Skip to main content
SpringerLink
Log in

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

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Albeverio, S., Karwowski, W.: A random walk on \(p\)-adics—the generator and its spectrum. Stoch. Process. Appl. 53, 1–22 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  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)

  6. Bakken, E., Weisbart, D.: Continuous time \(p\)-adic random walks and their path integrals. J. Theor. Probab. 32, 781–805 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)

    MATH  Google Scholar 

  9. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

  13. Hung, L., Lapidus, M.L.: Nonarchimedean Cantor set and string. J. Fixed Point Theory Appl. 3(1), 181–190 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  17. Kochubei, A.N.: Parabolic equations over the field of \(p\)-adic numbers. Math. USSR Izvestiya 39, 1263–1280 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  18. Varadarajan, V.S.: Path integrals for a class of \(p\)-adic Schrödinger equations. Lett. Math. Phys. 39(2), 97–106 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vladimirov, V.S.: Generalized functions over the field of \(p\)-adic numbers. Russ. Math. Surv. 43(5), 19–64 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vladimirov, V.S.: On the spectrum of some pseudo-differential operators over \(p\)-adic number field. Algebra Anal. 2, 107–124 (1990)

    MATH  Google Scholar 

  21. Vladimirov, V.S., Volovich, I.V.: \(p\)-adic quantum mechanics. Commun. Math. Phys. 123(4), 659–676 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Vladimirov, V.S., Volovich, I.V.: \(p\)-adic Schrödinger-type equation. Lett. Math. Phys. 18(1), 43–53 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Vladimirov, V.S., Volovich, I.V., Zelenov, E.l: \(p\)-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1994)

    Book  MATH  Google Scholar 

  24. Zelenov, E.I.: \(p\)-adic path integrals. J. Math. Phys. 32, 147–152 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Norwegian Defence Research Establishment, Kjeller, Norway

    Erik Bakken

  2. Department of Mathematics, University of California Riverside, Riverside, CA, USA

    David Weisbart

Authors
  1. Erik Bakken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Weisbart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Weisbart.

Additional information

Communicated by H. Duminil-Copin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bakken, E., Weisbart, D. p-Adic Brownian Motion as a Limit of Discrete Time Random Walks. Commun. Math. Phys. 369, 371–402 (2019). https://doi.org/10.1007/s00220-019-03447-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03447-y

Search

Navigation