Brownian Motion in a Vector Space over a Local Field is a Scaling Limit

TL;DR

This paper establishes that Brownian motion on a vector space over a local field can be obtained as a scaling limit of discrete random walks, extending previous results from p-adic numbers to more general local fields.

Contribution

It generalizes the connection between discrete random walks and Brownian motion from p-adic fields to arbitrary local fields, deepening the analogy with classical diffusion.

Findings

Brownian motion on local fields is a scaling limit of discrete random walks.

The Vladimirov-Taibleson operator acts as the generator of this Brownian motion.

Extension of previous p-adic results to general local fields.

Abstract

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.