Brownian Motion in a Vector Space over a Local Field is a Scaling Limit
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910434934652928 |
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| author | Pierce, Tyler Rajkumar, Rahul Stine, Andrea Weisbart, David Yassine, Adam M. |
| author_facet | Pierce, Tyler Rajkumar, Rahul Stine, Andrea Weisbart, David Yassine, Adam M. |
| contents | For any natural number $d$, the Vladimirov-Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a $d$-dimensional vector space $V$ over a local field $K$. Just as the Laplace operator on $L^2(\mathbb R^d)$ is the infinitesimal generator of Brownian motion with state space $\mathbb R^d$, the Vladimirov-Taibleson operator on $L^2(V)$ is the infinitesimal generator of real-time Brownian motion with state space $V$. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted $V$ to be the $p$-adic numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02502 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brownian Motion in a Vector Space over a Local Field is a Scaling Limit Pierce, Tyler Rajkumar, Rahul Stine, Andrea Weisbart, David Yassine, Adam M. Probability 60F17 (primary), 60G22, 11F85 (secondary) For any natural number $d$, the Vladimirov-Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a $d$-dimensional vector space $V$ over a local field $K$. Just as the Laplace operator on $L^2(\mathbb R^d)$ is the infinitesimal generator of Brownian motion with state space $\mathbb R^d$, the Vladimirov-Taibleson operator on $L^2(V)$ is the infinitesimal generator of real-time Brownian motion with state space $V$. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted $V$ to be the $p$-adic numbers. |
| title | Brownian Motion in a Vector Space over a Local Field is a Scaling Limit |
| topic | Probability 60F17 (primary), 60G22, 11F85 (secondary) |
| url | https://arxiv.org/abs/2405.02502 |