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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.05561 |
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| _version_ | 1866910517582364672 |
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| author | Pierce, Tyler Weisbart, David |
| author_facet | Pierce, Tyler Weisbart, David |
| contents | Vladimirov defined an operator on balls in $\mathbb Q_p$, the $p$-adic numbers, that is analogous to the Laplace operator in the real setting. Kochubei later provided a probabilistic interpretation of the operator. This Vladimirov-Kochubei operator generates a real-time diffusion process in the ring of $p$-adic integers, a Brownian motion in $\mathbb Z_p$. The current work shows that this process is a limit of discrete time random walks. It motivates the construction of the Vladimirov-Kochubei operator, provides further intuition about the properties of ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brownian Motion in the $p$-Adic Integers is a Limit of Discrete Time Random Walks Pierce, Tyler Weisbart, David Probability Mathematical Physics 60B10 Vladimirov defined an operator on balls in $\mathbb Q_p$, the $p$-adic numbers, that is analogous to the Laplace operator in the real setting. Kochubei later provided a probabilistic interpretation of the operator. This Vladimirov-Kochubei operator generates a real-time diffusion process in the ring of $p$-adic integers, a Brownian motion in $\mathbb Z_p$. The current work shows that this process is a limit of discrete time random walks. It motivates the construction of the Vladimirov-Kochubei operator, provides further intuition about the properties of ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group. |
| title | Brownian Motion in the $p$-Adic Integers is a Limit of Discrete Time Random Walks |
| topic | Probability Mathematical Physics 60B10 |
| url | https://arxiv.org/abs/2407.05561 |