Skip to main content
SpringerLink
Log in

Analysis on Ultra-Metric Spaces via Heat Kernels

  • Research Articles
  • Published:
p-Adic Numbers, Ultrametric Analysis and Applications Aims and scope Submit manuscript

Abstract

We give an overview of heat kernels on ultra-metric spaces based on the results of [9] and [11]. In particular, we present estimates of the heat kernel of the Vladimirov operator in \(\mathbb{Q}_{p}^{n}.\)

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.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.
Fig. 15.
Fig. 16.

References

  1. S. Albeverio and W. Karwowski, “Diffusion on \(p\)-adic numbers,” Gaussian Random Fields (Nagoya, 1990), Ser. Probab. Statist. 1, 86–99 (World Sci. Publ., River Edge, NJ, 1991).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Barlow, “Diffusions on fractals,” Lect. Notes Math. 1690, 1–121 (Springer, 1998).

    MATH  Google Scholar 

  4. M. Barlow, “Which values of the volume growth and escape time exponents are possible for graphs?,” Rev. Mat. Iberoamericana 20, 1–31 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Barlow and R. F. Bass, “The construction of Brownian motion on the Sierpinski carpet,” Ann. Inst. H. Poincaré Probab. Statist. 25 (3), 225–257 (1989).

    MathSciNet  MATH  Google Scholar 

  6. M. Barlow and R. F. Bass, “Transition densities for Brownian motion on the the Sierpínski carpet,” Probab. Theory Relat. Fields 91, 307–330 (1992).

    Article  MATH  Google Scholar 

  7. M. Barlow and R. F. Bass, “Brownian motion and harmonic analysis on Sierpínski carpets,” Canad. J. Math. 51, 673–744 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Barlow and E.A. Perkins, “Brownian motion on the Sierpínski gasket,” Probab. Theory Relat. Fields 79, 543–623 (1988).

    Article  MATH  Google Scholar 

  9. A. Bendikov, A. Grigor’yan, E. Hu and J. Hu, “Heat kernels and non-local dirichlet forms on ultrametric spaces,” Ann. Scuola Norm. Sup. Pisa 22, 399–461 (2021).

    MathSciNet  MATH  Google Scholar 

  10. A. Bendikov, A. Grigor’yan and Ch. Pittet, “On a class of Markov semigroups on discrete ultra-metric spaces,” Potent. Anal. 37, 125–169 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Bendikov, A. Grigor’yan, Ch. Pittet and W. Woess, “Isotropic Markov semigroups on ultra-metric spaces,” Uspekhi Matem. Nauk 69, 3–102 (2014).

    MathSciNet  MATH  Google Scholar 

  12. Z.-Q. Chen, T. Kumagai and J. Wang, “Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms,” Adv. Math. 374, art. 107269 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  13. Z.Q. Chen and T. Kumagai, “Heat kernel estimates for stable-like processes on \(d\)-sets,” Stoch. Process. Appl. 108, 27–62 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  14. St. N. Evans, “Local properties of Lévy processes on a totally disconnected group,” J. Theor. Probab. 2, 209–259 (1989).

    Article  MATH  Google Scholar 

  15. M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, 2nd ed. (Walter de Gruyter, Berlin, 2011).

    MATH  Google Scholar 

  16. A. Grigor’yan, E. Hu and J. Hu, “Two sides estimates of heat kernels of jump type Dirichlet forms,” Advan. Math. 330, 433–515 (2018).

    Article  MATH  Google Scholar 

  17. A. Grigor’yan, J. Hu and K.-S. Lau, “Heat kernels on metric-measure spaces and an application to semilinear elliptic equations,” Trans. Amer. Math. Soc. 355, 2065–2095 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Grigor’yan and T. Kumagai, “On the dichotomy in the heat kernel two sided estimates,” Proc. Symposia Pure Math. 77, 199–210 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  19. B. M. Hambly and T. Kumagai, “Transition density estimates for diffusion processes on post critically finite self-similar fractals,” Proc. London Math. Soc. 79, 431–458 (1997).

    MathSciNet  MATH  Google Scholar 

  20. J. Kigami, “Harmonic calculus on p.c.f. self-similar sets,” Trans. Amer. Math. Soc. 335, 721–755 (1993).

    MathSciNet  MATH  Google Scholar 

  21. J. Kigami, Analysis on Fractals (Cambridge Univ. Press, 2001).

    Book  MATH  Google Scholar 

  22. A. N. Kochubei, Pseudo-Differential Equations and Stochastics over non-Archimedean Fields, Monographs and Textbooks in Pure and Applied Mathematics 244 (Marcel Dekker Inc., New York, 2001).

    Book  MATH  Google Scholar 

  23. S. Kusuoka, “Lecture on diffusion processes on nested fractals,” Lect. Notes Math. 1567, 39–98 (1993).

    Google Scholar 

  24. S. Kusuoka and X. Y. Zhou, “Dirichlet form on fractals: Poincaré constant and resistance,” Probab. Theory Relat. Fields 93, 169–196 (1992).

    Article  MATH  Google Scholar 

  25. M. Del Muto and A. Figà-Talamanca, “Diffusion on locally compact ultrametric spaces,” Expo. Math. 22, 197–211 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  26. J. J. Rodríguez-Vega and W. A. Zúñiga-Galindo, “Taibleson operators,” \(p\)-Adic Parabolic Equations and Ultrametric Diffusion, Pacif. J. Math. 237, 327–347 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  27. M. H. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, 1975).

    MATH  Google Scholar 

  28. V. S. Vladimirov, “Generalized functions over the field of \(p\)-adic numbers,” Uspekhi Mat. Nauk 43 (239), 17–53 (1988).

    MathSciNet  Google Scholar 

  29. V. S. Vladimirov and I. V. Volovich, “\(p\)-Adic Schrödinger-type equation,” Lett. Math. Phys. 18, 43–53 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics (World Scientific Publishing Co., Inc., River Edge, NY, 1994).

    Book  MATH  Google Scholar 

  31. W. A. Zúñiga-Galindo, “Parabolic equations and Markov processes over \(p\)-adic fields,” Potent. Anal. 28, 185–200 (2008).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 317210226 - SFB 1283.

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, University of Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

    Alexander Grigor’yan

Authors
  1. Alexander Grigor’yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Grigor’yan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grigor’yan, A. Analysis on Ultra-Metric Spaces via Heat Kernels. P-Adic Num Ultrametr Anal Appl 15, 204–242 (2023). https://doi.org/10.1134/S2070046623030044

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070046623030044

Keywords

Search

Navigation