Geodesic
$p$-adic Brownian motion
Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1084-1093

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We define $p$-adic Brownian motion (Wiener process) and study its properties. We construct a presentation of the trajectories of this process by their series expansions with respect to van der Put's basis and show that they are nowhere differentiable functions satisfying the $p$-adic Lipschitz condition of order $1$. We define the $p$-adic Wiener measure on the space of continuous functions and study its properties.
Keywords: $p$-adic numbers, Wiener process, Brownian motion, van der Put's basis, trajectories of the Wiener process. 
@article{IM2_2016_80_6_a4,
     author = {E. I. Zelenov},
     title = {$p$-adic {Brownian} motion},
     journal = {Izvestiya. Mathematics },
     pages = {1084--1093},
     publisher = {mathdoc},
     volume = {80},
     number = {6},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_6_a4/}
}
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E. I. Zelenov. $p$-adic Brownian motion. Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1084-1093. http://geodesic.mathdoc.fr/item/IM2_2016_80_6_a4/