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

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.