Skip to main content
Log in

Probability density functions and the dynamics of complex systems associated to some classes of non-archimedean pseudo-differential operators

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

In this article, we study certain p-adic master equations which describe the dynamics of a large class of complex systems such as glasses, macromolecules and proteins. These equations are naturally associated to certain non-archimedean pseudo-differential operators whose symbols are connected via Fourier transform with radial probability density functions defined on the p-adic numbers. We show that the fundamental solutions of these equations are probability measures and determine a convolution semigroup on the p-adic numbers. Also, we show that the classical solution of this equations preserves the mass and satisfies the comparison principle. Moreover, we study some strong Markov processes corresponding to radial probability density functions of linear and logarithmic type.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Albeverio S., Khrennikov A. Yu., Shelkovich V.M.: Theory of \(p\)-adic distributions: linear and nonlinear models. London Mathematical Society Lecture Note Series, 370. Cambridge University Press, Cambridge (2010)

  2. Andreu-Vaillo Fuensanta, Mazón José M., Rossi Julio D., Toledo-Melero J. Julián: Nonlocal diffusion problems. Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI. Real Sociedad Matemática Española, Madrid (2010)

  3. Avetisov, V.A., Bikulov, A.K., Zubarev, A.P.: First passage time distribution and the number of returns for ultrametric random walks. J. Phys. A 42(8), 085003 (2009)

    Article  MathSciNet  Google Scholar 

  4. Avetisov, V.A., Bikulov, A.K.: On the ultrametricity of the fluctuation dynamicmobility of protein molecules. Proc. Steklov Inst. Math. 265(1), 75–81 (2009) [Tr.Mat. Inst. Steklova 265, 82–89 (2009) (Izbrannye VoprosyMatematicheskoy Fiziki i \(p\)-adicheskogo Analiza) (in Russian)]

  5. Avetisov, V.A., Bikulov, A.K., Osipov, V.A.: \(p\)-adic description of characteristic relaxation in complex systems. J. Phys. A 36(15), 4239–4246 (2003)

    Article  MathSciNet  Google Scholar 

  6. Avetisov, V.A., Bikulov, A.H., Kozyrev, S.V., Osipov, V.A.: \(p \)-adic models of ultrametric diffusion constrained by hierarchical energy landscapes. J. Phys. A 35(2), 177–189 (2002)

    Article  MathSciNet  Google Scholar 

  7. Avetisov, V.A., Bikulov, A., Osipov, V.A.: p-adic models of ultrametric diffusion in the conformational dynamics of macromolecules. Proc. Steklov Inst. Math. 245(2), 48–57 (2004) [Tr. Mat. Inst. Steklova 245, 55–64 (2004) (Izbrannye Voprosy Matematicheskoy Fiziki i p-adicheskogo Analiza) (in Russian)]

  8. Christian, B., Gunnar, F.: Potential Theory on Locally Compact Abelian Groups. Springer, New York (1975)

    MATH  Google Scholar 

  9. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York (1968)

    MATH  Google Scholar 

  10. Brawer, S.: Relaxation in Viscous Liquids and Glasses. American Ceramic Society, Columbus (1985)

    Google Scholar 

  11. Casas-Sánchez, O.F., Zúñiga-Galindo, W.A.: p-Adic elliptic quadratic forms, parabolic-type pseudodifferential equations with variable coefficients and Markov processes. p-Adic Numbers Ultrametr. Anal. Appl. 6(1), 1–20 (2014)

    Article  MathSciNet  Google Scholar 

  12. Chacón-Cortes, L.F., Zúñiga-Galindo, W.A.: Nonlocal operators, parabolic-type equations, and ultrametric random walks. J. Math. Phys. 54, 113503 (2013). https://doi.org/10.1063/1.4828857

    Article  MathSciNet  MATH  Google Scholar 

  13. Evans, S.N.: Local properties of Lévy processes on a totally disconnected group. J. Theor. Probab. 2(2), 209–259 (1989)

    Article  Google Scholar 

  14. Fife, P.: Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions. Trends in Non-linear Analysis, pp. 153–191. Springer, Berlin (2003)

    MATH  Google Scholar 

  15. Frauenfelder, H., Sligar, S.G., Wolynes, P.G.: The energy landscape and motions of proteins. Science 254, 1598–1603 (1991)

    Article  Google Scholar 

  16. Frauenfelder, H., McMahon, B.H., Fenimore, P.W.: Myoglobin: the hydrogen atom of biology and paradigm of complexity. PNAS 100(15), 8615–8617 (2003)

    Article  Google Scholar 

  17. Frauenfelder, H., Chan, S.S., Chan, W.S. (eds.): The Physics of Proteins. Springer, New York (2010)

    Google Scholar 

  18. Galeano-Peñaloza, J., Zuñiga-Galindo, W.: Pseudo-differential operators with semi-quasielliptic symbols over p-adic fields. J. Math. Anal. Appl. 386, 32–49 (2012)

    Article  MathSciNet  Google Scholar 

  19. Gutiérrez García, I., Torresblanca-Badillo, A.: Strong Markov processes and negative denite functions associated with non-Archimedean elliptic pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 11, 345–362 (2020). https://doi.org/10.1007/s11868-019-00293-3

  20. Gutiérrez, G.I., Torresblanca-Badillo, A.: Some classes of non-archimedean pseudo-differential operators related to Bessel potentials. J. Pseudo-Differ. Oper. Appl. (2020). https://doi.org/10.1007/s11868-020-00333-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Khrennikov, A., Oleschko, K., Correa, L.M.: Modeling fluid’s dynamics with master equations in ultrametric spaces representing the treelike structure of capillary networks. Entropy 18, 249 (2016). https://doi.org/10.3390/e18070249

