Skip to main content
SpringerLink
Log in

Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We construct p-adic Euclidean random fields \(\varvec{\Phi }\) over \(\mathbb {Q}_{p}^{N}\), for arbitrary N, these fields are solutions of p-adic stochastic pseudodifferential equations. From a mathematical perspective, the Euclidean fields are generalized stochastic processes parametrized by functions belonging to a nuclear countably Hilbert space, these spaces are introduced in this article, in addition, the Euclidean fields are invariant under the action of certain group of transformations. We also study the Schwinger functions of \(\varvec{\Phi }\).

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

Similar content being viewed by others

References

  1. Abdesselam, A., Chandra, A., Guadagni, G.: Rigorous quantum field theory functional integrals over the \(p\)-adics I: anomalous dimensions. arXiv:1302.5971

  2. Albeverio, S., Wu, J.L.: Euclidean random fields obtained by convolution from generalized white noise. J. Math. Phys. 36(10), 5217–5245 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Albeverio, S., Kozyrev, S.V.: Multidimensional basis of p-adic wavelets and representation theory. p-Adic Numbers Ultrametr. Anal. Appl. 1(3), 181–189 (2009)

  4. Albeverio, S., Gottschalk, H., Wu, J.-L., Euclidean random fields, pseudodifferential operators, and Wightman functions. Stochastic analysis and applications (Powys, 1995), pp. 20–37. World Sci. Publ, River Edge, NJ (1996)

  5. Albeverio, S., Gottschalk, H., Wu, J.-L.: Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions. Rev. Math. Phys. 8(6), 763–817 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Albeverio, S., Rüdiger, B., Wu, J.-L.: Analytic and probabilistic aspects of Lévy processes and fields in quantum theory. Lévy processes, pp. 187–224. Birkhäuser Boston, Boston, MA (2001)

  7. Albeverio, S., Gottschalk, H., Wu, J.-L.: SPDEs leading to local, relativistic quantum vector fields with indefinite metric and nontrivial S-matrix. Stochastic partial differential equations and applications (Trento, 2002), 21–38. Lecture Notes in Pure and Applied Mathematics, vol. 227, Dekker, New York (2002)

  8. Albeverio, S., Khrennikov, A.Y., Shelkovich, V.M.: Theory of \(p\)-Adic Distributions: Linear and Nonlinear Models. Cambridge University Press, Cambridge (2010)

  9. Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Springer, New York (1975)

    Book  MATH  Google Scholar 

  10. Brekke, L., Freund, P.G.O., Olson, M., Witten, E.: Nonarchimedean string dynamics. Nucl. Phys. B 302(3), 365–402 (1988)

    Article  MathSciNet  Google Scholar 

  11. Bruhat, F.: Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes \(p\)-adiques. Bull. Soc. Math. France 89, 43–75 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bikulov, A.K.: Stochastic equations of mathematical physics over the field of \(p\)-adic numbers. Theoret. Math. Phys. 119(2), 594–604 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bikulov, A.K., Volovich, I.V.: \(p\)-Adic Brownian motion. Izv. Math. 61(3), 537–552 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dragovich, B., Khrennikov, A.Y., Kozyrev, S.V., Volovich, I.V.: On p-adic mathematical physics. p-Adic Numbers Ultrametr. Anal. Appl. 1(1), 1–17 (2009)

  15. Evans, S.N., \(p\)-Adic white noise, chaos expansions, and stochastic integration. Probability measures on groups and related structures, vol. XI, pp. 102–115 (Oberwolfach, 1994). World Sci. Publ, River Edge, NJ (1995)

  16. Frampton, P.H., Okada, Y.: Effective scalar field theory of \(p\)-adic string. Phys. Rev. D 37, 3077–3084 (1988)

    Article  MathSciNet  Google Scholar 

  17. Frenkel, E.: Langlands Correspondence for Loop Groups. Cambridge Studies in Advanced Mathematics, vol. 103. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  18. Gaitsgory, D., Notes on 2D conformal field theory and string theory. In: Deligne, P. (eds.) Quantum Fields and Strings: A Course for Mathematicians, vol. 2 (Princeton, NJ, 1996/1997), pp. 1017–1089. American Math. Soc., Providence, RI (1999)

