, we also obtain the rate of convergence, under the Wasserstein distance, for this limit theorem.