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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.11147 |
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| _version_ | 1866909172602241024 |
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| author | Tudor, Ciprian A Zurcher, Jérémy |
| author_facet | Tudor, Ciprian A Zurcher, Jérémy |
| contents | 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 $F_{R}(t)= \frac{1}{σ_R}\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx$, where $σ^2_R= \mathbf{E} \left(\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx\right)^2$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11147 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The spatial average of solutions to SPDEs is asymptotically independent of the solution Tudor, Ciprian A Zurcher, Jérémy Probability 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 $F_{R}(t)= \frac{1}{σ_R}\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx$, where $σ^2_R= \mathbf{E} \left(\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx\right)^2$. 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. |
| title | The spatial average of solutions to SPDEs is asymptotically independent of the solution |
| topic | Probability |
| url | https://arxiv.org/abs/2404.11147 |