Non-archimedean pseudo-differential operators on Sobolev spaces related to negative definite functions

Abstract

In this article we study a large class of pseudo-differential operators on Sobolev spaces related to negative definite functions in the p-adic context and in arbitrary dimension. We show that these operators are m-dissipatives and generators of a strongly continuous contraction semigroup on the sobolev spaces mentioned above. Also, we study the convolution kernel, the Green function and the heat kernel attached to these operators. In addition, we study certain inhomogeneous equations and the Cauchy problem naturally associated to these operators.

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References

  1. 1.

    Aguilar-Arteaga, V.A., Estala-Arias, S.: Pseudodifferential operators and Markov processes on adèles p-Adic numbers ultrametric. Anal. Appl. 11(2), 89–113 (2019)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Aguilar-Arteaga, V., Cruz-López, M., Estala-Arias, S.: Non-Archimedean analysis and a wave-type pseudodifferential equation on finite adèles. J. Pseudo Differ. Oper. Appl. (2020). https://doi.org/10.1007/s11868-020-00343-1

    Article  MATH  Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

    Antoniouk, A.V., Oleschko, K., Kochubei, A.N., Khrennikov, A.Y.: A stochastic \(p\)-adic model of the capillary flow in porous random medium. Phys. A Stat. Mech. Appl. 505, 763–777 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Antoniouk, A.V., Khrennikov, A.Y., Kochubei, A.N.: Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables. J. Pseudo-Differ. Oper. Appl. (2019). https://doi.org/10.1007/s11868-019-00320-3

    Article  MATH  Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 7.

    Christian, B.: Gunnar, Forst: Potential theory on locally compact abelian groups. Springer, New York-Heidelberg (1975)

    Google Scholar 

  8. 8.

    Bikulov, A.K.: On solution properties of some types of p-adic kinetic equations of the form reactiondiffusion, \(p\)-Adic Numbers Ultrametric. Anal. Appl. 2(3), 187–206 (2010)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Thierry, Cazenave: Haraux Alain An Introduction to Semilinear Evolution Equations. Oxford University Press, Oxford (1998)

    Google Scholar 

  10. 10.

    Ethier, S.N., Kurtz, T.G.: Markov Processes Characterization and convergence, Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1986)

    Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Gutiérrez García I., Torresblanca-Badillo A., Strong Markov processes and negative definite functions associated with non-Archimedean elliptic pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. (2019), 1-18

  13. 13.

    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

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Jacob, N.: Pseudo Differential Operators and Markov Processes, Fourier Analysis and Semigroups., vol. I. Imperial College Press, London (2001)

    Google Scholar 

  15. 15.

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

    Google Scholar 

  16. 16.

    Khrennikov, A., Oleschko, K., Correa López, 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

    MathSciNet  Article  Google Scholar 

  17. 17.

    Khrennikov, AYu., Kozyrev, S.V., Zúñiga-Galindo, W.A.: Ultrametric Pseudodifferential Equations and Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2018). https://doi.org/10.1017/9781316986707

    Google Scholar 

  18. 18.

    Khrennikov, A.Y., Kochubei, A.N.: \(p-\)Adic analogue of the porous medium equation. J. Fourier Anal. Appl. 24, 1401–1424 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    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. MR 93e:35050 (1992)

  20. 20.

    Kochubei A. N., Pseudo-differential equations and stochastic over non-Archimedean fields, Pure and Applied Mathematics \(244,\) Marcel Dekker, New York, MR 2003b:35220 Zbl 0984.11063, 2001

  21. 21.

    Kochubei, A.N., SaitAmetov, 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)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kozyrev S. V., Methods and Applications of Ultrametric and \(p\)-Adic Analysis: From Wavelet Theory to Biophysics, Sovrem. Probl. Mat., 12, Steklov Math. Inst., RAS, Moscow, 2008, 3–168

  23. 23.

    Lunner, G., Phillips, R.S.: Dissipative operators in a Banach space. Pacific J. Math. 11, 679–698 (1961)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Google Scholar 

  25. 25.

    Pourhadi, E., Khrennikov, A., Saadati, R., Oleschko, K., Correa López, 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

    MathSciNet  Article  Google Scholar 

  26. 26.

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

    Google Scholar 

  27. 27.

    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  Google Scholar 

  28. 28.

    Torresblanca-Badillo, A., Zúñiga-Galindo, W.A.: Ultrametric Diffusion, exponential landscapes, and the first passage time problem. Acta Appl. Math. 157, 93 (2018)

    MathSciNet  Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 30.

    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 

  31. 31.

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

    Google Scholar 

  32. 32.

    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  Google Scholar 

  33. 33.

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

    Article  Google Scholar 

  34. 34.

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

    MathSciNet  Article  Google Scholar 

  35. 35.

    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 te preprint CERN-TH. 4781/87, Geneva, 1987, 11 pp

  36. 36.

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

  37. 37.

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

    MathSciNet  Article  Google Scholar 

  38. 38.

    Zúñiga-Galindo, W.A.: Pseudodifferential Equations Over Non-Archimedean Spaces. Lecture Notes in Mathematics 2174. Springer, Berlin (2016)

    Google Scholar 

  39. 39.

    Zúñiga-Galindo, W.A.: Non-archimedean white noise, pseudodifferential stochastic equations, and massive euclidean fields. J. Fourier Anal. Appl. 23(2), 288–323 (2017)

    MathSciNet  Article  Google Scholar 

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Correspondence to Anselmo Torresblanca-Badillo.

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Torresblanca-Badillo, A. Non-archimedean pseudo-differential operators on Sobolev spaces related to negative definite functions. J. Pseudo-Differ. Oper. Appl. 12, 7 (2021). https://doi.org/10.1007/s11868-021-00385-z

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Keywords

  • Sobolev Spaces
  • Pseudo-differential operators
  • Convolution kernel
  • Green function
  • m-dissipative operators
  • Heat kernel
  • Non-archimedean analysis