Geodesic
$p$-Adic Brownian Motion over $\mathbb Q_p$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 125-141

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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. 
@article{TM_2009_265_a9,
     author = {K. Kamizono},
     title = {$p${-Adic} {Brownian} {Motion} over $\mathbb Q_p$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {125--141},
     publisher = {mathdoc},
     volume = {265},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2009_265_a9/}
}
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K. Kamizono. $p$-Adic Brownian Motion over $\mathbb Q_p$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 125-141. http://geodesic.mathdoc.fr/item/TM_2009_265_a9/