$p$-Adic Brownian Motion over $\mathbb Q_p$

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a $p$-adic Brownian motion over $\mathbb Q_p$. First, we construct directly a $p$-adic white noise over $\mathbb Q_p$ by using a specific complete orthonormal system of $\mathbb L^2(\mathbb Q_p)$. A $p$-adic Brownian motion over $\mathbb Q_p$ is then constructed by the Paley–Wiener method. Finally, we introduce a $p$-adic random walk and prove a theorem on the approximation of a $p$-adic Brownian motion by a $p$-adic random walk.