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

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 L²( Rᵈ) is the infinitesimal generator of Brownian motion with state space Rᵈ, the Vladimirov-Taibleson operator on L²(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.