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Symmetric Stable Processes in Cones

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Abstract

We study exponents of integrability of the first exit time from generalized cones for conditioned rotation invariant stable Lévy processes. Along the way, we introduce the “spherical fractional Laplacian” and derive some of its spectral properties.

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References

  1. Bañuelos, R., Latała, R. and Méndez-Hernandez, P.J.: ‘A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes’, Proc. Amer. Math. Soc. 129(10) (2001), 2997–3008 (electronic).

    Google Scholar 

  2. Bañuelos, R. and Smits, R.: ‘Brownian motion in cones’, Probab. Theory Related Fields 108 (1997), 299–319.

    Google Scholar 

  3. Bañuelos, R., DeBlassie, R.D. and Smits, R.: ‘The first exit time of planar Brownian motion from the interior of a parabola’, Ann. Probab. 29 (2001), 882–901.

    Google Scholar 

  4. Bass, R.F. and Cranston, M.: ‘Exit times for symmetric stable processes in ℝn’, Ann. Probab. 11(3) (1983), 578–588.

    Google Scholar 

  5. Blumenthal, R.M. and Getoor, R.K.: Markov Processes and Potential Theory, Springer-Verlag, New York, 1968.

    Google Scholar 

  6. Blumenthal, R.M., Getoor, R.K. and Ray, D.B.: ‘On the distribution of first hits for the symmetric stable process’, Trans. Amer. Math. Soc. 99 (1961), 540–554.

    Google Scholar 

  7. Bogdan, K.: ‘The boundary Harnack principle for the fractional Laplacian’, Studia Math. 123(1) (1997), 43–80.

    Google Scholar 

  8. Bogdan, K.: ‘Representation of α-harmonic functions in Lipschitz domains’, Hiroshima Math. J. 29(2) (1999), 227–243.

    Google Scholar 

  9. Bogdan, K. and Byczkowski, T.: ‘Probabilistic proof of boundary Harnack principle for α-harmonic functions’, Potential Anal. 11 (1999), 135–156.

    Google Scholar 

  10. Bogdan, K. and Byczkowski, T.: ‘Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains’, Studia Math. 133(1) (1999), 53–92.

    Google Scholar 

  11. Bogdan, K., Burdzy, K. and Chen, Z.-Q.: ‘Censored stable processes’, to appear in Probab. Theory Related Fields (2001).

  12. Bogdan, K., Kulczycki, T. and Nowak, A.: ‘Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes’, Illinois J. Math. 46(2) (2002), 541–556.

    Google Scholar 

  13. Burdzy, K.: Multidimensional Brownian Excursions and Potential Theory, Pitman Research Notes in Mathematics 164, Longman Scientific & Technical, 1987.

  14. Burkholder, D.L.: ‘Exit times of Brownian motion, harmonic majorization, and Hardy spaces’, Adv. Math. 26(2) (1977), 182–205.

    Google Scholar 

  15. Chen, Z.Q. and Song, R.: ‘Estimates on Green functions and Poisson kernels for symmetric stable process’, Math. Ann. 312(3) (1998), 465–501.

    Google Scholar 

  16. Chen, Z.-Q. and Song, R.: ‘Martin boundary and integral representation for harmonic functions of symmetric stable processes’, J. Funct. Anal. 159(1) (1998), 267–294.

    Google Scholar 

  17. Chen, Z.Q. and Song, R.: ‘Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains’, Illinois J. Math. 44(1) (2000), 138–160.

    Google Scholar 

  18. Chung, K.L. and Zhao, Z.: From Brownian Motion to Schrödinger's Equation, Springer-Verlag, New York, 1995.

    Google Scholar 

  19. Davis, B. and Zhang, B.: ‘Moments of the lifetime of conditioned Brownian motion in cones’, Proc. Amer. Math. Soc. 121(3) (1994), 925–929.

    Google Scholar 

  20. DeBlassie, R.: ‘Exit times from cones in Rn of Brownian motion’, Probab. Theory Related Fields 74 (1987), 1–29.

    Google Scholar 

  21. DeBlassie, R.: ‘The first exit time of a two-dimensional symmetric stable process from a wedge’, Ann. Probab. 18 (1990), 1034–1070.

    Google Scholar 

  22. Erdélyi, A. (ed.): Higher Transcendental Functions, Vols. I, II, McGraw-Hill, New York, 1953.

    Google Scholar 

  23. Ikeda, N. and Watanabe, S.: ‘On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes’, J. Math. Kyoto Univ. 2(1) (1962), 79–95.

    Google Scholar 

  24. Kulczycki, T.: ‘Properties of Green function of symmetric stable process’, Probab. Math. Statist. 17(2) (1997), 339–364.

    Google Scholar 

  25. Kulczycki, T.: ‘Intrinsic ultracontractivity for symmetric stable processes’, Bull. Polish Acad. Sci. Math. 46(3) (1998), 325–334.

    Google Scholar 

  26. Kulczycki, T.: ‘Exit time and Green function of cone for symmetric stable processes’, Probab. Math. Statist. 19(2) (1999), 337–374.

    Google Scholar 

  27. Landkof, N.S.: Foundations of Modern Potential Theory, Springer-Verlag, New York, 1972.

    Google Scholar 

  28. Li, W.: ‘The first exit time of Brownian motion from unbounded domain’, Preprint.

  29. M. Lifshitz and Z. Shi, ‘The first exit time of Brownian motion from parabolic domain’, Bernoulli 8(6) (2002), 745–767.

    Google Scholar 

  30. Méndez-Hernández, P.J.: ‘Exit times from cones in ℝn of symmetric stable processes’, Preprint, 2000.

  31. Rubin, B., Fractional Integrals and Potentials, Pitman Monographs 82, Addison-Wesley, 1996.

  32. Ryznar, M.: ‘Estimates of Green function for relativistic α-stable process’, Potential Anal. 17 (2002), 1–23.

    Google Scholar 

  33. Song, R. and Wu, J.-M.: ‘Boundary Harnack principle for symmetric stable processes’, J. Funct. Anal. 168(2) (1999), 403–427.

    Google Scholar 

  34. van den Berg, M.: ‘Subexponential behavior of the Dirichlet heat kernel’, Preprint.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-1395, U.S.A. (e-mail

    Rodrigo Bañuelos

  2. Institute of Mathematics, Wrocław University of Technology, 50-370, Wrocław, Poland (e-mail

    Krzysztof Bogdan

Authors
  1. Rodrigo Bañuelos
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  2. Krzysztof Bogdan
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Bañuelos, R., Bogdan, K. Symmetric Stable Processes in Cones. Potential Analysis 21, 263–288 (2004). https://doi.org/10.1023/B:POTA.0000033333.72236.dc

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