$p$-adic Brownian motion

A $p$-adic analogue of Brownian motion is constructed and studied. The properties of the trajectories of a $p$-adic Wiener process are studied using Vladimirov's $p$-adic differentiation operator. To construct the $p$-adic Brownian motion we use a $p$-adic analogue of the Paley–Wiener method and a stochastic pseudodifferential equation.