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Strong Markov processes and negative definite functions associated with non-Archimedean elliptic pseudo-differential operators

  • Ismael Gutiérrez García
  • Anselmo Torresblanca-BadilloEmail author
Article
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Abstract

In this article we prove that the heat kernel attached to the non-Archimedean elliptic pseudodifferential operators determine a Feller semigroup and a uniformly stochastically continuous \(C_{0}\)-transition function of some strong Markov processes \({\mathfrak {X}}\) with state space \({\mathbb {Q}}_{p}^{n}.\) We explicitly write the Feller semigroup and the Markov transition function associated with the heat kernel. Also, we show that the symbols of these pseudo-differential operators are a negative definite function and moreover, that this symbols can be represented as a combination of a constant \(c\ge 0,\) a continuous homomorphism \(l: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}\) and a non-negative, continuous quadratic form \(q: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}.\)

Keywords

Pseudo-differential operators Feller semigroups Markov transition function Convolution semigroup Negative definite function Non-Archimedean analysis 

Notes

References

  1. 1.
    Albeverio, S., Khrennikov, A.Y., Shelkovich, V.M.: Theory of \(p\)-adic Distributions: Linear and Nonlinear Models. London Mathematical Society Lecture Note Series, vol. 370. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Albeverio, S., Khrennikov, A.Y., Shelkovich, V.M.: Harmonic analysis in the \(p\)-adic Lizorkin spaces: fractional operators, pseudo-differential equations, \(p\)-adic wavelets, Tauberian theorems. J. Fourier Anal. Appl. 12(4), 393–425 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christian, B., Gunnar, F.: Potential Theory on Locally Compact Abelian Groups. Springer, New York (1975)zbMATHGoogle Scholar
  4. 4.
    Chuong, N.M., Co, N.V.: The Cauchy problem for a class of pseudo-differential equations over \(p\)-adic field. J. Math. Anal. Appl. 340(1), 629–645 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chacón-Cortes L. F., Zúñiga-Galindo W. A., Nonlocal operators, parabolic-type equations, and ultrametric random walks. J. Math. Phys. 54, 113503 (2013) & Erratum 55 (2014), no. 10, 109901, 1 ppGoogle Scholar
  6. 6.
    Evans, S.N.: Local properties of Lévy processes on a totally disconnected group. J. Theor. Probab. 2(2), 209–259 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Forst, G.: Convolution semigroups of local type. Math. Scand. 34, 211–218 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hoh, W.: A symbolic calculus for pseudo differential operators generating Feller semigroups. Osaka J. Math. 35, 789–820 (1998)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hoh, W.: Pseudo Differential Operators Generating Markov Processes. Universität Bielefeld, Habilitationsschrift (1998)zbMATHGoogle Scholar
  10. 10.
    Jacob, N.: A class of Feller semigroups generated by pseudo differential operators. Math. Z. 215, 151–166 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jacob, N.: Feller semigroups, Dirichlet forms and pseudo differential operators. Forum Math. 4, 433–446 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jacob, N.: Further pseudodifferential operators generating Feller semigroups and Dirichlet forms. Rev. Matemática Iberoamericana 9(2), 373–407 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jacob, N.: Pseudo Differential Operators and Markov Processes. Fourier Analysis and Semigroups, vol. I. Imperial College Press, London (2001)zbMATHGoogle Scholar
  14. 14.
    Jacob, N.: Pseudo Differential Operators and Markov Processes. Generators and Their Potential Theory, vol. II. Imperial College Press, London (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Jacob, N.: Pseudo Differential Operators and Markov Processes. Markov Processes and Applications, vol. III. Imperial College Press, London (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Khrennikov, A.Y.: Fundamental solutions over the field of \(p\)-adic numbers. Algebra i Analiz 4:3 (1992), 248–266. In Russian; translated in St. Petersburg Math. J. 4:3, 613–628 (1993)Google Scholar
  17. 17.
    Kochubei, A.N.: A non-Archimedean wave equation. Pac. J. Math. 235(2), 245–261 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kochubei, A.N.: A Schrödinger-type equation over the field of \(p\)-adic numbers. J. Math. Phys. 34(8), 3420–3428 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kochubei A. N., Fundamental solutions of pseudodifferential equations associated with \(p\)-adic quadratic forms. Izv. Ross. Akad. Nauk Ser. Mat. 62(6), 103–124 (1998). In Russian; translated in Izvestiya Math. 62(6), 1169–1188 (1998)Google Scholar
  20. 20.
    Kochubei, A. N.: Parabolic equations over the field of \(p\)-adic numbers. Izv. Akad. Nauk SSSR Ser. Mat. 55(6), 1312–1330 (1991). In Russian; translated in Math. USSR Izvestiya 39, 1263–1280 (1992)Google Scholar
  21. 21.
    Kochubei, A.N.: Pseudo-Differential Equations and Stochastic Over non-Archimedean Fields, Pure and Applied Mathematics, vol. 244. Marcel Dekker, New York (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    Rodríguez-Vega, J.J., Zúñiga-Galindo, W.A.: Elliptic pseudodifferential equations and Sobolev spaces over \(p\)-adic fields. Pac. J. Math. 246, 407–420 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Taibleson, M.H.: Fourier Analysis on Local Fields. Princeton University Press, Princeton (1975)zbMATHGoogle Scholar
  25. 25.
    Taira, K.: Boundary Value Problems and Markov Processes. Lecture Notes in Mathematics. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  26. 26.
    Torresblanca-Badillo, A., Zúñiga-Galindo, W.A.: Non-Archimedean pseudodifferential operators and feller semigroups, \(p\)-adic numbers. Ultrametric Anal. Appl. 10(1), 60–76 (2018)zbMATHGoogle Scholar
  27. 27.
    Torresblanca-Badillo, A., Zúñiga-Galindo, W.A.: Ultrametric diffusion, exponential landscapes, and the first passage time problem. Acta Appl. Math. 157, 93 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: \(p\)-adic Analysis and Mathematical Physics. World Scientific, Singapore (1994)CrossRefzbMATHGoogle Scholar
  29. 29.
    Zúñiga-Galindo, W.A.: Fundamental solutions of pseudo-differential operators over \(p\)-adic fields. Rend. Sem. Mat. Univ. Padova 109, 241–245 (2003)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Zúñiga-Galindo, W.A.: Parabolic equations and Markov processes over \(p\)-adic fields. Potential Anal. 28(2), 185–200 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zúñiga-Galindo, W.A.: Pseudo-differential equations connected with \(p\)-adic forms and local zeta functions. Bull. Aust. Math. Soc. 70(1), 73–86 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zúñiga-Galindo, W.A.: Pseudodifferential Equations Over Non-Archimedean Spaces. Lecture Notes in Mathematics, vol. 2174. Springer, Berlin (2016)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ismael Gutiérrez García
    • 1
  • Anselmo Torresblanca-Badillo
    • 1
    Email author
  1. 1.Department of Mathematic and StatisticUniversidad del NorteBarranquillaColombia

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