Communications in Mathematical Physics

, Volume 369, Issue 2, pp 371–402 | Cite as

p-Adic Brownian Motion as a Limit of Discrete Time Random Walks

  • Erik Bakken
  • David WeisbartEmail author


The p-adic diffusion equation is a pseudo differential equation that is formally analogous to the real diffusion equation. The fundamental solutions to pseudo differential equations that generalize the p-adic diffusion equation give rise to p-adic Brownian motions. We show that these stochastic processes are similar to real Brownian motion in that they arise as limits of discrete time random walks on grids. While similar to those in the real case, the random walks in the p-adic setting are necessarily non-local. The study of discrete time random walks that converge to Brownian motion provides intuition about Brownian motion that is important in applications and such intuition is now available in a non-Archimedean setting.



  1. 1.
    Albeverio, S., Karwowski, W.: A random walk on \(p\)-adics—the generator and its spectrum. Stoch. Process. Appl. 53, 1–22 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avetisov, V.A., Bikulov, A.Kh.: On the ultrametricity of the uctuation dynamic mobility of protein molecules. Proc. Steklov Inst. Math. 265(1), 75–81 (2009)Google Scholar
  3. 3.
    Avetisov, V.A., Bikulov, A.Kh., Kozyrev, S.V.: Application of \(p\)-adic analysis to models of breaking of replica symmetry. J. Phys. A 32(50), 8785–8791 (1999)Google Scholar
  4. 4.
    Avetisov, V.A., Bikulov, A.Kh., Kozyrev, S.V.: Description of logarithmic relaxation by a model of a hierarchical random walk. Dokl. Akad. Nauk 368(2), 164–167 (1999)Google Scholar
  5. 5.
    Avetisov, V.A., Bikulov, A.Kh., Osipov, V.Al.: \(p\)-adic description of characteristic relaxation in complex systems. J. Phys. A 36(15), 4239–4246 (2003)Google Scholar
  6. 6.
    Bakken, E., Weisbart, D.: Continuous time \(p\)-adic random walks and their path integrals. J. Theor. Probab. 32, 781–805 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bakken, E., Digernes, T., Weisbart, D.: Brownian motion and finite approximations of quantum systems over local fields. Rev. Math. Phys. 29(5), 1750016 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)zbMATHGoogle Scholar
  9. 9.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Chentsov, N.N.: Weak convergence of stochastic processes whose trajectories have no discontinuities of the second kind and the “heuristic” approach to the Kolmogorov-Smirnov tests. Theory Probab. Appl. 1(1), 140–144 (1956)CrossRefGoogle Scholar
  11. 11.
    Dragovich, B., Khrennikov, A.Yu., Kozyrev, S.V., Volovich, I.V.: On \(p\)-adic mathematical physics. \(p\)-Adic Numb. Ultr. Anal. Appl. 1(1), 1–17 (2009)Google Scholar
  12. 12.
    Digernes, T., Varadarajan, V.S., Weisbart, D.: Schrödinger operators on local fields: self-adjointness and path integral representations for propagators. Infinite Dimens. Anal. Quantum Probab. Related Top. 11(4), 495–512 (2008)Google Scholar
  13. 13.
    Hung, L., Lapidus, M.L.: Nonarchimedean Cantor set and string. J. Fixed Point Theory Appl. 3(1), 181–190 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hung, L., Lapidus, M.L.: Self-similar \(p\)-adic fractal strings and their complex dimensions. \(p\)-Adic Numbers Ultrametr. Anal. Appl. 1(2), 167–180 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ismagilov, R.S.: On the spectrum of the self adjoint operator in \(L^2(K)\) where \(K\) is a local field; an analog of the Feynman–Kac formula. Theor. Math. Phys. 89, 1024–1028 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Khrennikov, A., Kozyrev, S., Zúñiga-Galindo, W.A.: Ultrametric Equations and Its Applications. Encyclopedia of Mathematics and Its Applications, 168th edn. Cambridge University Press, Cambridge (2018)zbMATHGoogle Scholar
  17. 17.
    Kochubei, A.N.: Parabolic equations over the field of \(p\)-adic numbers. Math. USSR Izvestiya 39, 1263–1280 (1992)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Varadarajan, V.S.: Path integrals for a class of \(p\)-adic Schrödinger equations. Lett. Math. Phys. 39(2), 97–106 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vladimirov, V.S.: Generalized functions over the field of \(p\)-adic numbers. Russ. Math. Surv. 43(5), 19–64 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vladimirov, V.S.: On the spectrum of some pseudo-differential operators over \(p\)-adic number field. Algebra Anal. 2, 107–124 (1990)zbMATHGoogle Scholar
  21. 21.
    Vladimirov, V.S., Volovich, I.V.: \(p\)-adic quantum mechanics. Commun. Math. Phys. 123(4), 659–676 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vladimirov, V.S., Volovich, I.V.: \(p\)-adic Schrödinger-type equation. Lett. Math. Phys. 18(1), 43–53 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vladimirov, V.S., Volovich, I.V., Zelenov, E.l: \(p\)-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1994)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zelenov, E.I.: \(p\)-adic path integrals. J. Math. Phys. 32, 147–152 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Norwegian Defence Research EstablishmentKjellerNorway
  2. 2.Department of MathematicsUniversity of California RiversideRiversideUSA

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