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Some Geometric Observations on Heat Kernels of Markov Semigroups with Non-local Generators

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Recent Advances in Mathematical Analysis

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Abstract

In this paper we study the possibility to represent the heat kernel associated with a pseudo differential operator with negative definite symbol with the help of a combination of two metrics associated with the symbol. We discuss in great detail two examples and we outline the consequences of this approach.

Dedicated to Professor Francesco Altomare

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References

  1. Bakry, D., Gentil, J., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  2. Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  3. Böttcher, B., Schilling, R., Wang, J.: Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer, Berlin (2013)

    Google Scholar 

  4. Bray, L., Jacob, N.: Some considerations on the structure of transition densities of symmetric Lévy processes. Commun. Stoch. Anal. 10, 405–420 (2016)

    MathSciNet  Google Scholar 

  5. Courrège, P.: Sur la forme intégro-différentielle des opérateurs de \(C_k^\infty \) dans C satisfaiscent au principe du maximum. In: Sém. Theorie du Potential 1965/66. Exposé 2, 38pp., Paris (1966)

    Google Scholar 

  6. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  7. Evans, K., Jacob, N.: On adjoint additive processes. Probab. Math. Stat. 40, 205–223 (2020)

    MathSciNet  MATH  Google Scholar 

  8. Hoh, W.: The martingale problem for a class of pseudo-differential operators. Math. Ann. 300, 121–147 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hoh, W.: Pseudo differential operators with negative definite symbols and the martingale problem. Stoch. Stoch. Rys. 55, 225–252 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hoh, W.: A symbolic calculus for pseudo-differential operators generating Feller semigroups. Osaka J. Math. 35, 798–820 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Hoh, W.: Pseudo Differential Operators Generating Markov Processes. Habilitationschrift, Bielefeld (1998)

    Google Scholar 

  12. Jacob, N.: A class of Feller semigroups generated by pseudo-differential operators. Math. Z. 215, 151–166 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  14. Jacob, N.: Pseudo Differential Operators and Markov Processes. Vol. II: Generators and Their Potential Theory. Imperial College Press, London (2002)

    Book  MATH  Google Scholar 

  15. Jacob, N., Knopova, V., Landwehr, S., Schilling, R.: A geometric interpretation of the transition density of a symmetric Lévy process. Sci. China Math. 55, 1099–1126 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jacob, N., Rhind, E.O.T.: Aspects of micro-local analysis and geometry in the study of Lévy-type generators. In: Bahns, D., Pohl, A., Witt, I. (eds.) Open Quantum Systems, pp. 77–140. Springer, Berlin (2019)

    Chapter  Google Scholar 

  17. Jacob, N., Schilling, R.: Estimates for Feller semigroups generated by pseudo-differential operators. In: J. Rakosnik (ed.) Function Spaces, Differential Operators and Nonlinear Analysis, pp. 27–49. Prometheus Publishing House, Buffalo (1996)

    MATH  Google Scholar 

  18. Kilbas, A., Srivastava, H., Trujello, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  19. Komatsu, T.: Markov processes associated with certain intergo-differential operators. Osaka J. Math. 10, 271–303 (1973)

    MathSciNet  MATH  Google Scholar 

  20. Komatsu, T.: On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21, 113–132 (1984)

    MathSciNet  MATH  Google Scholar 

  21. Komatsu, T.: Pseudo-differential operators and Markov processes. J. Math Soc. Japan 36, 387–418 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kwasnicki, M.: Ten equivalent definitions of the fractional Laplace operators. Fract Calc. Appl. Anal. 20, 7–51 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Meyer, P.-A.: Démonstrations probabiliste de certaines inéqalités de Littlewood-Paley. Exposé 2: L’operateur carré du champ. In: Séminaire de Probabilités, vol. X. Springer, Berlin (1976)

    Google Scholar 

  24. Schilling, R.: Subordination in the sense of Bochner and a related functional calculus. J. Austr. Math. Soc. (Ser. A) 64, 368–396 (1998)

    Google Scholar 

  25. Schilling, R., Schnurr, A.: The symbol associated with the solution of a stochastic differential equation. Electron. J. Probab. 15, 1369–1393 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schilling, R.L., Song, R., VondraÄŤek, Z.: Bernstein Functions, 2nd edn. De Gruyter Verlag, Berlin (2012)

    Book  MATH  Google Scholar 

  27. Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522–536 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  28. Spener, A., Weber, I., Zacher, R.: The fractional Laplacian has infinite dimension. Commun. P.D.E. 45, 57–75 (2020)

    Google Scholar 

  29. Sturm, K.T.: Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26, 1–55 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Varopoulos, N., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Swansea University, Swansea, Wales, UK

    Kristian P. Evans & Niels Jacob

Authors
  1. Kristian P. Evans
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  2. Niels Jacob
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Corresponding author

Correspondence to Niels Jacob .

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Editors and Affiliations

  1. Dipartimento di Matematica, UniversitĂ  degli Studi di Bari Aldo Moro, Bari, Italy

    Anna Maria Candela

  2. Dipartimento di Matematica, UniversitĂ  degli Studi di Bari Aldo Moro, Bari, Italy

    Mirella Cappelletti Montano

  3. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Lecce, Italy

    Elisabetta Mangino

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Evans, K.P., Jacob, N. (2023). Some Geometric Observations on Heat Kernels of Markov Semigroups with Non-local Generators. In: Candela, A.M., Cappelletti Montano, M., Mangino, E. (eds) Recent Advances in Mathematical Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-20021-2_17

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