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

Abstract: Let $\left(u(t,x), t\geq 0, x\in \mathbb{R}^d\right)$ 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 $F_{R}(t)= \frac{1}{\sigma_R}\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx$, where $\sigma^2_R= \mathbf{E} \left(\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx\right)^2$. 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.