Brownian Motion in the \({\varvec{p}}\) -Adic Integers is a Limit of Discrete Time Random Walks

Abstract

Vladimirov defined an operator on balls in \(\mathbb {Q}_{p}\), the p-adic numbers, analogous to the Laplace operator in the real setting. Kochubei later gave a probabilistic interpretation of this operator. The 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 proves that this process is a limit of discrete-time random walks. It motivates the construction of the Vladimirov–Kochubei operator, provides further intuition about ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group.