    Article  MathSciNet  Google Scholar 

  22. Khrennikov, A., Kozyrev, S., Zúñiga-Galindo, W.A.: Ultrametric Equations and Its Applications. Encyclopedia of Mathematics and Its Applications, vol. 168. Cambridge University Press, Cambridge (2018)

    MATH  Google Scholar 

  23. Kozyrev, S.V.: Dynamics on rugged landscapes of energy and ultrametric diffusion. p-Adic Numbers Ultrametr. Anal. Appl. 2, 122–132 (2010)

    Article  MathSciNet  Google Scholar 

  24. Kozyrev, S.V.: Methods and applications of ultrametric and \(p\)-adic analysis: from wavelet theory to biophysics. In: Sovrem. Probl. Mat., vol. 12, pp. 3–168. Steklov Math. Inst., RAS (2008)

  25. Kozyrev, S.V.: Ultrametric dynamics as a model of interbasin kinetics. J. Comput. Math. Anal. 41, 38–48 (2006)

    MathSciNet  Google Scholar 

  26. Kozyrev, S.V.: Ultrametric analysis and interbasin kinetics. AIP Conf. Proc. 826, 121–128 (2006)

    Article  MathSciNet  Google Scholar 

  27. Matsuoka, S.: Relaxation Phenomena in Polymers. Hanser, Munich (1992)

    Google Scholar 

  28. Ngai, K.W., Wright, B. (eds.): Relaxation in Complex Systems. Naval Research Laboratories and Office of Naval Research, Washington (1984)

    Google Scholar 

  29. Ogielski, A.T., Stein, D.L.: Dynamics on ultrametric spaces. Phys. Rev. Lett. 55(15), 1634–1637 (1985)

    Article  MathSciNet  Google Scholar 

  30. Oleschko, K., Khrennikov, A.Y.: Applications of \(p\)-adics to geophysics: Linear and quasilinear diffusion of water-in-oil and oil-in-water emulsions. TMF 190(1), 179–190 (2017); Theor. Math. Phys. 190(1), 154–163 (2017)

  31. Pourhadi, E., Khrennikov, A., Saadati, R., Oleschko, K., Correa, L.M.: Solvability of the \(p\)-adic analogue of Navier–Stokes equation via the wavelet theory. Entropy 21, 1129 (2019). https://doi.org/10.3390/e21111129

    Article  MathSciNet  Google Scholar 

  32. Rammal, R., Toulouse, G., Virasoro, M.A.: Ultrametricity for physicists. Rev. Mod. Phys. 58(3), 765–788 (1986)

    Article  MathSciNet  Google Scholar 

  33. Rodríguez-Vega, J.J., Zúñiga-Galindo, W.A.: Taibleson operators, p-adic parabolic equations and ultrametric diffusion. Pac. J. Math. 237(2), 327–347 (2008)

    Article  MathSciNet  Google Scholar 

  34. Stillinger, F.H., Weber, T.A.: Hidden structure in liquids. Phys. Rev. A 25, 978–989 (1982)

    Article  Google Scholar 

  35. Stillinger, F.H., Weber, T.A.: Packing structures and transitions in liquids and solids. Science 225, 983–989 (1984)

    Article  Google Scholar 

  36. Taibleson, M.H.: Fourier Analysis on Local Fields. Princeton University Press, Princeton (1975)

    Book  Google Scholar 

  37. Taira, K: Boundary value problems and Markov processes, 2nd edn. Lecture Notes in Mathematics, 1499. Springer, Berlin (2009)

  38. Torresblanca-Badillo, A., Zúñiga-Galindo, W.A.: Non-archimedean pseudodifferential operators and Feller semigroups. p-Adic Numbers Ultrametr. Anal. Appl. 10(1), 57–73 (2018)

    Article  MathSciNet  Google Scholar 

  39. Torresblanca-Badillo, A., Zúñiga-Galindo, W.A.: Ultrametric diffusion, exponential landscapes, and the first passage time problem. Acta Appl. Math. 157, 93–116 (2018). https://doi.org/10.1007/s10440-018-0165-2

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

    Book  Google Scholar 

  41. Vladimirov, V.S., Volovich, I.V.: Superanalysis. Theor. Mat. Fiz. 59, 317–335 (1984)

  42. Volovich, I.V.: Number theory as the ultimate physical theory, preprint CERN-TH. 4781/87, CERN, Geneva (1987), Reproduced in \(p\)-Adic Numbers Ultrametric Anal. Appl. 2(1), 77–87 (2010)

  43. Zúñiga-Galindo, W.A.: Parabolic equations and Markov processes over p-adic fields. Potential Anal. 28(2), 185–200 (2008)

    Article  MathSciNet  Google Scholar 

  44. Zúñiga-Galindo, W.A.: The non-Archimedean stochastic heat equation driven by Gaussian noise. J. Fourier Anal. Appl. 21(3), 600–627 (2015)

    Article  MathSciNet  Google Scholar 

  45. Zúñiga-Galindo, W.A.: Pseudodifferential Equations over Non-Archimedean Spaces. Lectures Notes in Mathematics, vol. 2174. Springer, New York (2016)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anselmo Torresblanca-Badillo.

Additional information

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

Gutiérrez-García, I., Torresblanca-Badillo, A. Probability density functions and the dynamics of complex systems associated to some classes of non-archimedean pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 12, 12 (2021). https://doi.org/10.1007/s11868-021-00381-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11868-021-00381-3

Keywords

Navigation