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

Abstract Vladimirov defined an operator on balls in $$\mathbb {Q}_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> , 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 $$\mathbb {Z}_{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> . 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.