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. 
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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/

[1] S. Albeverio, W. Karwowski, “Diffusion on $p$-adic numbers”, Gaussian random fields (Nagoya, 1990), Ser. Probab. Statist., 1, World Sci. Publ., River Edge, NJ, 1991, 86–99 | MR | Zbl

[2] A. Kh. Bikulov, I. V. Volovich, “$p$-adic Brownian motion”, Izv. Math., 61:3 (1997), 537–552 | DOI | DOI | MR | Zbl

[3] A. Yu. Khrennikov, M. Nilson, $p$-adic deterministic and random dynamics, Math. Appl., 574, Kluwer Acad. Publ., Dordrecht, 2004, xviii+270 pp. | DOI | MR | Zbl

[4] S. N. Evans, “Local fields, Gaussian measures, and Brownian motions”, Topics in probability and Lie groups: boundary theory, CRM Proc. Lecture Notes, 28, Amer. Math. Soc., Providence, RI, 2001, 11–50 | MR | Zbl

[5] A. N. Kochubei, Pseudo-differential equations and stochastics over non-Archimedean fields, Monogr. Textbooks Pure Appl. Math., 244, Marcel Dekker, Inc., New York, 2001, xii+316 pp. | MR | Zbl

[6] V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev, V. A. Osipov, “$p$-adic models of ultrametric diffusion constrained by hierarchical energy landscapes”, J. Phys. A, 35:2 (2002), 177–189 | DOI | MR | Zbl

[7] I. V. Volovich, “$p$-adic space-time and string theory”, Theoret. and Math. Phys., 71:3 (1987), 574–576 | DOI | MR | Zbl

[8] V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-adic analysis and mathematical physics, Series on Soviet and East European Mathematics, 1, World Scientific Publishing Co., Inc., River Edge, NJ, 1994, xx+319 pp. | DOI | MR | MR | Zbl | Zbl

[9] B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, “On $p$-adic mathematical physics”, $p$-Adic Numbers Ultrametric Anal. Appl., 1:1 (2009), 1–17 | DOI | MR | Zbl

[10] V. Anashin, A. Khrennikov, Applied algebraic dynamics, de Gruyter Exp. Math., 49, Walter de Gruyter Co., Berlin, 2009, xxiv+533 pp. | DOI | MR | Zbl

[11] V. Anashin, “The non-Archimedean theory of discrete systems”, Math. Comput. Sci., 6:4 (2012), 375–393 | DOI | MR | Zbl

[12] V. Anashin, A. Khrennikov, E. Yurova, “Ergodicity criteria for non-expanding transformations of 2-adic spheres”, Discrete Contin. Dyn. Syst., 34:2 (2014), 367–377 | DOI | MR | Zbl

[13] A. V. Bulinskii, A. N. Shiryaev, Teoriya sluchainykh protsessov, Fizmatlit, M., 2003, 400 pp.

[14] W. H. Schikhof, Ultrametric calculus. An introduction to $p$-adic analysis, Cambridge Stud. Adv. Math., 4, Cambridge Univ. Press, Cambridge, 1984, viii+306 pp. | MR | Zbl