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Path regularity for Feller semigroups via Gaussian kernel estimates and generalizations to arbitrary semigroups on C0

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Abstract

Given γ ∈ (−1,1), we present a dyadic growth condition on the finite dimensional distributions of operator semigroups on C0(E which - for γ>0 and Feller semigroups - assures that the corresponding Feller process has paths in local Hölder spaces and in weighted Besov spaces of order γ. We show that, for operator semigroups satisfying Gaussian kernel estimates of order m>1, condition holds for all and even for all in the case of Feller semigroups. Such Gaussian kernel estimates are typical for Feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on ℝD of order mD.

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References

  1. 1.

    Auscher, P., Tchamitchian, Ph.: Square root problem for divergence operators and related topics. Astérisque 249, Soc. Math. de France, 1988

  2. 2.

    Barlow, M.T., Bass, R.F.: Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51, 673–744 (1999)

  3. 3.

    Blumenthal, R.M., Getoor, R.K.: Sample functions of stochastic processes with stationary independent increments. J. Math. Tech. 10, 493–516 (1961)

  4. 4.

    Blunck, S.: A Hörmander-type spectral multiplier 7theorem for operators without heat kernel. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. II, 449–459 (2003)

  5. 5.

    Blunck, S., Kunstmann, P.C.: Weighted norm estimates and maximal regularity. Adv. in Diff. Eq. 7, 1513–1532 (2002)

  6. 6.

    Blunck, S., Kunstmann, P.C.: Calderon-Zygmund theory for non-integral operators and the H functional calculus. Revista Mat. Iberoam. 19, 919–942 (2003)

  7. 7.

    Conway, J.B.: A course in functional analysis. Springer New York 1990

  8. 8.

    Coulhon, T., Duong, X.T.: Maximal regularity and kernel bounds; observations on a theorem by Hieber and Prüss. Adv. in Diff. Eq. 5, 343–368 (2000)

  9. 9.

    Ciesielski, Z., Kerkyacharian, G., Roynette, B.: Quelques espaces fonctionnels associés à des processus Gaussiens. Studia Math. 107, 171–204 (1993)

  10. 10.

    Davies, E.B.: Uniformly elliptic operators with measurable coefficients. J. Funct. Anal. 132, 141–169 (1995)

  11. 11.

    Dellacherie, C., Meyer, P.A.: Probabilités et potentiel. Chapitres I à IV. Public. de l’Institut de Math. de l’Univ. de Strasbourg No. XV. Hermann Paris 1975

  12. 12.

    Duong, X.T., Ouhabaz, E.M., Sikora, A.: Plancherel type estimates and sharp spectral multipliers. J. Funct. Anal. 196, 443–485 (2002)

  13. 13.

    Duong, X.T., McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15 (2), 233–265 (1999)

  14. 14.

    Ethier, S.N., Kurtz, T.G.: Markov processes. Characterization and convergence. John Wiley & Sons New York 1986

  15. 15.

    Grigor’yan, A.: Gaussian upper bounds for the heat kernel on arbitrary manifolds. J. Diff. Geom. 45, 33–52 (1997)

  16. 16.

    Herren, V.: Lévy type processes and Besov spaces. Potential Abalysis 7, 689–704 (1997)

  17. 17.

    Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Springer- Verlag New York 1988

  18. 18.

    Parthasarathy, K.R.: Probability measures on metric spaces. Academic Press New York 1967

  19. 19.

    Peetre, J.: New thoughts on Besov spaces. Duke University Math. Series No. 1, Durham N.C. 1976

  20. 20.

    Pruitt, W.E.: The growth of random walks and Lévy processes. Ann. Probab. 9, 948–956 (1981)

  21. 21.

    Schilling, R.L.: On Feller processes with sample paths in Besov spaces. Math. Ann. 309, 663–675 (1997)

  22. 22.

    Schilling, R.L.: Growth and Hölder conditions for the sample paths of Feller processes. Probab. Theor. Rel. Fields 112, 565–611 (1998)

  23. 23.

    Schilling, R.L.: Function spaces as path spaces of Feller processes. Math. Nach. 217, 147–174 (2000)

  24. 24.

    Simon, B.: Schrödinger semigroups. Bull. Amer. Math. Soc. 7, 447–526 (1982)

  25. 25.

    Schmeisser, H.J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Akademische Verlagsgesellschaft Geest & Portig, Leipzig 1987, and Wiley, J., Chichester, 1987

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Correspondence to Sönke Blunck.

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Blunck, S. Path regularity for Feller semigroups via Gaussian kernel estimates and generalizations to arbitrary semigroups on C0. Probab. Theory Relat. Fields 133, 71–97 (2005). https://doi.org/10.1007/s00440-004-0415-2

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Keywords

  • Assure
  • Growth Condition
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology