Schottky-Invariant $p$-Adic Diffusion Operators
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910725995233280 |
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| author | Bradley, Patrick Erik |
| author_facet | Bradley, Patrick Erik |
| contents | A parametrised diffusion operator on the regular domain $Ω$ of a $p$-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on $Ω$ which are invariant under the Schottky group $Γ$, where integration is against the measure defined by an invariant regular differential 1-form $ω$. It is proven that the space of Schottky invariant $L^2$-functions on $Ω$ outside the zeros of $ω$ has an orthonormal basis consiting of $Γ$-invariant extensions of Kozyrev wavelets which are eigenfunctions of the operator. The eigenvalues are calculated, and it is shown that the heat equation for this operator provides a unique solution for its Cauchy problem with Schottky-invariant continuous initial conditions supportes outside the zero set of $ω$, and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are càdlàg. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17586 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schottky-Invariant $p$-Adic Diffusion Operators Bradley, Patrick Erik Algebraic Geometry Analysis of PDEs Number Theory 14H30 A parametrised diffusion operator on the regular domain $Ω$ of a $p$-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on $Ω$ which are invariant under the Schottky group $Γ$, where integration is against the measure defined by an invariant regular differential 1-form $ω$. It is proven that the space of Schottky invariant $L^2$-functions on $Ω$ outside the zeros of $ω$ has an orthonormal basis consiting of $Γ$-invariant extensions of Kozyrev wavelets which are eigenfunctions of the operator. The eigenvalues are calculated, and it is shown that the heat equation for this operator provides a unique solution for its Cauchy problem with Schottky-invariant continuous initial conditions supportes outside the zero set of $ω$, and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are càdlàg. |
| title | Schottky-Invariant $p$-Adic Diffusion Operators |
| topic | Algebraic Geometry Analysis of PDEs Number Theory 14H30 |
| url | https://arxiv.org/abs/2405.17586 |