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We give sharp global estimates for the Green function, Martin kernel and Poisson kernel in Lipschitz domains for symmetric α-stable processes. We give some applications of the estimates.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
419--441
Opis fizyczny
Biblogr. 22 poz.
Twórcy
autor
- Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego, 27 50-370 Wrocław, Poland
Bibliografia
- [1] R. M. Blumenthal and R. K. Geto or, Markov Processes and Potential Theory, Springer, New York 1968.
- [2] R. M. Blumenthal, R. K. Getoor and D. B. Ray, On the distribution of first hits for the symmetric stable process, Trans. Amer. Math. Soc. 99 (1961), pp. 540-554.
- [3] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), pp. 43-80.
- [4] K. Bogdan, Representation of a-harmonic functions in Lipschitz domains, Hiroshima Math. J. 29 (1999), pp. 227-243.
- [5] K. Bogdan, Sharp estimates for the Green function in Lipschitz domains, J. Math. Anal. Appl. 243 (2000), pp. 326-337.
- [6] K. Bogdan and T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1) (1999), pp. 53-92.
- [7] K. Bogdan and T. Byczkowski, Probabilistic proof of the boundary Harnack principle for cc-harmonic functions, Potential Anal. 11 (1999), pp. 135-156.
- [8] K. Bogdan and T. Byczkowski, Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Statist. 20 (2) (2000), pp. 293-335.
- [9] K. Bogdan and B. Dyda, Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domains, preprint.
- [10] K. Burdzy and T. Kulczycki, Stable processes have thorns, Ann. Probab. (2002).
- [11] Z.-Q. Chen and P. Kim, Green function estimate for censored stable processes, Probab. Theory Related Fields (2002).
- [12] Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable process, Math. Ann. 312 (3) (1998), pp. 465-501.
- [13] Z.-Q. Chen and R. Song, A note on the Green function estimates for symmetric stable processes, preprint.
- [14] Z.-Q. Chen and R. Song, Martin boundary and integral representation for harmonic functions of symmetric stable processes, J. Funct. Anal. 159 (1998), pp. 267-294.
- [15] K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger s Equation, Springer, New York 1995.
- [16] B. Dahl berg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (3) (1977), pp. 275-288.
- [17] N. Ikeda and S. Watanabe, On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes, J. Math. Kyoto Univ. 2 (1962), pp. 79-95.
- [18] D. S. Jerison and C. E. Kenig, Boundary value problems on Lipschitz domains, in: Studies in Partial Differential Equations, W. Littman (Ed.), Vol. 23, Studies in Mathematics, Math. Assoc, of America, Washington, DC, 1982, pp. 1-68.
- [19] T. Kulczycki, Properties of Green function of symmetric stable process, Probab. Math. Statist 17 (2) (1997), pp. 339-364.
- [20] K. Michalik and K. Samotij, Martin representation for OL-harmonic functions, Probab. Math. Statist. 20 (1) (2000), pp. 75-91.
- [21] R. Song and J. M. Wu, Boundary Harnack principle for symmetric stable processes, J. Funct. Anal. 168 (1999), pp. 403-427.
- [22] Z. Zhao, Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), pp. 309-334.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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