$p$-Adic Brownian Motion is a Scaling Limit

TL;DR

This paper proves that $p$-adic Brownian motions can be obtained as scaling limits of discrete random walks on groups, extending previous results to all positive exponents and non-compact time intervals.

Contribution

It simplifies the proof of convergence, removes previous restrictions on the Vladimirov operator's exponent, and broadens the scope to include all positive exponents and unbounded time intervals.

Findings

Established convergence of discrete random walks to $p$-adic Brownian motion for all positive exponents.

Removed the restriction to compact time intervals in the convergence proof.

Provided moment estimates for the discrete processes of independent interest.

Abstract

A $p$-adic Brownian motion is a continuous time stochastic process in a $p$-adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in $(1, \infty)$, and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest.