Censored stable processes

Probability Theory and Related Fields

Volumn 127, Issue 1, 2003, Pages 89-152

  Bogdan, Krzysztof ^a   Burdzy, Krzysztof ^b   Chen, Zhen Qing ^b  

^a WROCLAW UNIVERSITY OF TECHNOLOGY   (Poland)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0141457364     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-003-0275-1     Document Type: Article

References (47)

1. Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. Springer, 1996
2. Ancona, A.: Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann, Inst. Fourier (Grenoble) 28, 169-213 (1978)
3. Bass, R., Burdzy, K.: A probabilistic proof of the boundary Harnack principle. In seminar on stochastic processes, 1989, Progr. Probab. 18, 1-16 (1990), Birkhäuser
4. Bertoin, J: Lévy Processes. Cambridge University Press, 1996
5. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, 1968
6. 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)
7. Bogdan, K.: The boundary Harnack principle for the fractional Laplacian. Studia Math. 123(1), 43-80 (1997)
8. Bogdan, K., Byczkowski, T.: Probabilistic proof of boundary Harnack principle for α-harmonic functions. Potential Anal. 11, 135-156 (1999)
9. Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Statist. 20(2), 293-335 (2000)
10. Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133(1), 53-92 (1999)
11. Burdzy, K., Chen, Z.-Q., Sylvester, J.: Heat equation and reflected Brownian motion in time-dependent domains. To appear in Ann. Probab. 2003
12. Caetano, A.M.: Approximation by functions of compact support in Besov-Triebel-Lizorkin spaces on irregular domains. Studia Math. 142, 47-63 (2000)
13. Chen, Z.-Q.: Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6(2), 227-266 (1999)
14. Chen, Z.-Q.: On reflected Dirichlet spaces. Probab. Theory Relat. Fields 94, 135-162 (1992)
15. Chen, Z.-Q., Fitzsimmons, P.J., Takeda, M., Ying, J., Zhang, T.-S.: Absolute continuity of symmetric Markov processes. To appear in Ann. Probab. 2003
16. Chen, Z.-Q., Kim, P.: Green function estimate for censored stable processes. Probab. Theory Relat. Fields 124, 595-610 (2002)
17. Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Preprints, 2002
18. Chen, Z.-Q., Song, R.: Estimates on Green functions and Poisson kernels of symmetric stable processes. Math. Ann. 312, 465-601 (1998)
19. Chen, Z.-Q., Song, R.: A note on Green function estimates for symmetric stable processes. Preprint, 2003
20. Chen, Z.-Q., Song, R.: Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains. Illinois J. Math. 44(1), 138-160 (2000)
21. Chen, Z.-Q., Song, R.: Martin boundary and integral representation for harmonic functions of symmetric stable processes. J. Funct. Anal. 159, 267-294 (1998)
22. Chen, Z.-Q., Song, R.: Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. J. Funct. Anal. 150, 204-239 (1997)
23. Chung, K.L.: Green's function for a ball. In: Seminar on Stochastic Processes, 1986, pages 1-13, Boston, 1987. Progr. Probab. Statist., 13, Birkhäuser.
24. Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger's Equation. Springer-Verlag, New York, 1995
25. Dahlberg, B.: Estimates of harmonic measure. Arch. Rat. Mech. Anal. 65, 275-288 (1977)
26. Ethier, S.N., Kurtz, T.G.: Markov Processes. Characterization and Convergence. Wiley, New York, 1986
27. Farkas, W., Jacob, N.: Sobolev spaces on non-smooth domains and Dirichlet forms related to subordinate reflecting diffusions. Math. Nachr. 224, 75-104 (2001)
28. Friedman, A.: Foundations of modern analysis. Dover Publ. Inc., New York, 1982
29. Fukushima, M., Oshima, Y., Takeda, M: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin, 1994
30. Fukushima, M., Uemura, T.: On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets. To appear in Potential Analysis, 2003
31. Getoor, R.K. : First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101, 75-90 (1961)
32. Gong, G., Qian, M.-P, Zhao, Z.: Killed diffusions and their conditioning. Probab. Theory Relat. Fields 80, 151-167 (1988)
33. Hoh, W.: Pseudo differential operators with negative definite symbols of variable order. Rev. Mat. Iberoamericana 16(2), 219-241 (2000)
34. Ikeda, N., Nagasawa, M., Watanabe, S.: A construction of Markov process by piecing out. Proc. Japan Academy 42, 370-375 (1966)
35. Ikeda, N., Watanabe, S.: On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2(1), 79-95 (1962)
36. Jerison, D., Kenig, C.: Boundary value problems on Lipschitz domains. In: Studies in Partial Differential Equations, ed. W. Littman, Math. Assoc. Amer., 1982
37. Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71-88 (1981)
38. n. Math. Reports, Vol.2, Part 1, Harwood Acad. Publ., London, 1984
39. Komatsu, T.: Uniform estimates for fundamental solutions associated with non-local Dirichlet forms. Osaka J. Math. 32, 833-860 (1995)
40. Kulczycki, T.: Properties of Green function of symmetric stable processes. Probab. Math. Statist. 17(2), 339-364 (1997)
41. Landkof, N.S.: Foundations of Modern Potential Theory. Springer-Verlag, New York, 1972
42. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York-London, 1994
43. Sharpe, M.: General Theory of Markov Processes. Academic Press, Inc. 1988
44. Silverstein, M.L.: Symmetric Markov Processes. Lecture Notes in Math. vol. 426, Springer-Verlag, 1974
45. Song, R., Wu, J.M.: Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168(2), 403-427 (1999)
46. Triebel, H.: Interpolation theory, function spaces, differential operators, second edition. Johann Ambrosius Barth, 1995
47. Wu, J.M.G.: Comparisons of kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Ann. Inst. Fourier (Grenoble) 28(4), 147-167 (1978)