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Izv. RAN. Ser. Mat., 1997, Volume 61, Issue 3, Pages 75–90 (Mi izv126)  

This article is cited in 21 scientific papers (total in 21 papers)

$p$-adic Brownian motion

A. Kh. Bikulova, I. V. Volovichb

a Togliatti State Polytechnic Institute
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract:$p$-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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English version:
Izvestiya: Mathematics, 1997, 61:3, 537–552

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Document Type: Article
MSC: 60J65, 60G20, 60G17
Received: 17.10.1996

Citation: A. Kh. Bikulov, I. V. Volovich, “$p$-adic Brownian motion”, Izv. RAN. Ser. Mat., 61:3 (1997), 75–90; Izv. Math., 61:3 (1997), 537–552

Citation in format AMSBIB
\by A.~Kh.~Bikulov, I.~V.~Volovich
\paper $p$-adic Brownian motion
\jour Izv. RAN. Ser. Mat.
\yr 1997
\vol 61
\issue 3
\pages 75--90
\jour Izv. Math.
\yr 1997
\vol 61
\issue 3
\pages 537--552

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    This publication is cited in the following articles:
    1. A. Kh. Bikulov, “Stochastic $p$-adic equations of mathematical physics”, Theoret. and Math. Phys., 119:2 (1999), 594–604  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. S. V. Lyudkovskii, “Quasi-invariant and pseudo-differentiable measures with values in non-Archimedean fields on a non-Archimedean Banach space”, J. Math. Sci., 128:6 (2005), 3428–3460  mathnet  crossref  mathscinet  zmath  elib  elib
    3. V. S. Vladimirov, “Tables of Integrals of Complex-Valued Functions of $p$-Adic Arguments”, Proc. Steklov Inst. Math., 284, suppl. 2 (2014), S1–S59  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. S. V. Lyudkovskii, “Topological Transformation Groups of Manifolds over Non-Archimedean Fields, Their Representations, and Quasi-Invariant Measures, II”, Journal of Mathematical Sciences, 150:4 (2008), 2123–2223  mathnet  crossref  mathscinet  elib
    5. Albeverio S., Khrennikov A.Yu., Shelkovich V.M., “Harmonic analysis in the p-adic Lizorkin spaces: Fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems”, Journal of Fourier Analysis and Applications, 12:4 (2006), 393–425  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    6. Khrennikov A.Y., Shelkovich V.M., “Distributional asymptotics and p-adic Tauberian and Shannon-Kotelnikov theorems”, Asymptotic Analysis, 46:2 (2006), 163–187  mathscinet  zmath  isi  elib
    7. Khrennikov A.Y., Kozyrev S.V., “Ultrametric random field”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 9:2 (2006), 199–213  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. S. A. Albeverio, Ya. I. Belopol'skaya, “Jump processes in $Q_p$ associated with nonlinear pseudo-differential equations”, J. Math. Sci. (N. Y.), 152:6 (2008), 799–816  mathnet  crossref  elib  elib
    9. Kenji Kamizono, “Symmetric stochastic integrals with respect top-adic Brownian motion†”, Stochastics An International Journal of Probability and Stochastic Processes, 79:6 (2007), 523  crossref  mathscinet  zmath  scopus  scopus
    10. S. V. Kozyrev, “Methods and Applications of Ultrametric and $p$-Adic Analysis: From Wavelet Theory to Biophysics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S1–S84  mathnet  crossref  crossref  zmath  isi  elib
    11. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, “On p-adic mathematical physics”, P-Adic Num Ultrametr Anal Appl, 1:1 (2009), 1  crossref  mathscinet  scopus  scopus
    12. Proc. Steklov Inst. Math., 265 (2009), 115–130  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    13. Shelkovich, V, “p-Adic Haar Multiresolution Analysis and Pseudo-Differential Operators”, Journal of Fourier Analysis and Applications, 15:3 (2009), 366  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    14. V. M. Shelkovich, “$p$-adic evolution pseudo-differential equations and $p$-adic wavelets”, Izv. Math., 75:6 (2011), 1249–1278  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. A. YU. KHRENNIKOV, S. V. KOZYREV, K. OLESCHKO, A. G. JARAMILLO, M. de JESÚS CORREA LÓPEZ, “APPLICATION OF p-ADIC ANALYSIS TO TIME SERIES”, Infin. Dimens. Anal. Quantum. Probab. Relat. Top, 16:04 (2013), 1350030  crossref  mathscinet  zmath  isi  scopus  scopus
    16. W. A. Zúñiga-Galindo, “The Non-Archimedean Stochastic Heat Equation Driven by Gaussian Noise”, J Fourier Anal Appl, 21:3 (2015), 600  crossref  mathscinet  scopus  scopus
    17. E. I. Zelenov, “The $p$-adic law of large numbers”, Izv. Math., 80:3 (2016), 489–499  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. E. I. Zelenov, “$p$-adic Brownian motion”, Izv. Math., 80:6 (2016), 1084–1093  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    19. A. Kh. Bikulov, A. P. Zubarev, “Polnye sistemy sobstvennykh funktsii operatora Vladimirova v $L^{2}(B_r)$ i $L^{2}(\mathbb{Q}_{p})$”, Fundament. i prikl. matem., 21:3 (2016), 39–56  mathnet
    20. Zuniga-Galindo W.A., “Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields”, J. Fourier Anal. Appl., 23:2 (2017), 288–323  crossref  mathscinet  zmath  isi  scopus
    21. Dragovich B., Khrennikov A.Yu., Kozyrev S.V., Volovich I.V., Zelenov E.I., “P-Adic Mathematical Physics: the First 30 Years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121  crossref  mathscinet  zmath  isi  scopus  scopus
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