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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.20147 |
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| _version_ | 1866911415344824320 |
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| author | Geng, Xi Wang, Sheng Xu, Weijun |
| author_facet | Geng, Xi Wang, Sheng Xu, Weijun |
| contents | We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability one, the solution $u$ to PAM with constant initial data has pointwise growth asymptotics \[ u(t,x)\sim e^{L^{*}t^{5/3}+o(t^{5/3})} \] as $t \rightarrow +\infty$. Both the power $t^{5/3}$ on the exponential and the exact value of $L^*$ are different from their counterparts in the Euclidean situation. They are determined through an explicit optimisation procedure. Our proof relies on certain fine localisation techniques, which also reveals a stronger non-Euclidean localisation mechanism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics Geng, Xi Wang, Sheng Xu, Weijun Probability Analysis of PDEs We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability one, the solution $u$ to PAM with constant initial data has pointwise growth asymptotics \[ u(t,x)\sim e^{L^{*}t^{5/3}+o(t^{5/3})} \] as $t \rightarrow +\infty$. Both the power $t^{5/3}$ on the exponential and the exact value of $L^*$ are different from their counterparts in the Euclidean situation. They are determined through an explicit optimisation procedure. Our proof relies on certain fine localisation techniques, which also reveals a stronger non-Euclidean localisation mechanism. |
| title | Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2506.20147 |