  19. Gel’fand, I.M., Vilenkin, N.Y.: Generalized Functions. Applications of Harmonic Analysis, vol. 4. Academic Press, New York (1964)

    MATH  Google Scholar 

  20. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn. Springer, New York (1987)

    MATH  Google Scholar 

  21. Ghoshal, D., Sen, A.: Tachyon condensation and brane descent relations in \(p\)-adic string theory. Nucl. Phys. B 584, 300–312 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hida, T., Kuo, H.H., Potthoff, J., Streit, L.: White Noise. An Infinite-dimensional Calculus. Kluwer Academic Publishers Group, Dordrecht (1993)

    MATH  Google Scholar 

  23. Kamizono, K.: \(p\)-Adic Brownian motion over \(\mathbb{Q}_{p} \). Proc. Steklov Inst. Math. 265(1), 115–130 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kochubei, A.N.: Pseudo-Differential Equations and Stochastics Over Non-Archimedean Fields. Marcel Dekker Inc, New York (2001)

    Book  MATH  Google Scholar 

  26. Kochubei, A.N., Sait-Ametov, M.R.: Interaction measures on the space of distributions over the field of \(p\)-adic numbers. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6(3), 389–411 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Khrennikov, A.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer, Dordreht (1997)

    Book  MATH  Google Scholar 

  28. Khrennikov, A., Huang, Z.: \(p\)-adic valued white noise. Quantum Probab. Relat. Top. 9, 273–294 (1994)

    Article  Google Scholar 

  29. Khrennikov, A.Y.: Generalized functions and Gaussian path integrals over non-Archimedean function spaces. USSR-Izv. 39(1), 761–794 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Khrennikov, A.Y.: Generalized functions on the non-Archimedean superspace. USSR-Izv. 39(3), 1209–1238 (1992)

    Article  MathSciNet  Google Scholar 

  31. Khrennikov, A.Y., Kozyrev, S.V.: Ultrametric random field. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(2), 199–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Khrennikov, A.Y., Kozyrev, S.V., Oleschko, K., Jaramillo, A.G., Correa López, M.: Application of \(p\)-adic analysis to time series. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16(4), 1350030 (15 pages) (2013)

  33. Khrennikov, A.: Zhiyuan Huang, Generalized functionals of \(p\)-adic white noise. Dokl. Akad. Nauk 344(1), 23–26 (1995)

    MathSciNet  Google Scholar 

  34. Kolmogorov, A.N.: The Wiener spiral and some other interesting curves in Hilbert space. Dokl. Akad. Nauk S.S.S.R. 26(2), 115–118 (1940)

    Google Scholar 

  35. Lax Peter, D., Phillips Ralph, S.: Scattering Theory for Automorphic Functions. Annals of Mathematics Studies, No. 87. Princeton Univ. Press, Princeton, NJ (1976)

  36. Minlos, R.A.: Generalized random processes and their extension to a measure. In: Selected Translations of Mathematical, Statistics and Probility, vol. 3 pp. 291–313. Amer. Math. Soc., Providence, RI (1963)

  37. Missarov, M.D.: Random fields on the adele ring and Wilson’s renormalization group. Annales de l’institut Henri Poincare (A): Physique Theorique 50(3), 357–367 (1989)

    MathSciNet  MATH  Google Scholar 

  38. Missarov, M.D.: \(p\)-adic \(\varphi ^{4}\)-theory as a functional equation problem. Lett. Math. Phys. 39(3), 253–260 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. Missarov, M.D.: \(p\)-adic renormalization group solutions and the Euclidean renormalization group conjectures. p-Adic Numbers Ultrametr. Anal. Appl. 4(2), 109–114 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Moeller, N., Zwiebach, B.: Dynamics with in finitely many time derivatives and rolling tachyons. JHEP 10, 034 (2002)

