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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.20146 |
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| _version_ | 1866908421001838592 |
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| author | Geng, Xi Xu, Weijun |
| author_facet | Geng, Xi Xu, Weijun |
| contents | We establish the second-order moment asymptotics for a parabolic Anderson model $\partial_{t}u=(Δ+ξ)u$ in the hyperbolic space with a regular, stationary Gaussian potential $ξ$. It turns out that the growth and fluctuation asymptotics both are identical to the Euclidean situation. As a result, the solution exhibits the same moment intermittency property as in the Euclidean case. An interesting point here is that the fluctuation exponent is determined by a variational problem induced by the Euclidean (rather than hyperbolic) Laplacian. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parabolic Anderson Model in the Hyperbolic Space. Part I: Annealed Asymptotics Geng, Xi Xu, Weijun Probability We establish the second-order moment asymptotics for a parabolic Anderson model $\partial_{t}u=(Δ+ξ)u$ in the hyperbolic space with a regular, stationary Gaussian potential $ξ$. It turns out that the growth and fluctuation asymptotics both are identical to the Euclidean situation. As a result, the solution exhibits the same moment intermittency property as in the Euclidean case. An interesting point here is that the fluctuation exponent is determined by a variational problem induced by the Euclidean (rather than hyperbolic) Laplacian. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics. |
| title | Parabolic Anderson Model in the Hyperbolic Space. Part I: Annealed Asymptotics |
| topic | Probability |
| url | https://arxiv.org/abs/2506.20146 |