Geodesic
$p$-adic Brownian motion
Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 537-552.

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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. 
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A. Kh. Bikulov; I. V. Volovich. $p$-adic Brownian motion. Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 537-552. http://geodesic.mathdoc.fr/item/IM2_1997_61_3_a2/

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