p-Adic Brownian Motion as a Limit of Discrete Time Random Walks

The p-adic diffusion equation is a pseudo differential equation that is formally analogous to the real diffusion equation. The fundamental solutions to pseudo differential equations that generalize the p-adic diffusion equation give rise to p-adic Brownian motions. We show that these stochastic processes are similar to real Brownian motion in that they arise as limits of discrete time random walks on grids. While similar to those in the real case, the random walks in the p-adic setting are necessarily non-local. The study of discrete time random walks that converge to Brownian motion provides intuition about Brownian motion that is important in applications and such intuition is now available in a non-Archimedean setting.