The spatial average of solutions to SPDEs is asymptotically independent of the solution
Résumé
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 $\sigma ^{2}_{R}= \E \left( \int_{ \vert x\vert \leq R} \left( u(t,x)-1\right) dx\right) ^{2}.$ 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.
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