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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.11323 |
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| _version_ | 1866916465089708032 |
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| author | Zhang, Wanying Zhang, Yong Li, Jingyu |
| author_facet | Zhang, Wanying Zhang, Yong Li, Jingyu |
| contents | Let $\{u(t,x)\}_{t>0,x\in{{\mathbb R}^{d}}}$ denote the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a multiplicative noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$. Let $S_{N,t}:=N^{-d}\int_{{[0,N]}^d}{[U(t,x)-1]}{\rm d}x$ denote the spatial average on ${{\mathbb R}^{d}}$. We obtain various functional central limit theorems (CLTs) for spatial averages based on the quantitative analysis of $f$ and spatial dimension $d$. In particular, when $f$ is given by Riesz kernel, that is, $f({\rm x})={\Vert x \Vert}^{-β}{\rm d}x$, $β\in(0,2\wedge d)$, the functional CLT is also based on the index $β$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11323 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Functional central limit theorems for spatial averages of the parabolic Anderson model with delta initial condition in dimension $d\geq 1$ Zhang, Wanying Zhang, Yong Li, Jingyu Probability Let $\{u(t,x)\}_{t>0,x\in{{\mathbb R}^{d}}}$ denote the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a multiplicative noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$. Let $S_{N,t}:=N^{-d}\int_{{[0,N]}^d}{[U(t,x)-1]}{\rm d}x$ denote the spatial average on ${{\mathbb R}^{d}}$. We obtain various functional central limit theorems (CLTs) for spatial averages based on the quantitative analysis of $f$ and spatial dimension $d$. In particular, when $f$ is given by Riesz kernel, that is, $f({\rm x})={\Vert x \Vert}^{-β}{\rm d}x$, $β\in(0,2\wedge d)$, the functional CLT is also based on the index $β$. |
| title | Functional central limit theorems for spatial averages of the parabolic Anderson model with delta initial condition in dimension $d\geq 1$ |
| topic | Probability |
| url | https://arxiv.org/abs/2208.11323 |