Global Geometry within an SPDE Well-Posedness Problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914202186153984 |
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| author | Chen, Hongyi Ouyang, Cheng |
| author_facet | Chen, Hongyi Ouyang, Cheng |
| contents | On a closed Riemannian manifold, we construct a family of intrinsic Gaussian noises indexed by a regularity parameter $α\geq0$ to study the well-posedness of the parabolic Anderson model. We show that with rough initial conditions, the equation is well-posed assuming non-positive curvature with a condition on $α$ similar to that of Riesz kernel-correlated noise in Euclidean space. Non-positive curvature was used to overcome a new difficulty introduced by non-uniqueness of geodesics in this setting, which required exploration of global geometry. The well-posedness argument also produces exponentially growing in time upper bounds for the moments. Using Feynman-Kac formula for moments, we also obtain exponentially growing in time second moment lower bounds for our solutions with bounded initial condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_04572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Geometry within an SPDE Well-Posedness Problem Chen, Hongyi Ouyang, Cheng Probability Analysis of PDEs Differential Geometry On a closed Riemannian manifold, we construct a family of intrinsic Gaussian noises indexed by a regularity parameter $α\geq0$ to study the well-posedness of the parabolic Anderson model. We show that with rough initial conditions, the equation is well-posed assuming non-positive curvature with a condition on $α$ similar to that of Riesz kernel-correlated noise in Euclidean space. Non-positive curvature was used to overcome a new difficulty introduced by non-uniqueness of geodesics in this setting, which required exploration of global geometry. The well-posedness argument also produces exponentially growing in time upper bounds for the moments. Using Feynman-Kac formula for moments, we also obtain exponentially growing in time second moment lower bounds for our solutions with bounded initial condition. |
| title | Global Geometry within an SPDE Well-Posedness Problem |
| topic | Probability Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2502.04572 |