On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917696721911808 |
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| author | Weisbart, David |
| author_facet | Weisbart, David |
| contents | For each prime $p$, a Vladimirov operator with a positive exponent specifies a $p$-adic diffusion equation and a measure on the Skorokhod space of $p$-adic paths. The product, $P$, of these measures with fixed exponent is a probability measure on the product of the $p$-adic path spaces. The adelic paths have full measure if and only if the sum, $σ$, of the diffusion constants is finite. Finiteness of $σ$ implies that there is an adelic Vladimirov operator, $Δ_{\mathbb A}$, and an associated diffusion equation whose fundamental solution gives rise to the measure induced by $P$ on an adelic Skorokhod space. For a wide class of potentials, the dynamical semigroups associated to adelic Schrödinger operators with free part $Δ_{\mathbb A}$ have path integral representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_07809 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion Weisbart, David Probability 60B05, 60G22, 60H05, 11R56 For each prime $p$, a Vladimirov operator with a positive exponent specifies a $p$-adic diffusion equation and a measure on the Skorokhod space of $p$-adic paths. The product, $P$, of these measures with fixed exponent is a probability measure on the product of the $p$-adic path spaces. The adelic paths have full measure if and only if the sum, $σ$, of the diffusion constants is finite. Finiteness of $σ$ implies that there is an adelic Vladimirov operator, $Δ_{\mathbb A}$, and an associated diffusion equation whose fundamental solution gives rise to the measure induced by $P$ on an adelic Skorokhod space. For a wide class of potentials, the dynamical semigroups associated to adelic Schrödinger operators with free part $Δ_{\mathbb A}$ have path integral representations. |
| title | On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion |
| topic | Probability 60B05, 60G22, 60H05, 11R56 |
| url | https://arxiv.org/abs/2007.07809 |