    Article  Google Scholar 

  41. Nobuaki, O.: White Noise Calculus and Fock Space. Lecture Notes in Mathematics. Springer, Berlin (1994)

    MATH  Google Scholar 

  42. Pavlov, B.S., Faddeev, L.D.: Scattering theory and automorphic functions. (Russian) Boundary value problems of mathematical physics and related questions in the theory of functions, 6. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 27, 161–193 (1972)

    Google Scholar 

  43. Rodríguez-Vega, J.J., Zúñiga-Galindo, W.A.: Elliptic pseudodifferential equations and Sobolev spaces over p-adic fields. Pacific J. Math. 246(2), 407–420 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Weiyi, Su, Hua, Qiu: \(p\)-adic calculus and its applications to fractal analysis and medical science. Facta Universitatis (Niš): Ser. Elec. Energ. 21(3), 339–347 (2008)

    Google Scholar 

  45. Smirnov, V.A.: Calculation of general \(p\)-adic Feynman amplitude. Commun. Math. Phys. 149(3), 623–636 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  47. Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)

    MATH  Google Scholar 

  48. Varadarajan, V.S.: Non-Archimedean models for space-time. Modern Phys. Lett. A 16(4–6), 387–395 (2001)

    Article  MathSciNet  Google Scholar 

  49. Varadarajan, V.S.: Reflections on Quanta, Symmetries, and Supersymmetries. Springer, New York (2011)

    Book  MATH  Google Scholar 

  50. Vladimirov, V.S.: On the non-linear equation of a \(p\)-adic open string for a scalar field. Russ. Math. Surv. 60, 1077–1092 (2005)

    Article  MATH  Google Scholar 

  51. Vladimirov, V.S.: On the equations for p-adic closed and open strings. p-Adic Numbers Ultrametr. Anal. Appl. 1(1), 79–87 (2009)

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

  53. Vladimirov, V.S., Volovich, Y.I.: Nonlinear dynamics equation in \(p\)-adic string theory. Theor. Math. Phys. 138, 297–307 (2004). arXiv:math-ph/0306018

  54. Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: Spectral theory in \(p\)-adic quantum mechanics and representation theory. Soviet Math. Dokl. 41(1), 40–44 (1990)

    MathSciNet  MATH  Google Scholar 

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

  56. Volovich, I.V.: \(p\)-Adic string. Class. Quantum Grav. 4(4), L83–L87 (1987)

    Article  MathSciNet  Google Scholar 

  57. Volovich, I.V.: Number theory as the ultimate physical theory. p-Adic Numbers Ultrametric Anal. Appl. 2(1), 77–87 (2010). This paper corresponds to the preprint CERN-TH.4781/87, Geneva, July 1987

  58. Weil, A.: Basic Number Theory. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  59. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  60. Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1(6), 769–796 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zelenov, E.I.: A \(p\)-adic infinite-dimensional symplectic group. Russ. Acad. Sci. Izv. Math. 43(3), 421–441 (1994)

    MathSciNet  Google Scholar 

  62. Zelenov, E.I.: Quantum approximation theorem. p-Adic Numbers Ultrametr. Anal. Appl. 1(1), 88–90 (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

  64. Zúñiga-Galindo, W.A.: The Non-Archimedean stochastic heat equation driven by Gaussian noise. J. Fourier Anal. Appl. doi: 10.1007/s00041-014-9383-9

Download references

Acknowledgments

The author wishes to thank to Sergii Torba and the anonymous referees for many useful comments and discussions, which led to an improvement of this work.

Author information

Authors and Affiliations

  1. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Departamento de Matemáticas, Unidad Querétaro, Libramiento Norponiente #2000 Fracc. Real de Juriquilla, 76230, Santiago de Querétaro, Qro, Mexico

    W. A. Zúñiga-Galindo

Authors
  1. W. A. Zúñiga-Galindo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. A. Zúñiga-Galindo.

Additional information

Communicated by Michael Ruzhansky.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zúñiga-Galindo, W.A. Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields. J Fourier Anal Appl 23, 288–323 (2017). https://doi.org/10.1007/s00041-016-9470-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-016-9470-1

Keywords

Mathematics Subject Classification

Search

Navigation