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

Abstract

Let $\left(u\right(t,x),t\ge 0,x\in R^d)$ 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 $F_R\left(t\right)=\frac{1}{{\sigma }_R}{\int }_{\left|x\right|\le R}\left(u\right(t,x)-1)dx$, where ${\sigma }_R^2=E{\left({\int }_{\left|x\right|\le R}\left(u\right(t,x)-1)dx\right)}^2$. It is known that, when R goes to infinity, $F_R\left(t\right)$ converges in law to a standard Gaussian random variable Z. We show that the spatial average $F_R\left(t\right)$ is actually asymptotic independent by the solution itself, at any time and at any point in space, meaning that the random vector $\left(F_R\right(t),u(t,x_0\left)\right)$ converges in distribution, as $R\to \infty$, to $(Z,u(t,x_0\left)\right)$, 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.