Two-sided optimal bounds for green functions of half-spaces for relativistic α-stable process

Potential Analysis

Volumn 28, Issue 3, 2008, Pages 201-239

  Grzywny, Tomasz ^a   Ryznar, Michał ^a  

^a WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

First exit time from a ball; Green function; Stable relativistic process; Tail function

Indexed keywords

EID: 43349088493     PISSN: 09262601     EISSN: 1572929X     Source Type: Journal    
DOI: 10.1007/s11118-007-9071-3     Document Type: Article

References (23)

1. Bass, R.F.: Probabilistic Techniques in Analysis. Springer, New York (1995)
2. Billingsley, P.: Probability and Measure. J. Wiley, New York (1979)
3. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Springer, New York (1968)
4. Blumenthal, R.M., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540-554 (1961)
5. Bogdan, K., Stós, A., Sztonyk, P.: Harnack inequality for symmetric stable processes on d-sets. Studia Math. 158(2), 163-198 (2003)
6. Byczkowski, T., Ryznar, M., Małecki, J.: Bessel potentials, hitting distributions and green functions. Trans. Amer. Math. Soc. (2006) (to appear). http://arxiv.org/abs/math/0612176
7. Byczkowski, T., Ryznar, M., Byczkowska, H.: Bessel potentials, Green functions and exponential functionals on half-spaces. Probab. Math. Statist. 26, 155-173 (2006)
8. Carmona, R., Masters, W.C., Simon, B.: Relativistic Schrödinger operators; Asymptotic behaviour of the eigenfunctions. J. Funct. Anal. 91, 117-142 (1990)
9. Chen, Z.-Q., Song, R.: Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201, 262-281 (2003)
10. Chung, K.L., Zhao, Z.: From Brownian motion to Schrödinger's equation. Springer, New York (1995)
11. Durrett, R.: Brownian Motion and Martingales in Analysis. Wadsworth, Belmont (1984)
12. Erdelyi et al. (ed.): Higher Transcendental Functions, vol. II, pp. 1953-1955. McGraw-Hill, New York (1953)
13. Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101, 75-90 (1961)
14. Grzywny, T.: Potential theory for α-stable relativistic process. Master Thesis, Institut of Mathematics and Computer Sciences, Wroclaw University of Technology (2005)
15. Grzywny, T., Ryznar, M.: Estimates of Green function for some perturbations of fractional Laplacian. Illinois J. Math. (to appear), http://www.math.uiuc.edu/hildebr/ijmwww/accepted.html
16. Hörmander, L.: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd edn. Springer, New York (1983)
17. Ikeda, N., 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, 79-95 (1962)
18. Kim, P., Lee, Y.-R.: Generalized 3G theorem, non-local Schrödinger operator and application to relativistic stable process on non-smooth open sets. J Funct. Anal. 246(1), 113-134 (2007)
19. Kulczycki, T., Siudeja, B.: Intrinsic ultracontractivity of Feynman-Kac semigroup for relativistic stable processes. Trans. Amer. Math. Soc. 358, 5025-5057 (2006)
20. Lieb, E.: The stability of matter. Rev. Modern Phys. 48, 553-569 (1976)
21. Rao, M., Song, R., Vondracek, Z.: Green function estimates and Harnack inequality for subordinate Brownian motions. Potential Anal. 25, 1-27 (2006)
22. Ryznar, M.: Estimate of Green function for relativistic α-stable processes. Potential Anal. 17, 1-23 (2002)
23. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30. Princeton, New Jork (1970)