Boundary Harnack principle for subordinate Brownian motions

Stochastic Processes and their Applications

Volumn 119, Issue 5, 2009, Pages 1601-1631

  Kim, Panki ^a   Song, Renming ^b   Vondraček, Zoran ^c  

^a Seoul National University   (South Korea)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^c UNIVERSITY OF ZAGREB   (Croatia)

Author keywords

Bernstein functions; Boundary Harnack principle; Complete Bernstein functions; Green functions; Harmonic functions; Harnack inequality; Martin boundary; Mixture of symmetric stable processes; Poisson kernels; Subordinate Brownian motion; Subordinator; Symmetric stable processes

Indexed keywords

AMBER; FARM BUILDINGS; FOURIER SERIES; GREEN'S FUNCTION; HARMONIC ANALYSIS; MIXTURES; POISSON DISTRIBUTION; POISSON EQUATION; STABILITY CRITERIA;

BERNSTEIN FUNCTIONS; BOUNDARY HARNACK PRINCIPLE; COMPLETE BERNSTEIN FUNCTIONS; GREEN FUNCTIONS; HARMONIC FUNCTIONS; HARNACK INEQUALITY; MARTIN BOUNDARY; POISSON KERNELS; SUBORDINATE BROWNIAN MOTION; SUBORDINATOR; SYMMETRIC STABLE PROCESSES;

BROWNIAN MOVEMENT;

EID: 62249180786     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2008.08.003     Document Type: Article

References (29)

1. Bertoin J. Lévy Processes (1996), Cambridge University Press, Cambridge
2. Blumenthal R.M., and Getoor R.K. Markov Processes and Potential Theory (1968), Academic Press, New York
3. Bingham N.H., Goldie C.M., and Teugels J.L. Regular Variation (1987), Cambridge University Press, Cambridge
4. Bogdan K. The boundary Harnack principle for the fractional Laplacian. Studia Math. 123 1 (1997) 43-80
5. Bogdan K. Representation of α-harmonic functions in Lipschitz domains. Hiroshima Math. J. 29 (1999) 227-243
6. Bogdan K., Kulczycki T., and Kwasnicki M. Estimates and structure of α-harmonic functions. Probab. Theory Related Fields 140 (2008) 345-381
7. Bogdan K., Stos A., and Sztonyk P. Potential theory for Lévy stable processes. Bull. Polish Acad. Sci. Math. 50 (2002) 361-372
8. Chen Z.-Q., and Song R. Martin boundary and integral representation for harmonic functions of symmetric stable processes. J. Funct. Anal. 159 (1998) 267-294
9. Chen Z.-Q., and Song R. Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201 (2003) 262-281
10. Fristedt B.E. Sample functions of stochastic processes with stationary, independent increments. Advances in Probability and Related Topics vol. 3 (1974), Dekker, New York 241-396
11. Fukushima M., Oshima Y., and Takeda M. Dirichlet Forms and Symmetric Markov Processes (1994), Walter De Gruyter, Berlin
12. Grzywny T., and Ryznar M. Estimates of Green functions for some perturbations of fractional Laplacians. Illinois J. Math. 51 4 (2007) 1409-1438
13. Jacob N. Pseudo Differential Operators and Markov Processes vol. 1 (2001), Imperial College Press, London
14. Jerison D.S., and Kenig C.E. Boundary behavior of harmonic functions in non-tangentially accessible domains. Adv. Math. 46 (1982) 80-147
15. Kim P., and Lee Y.-R. Generalized 3G theorem and application to relativistic stable process on non-smooth open sets. J. Funct. Anal. 246 1 (2007) 113-134
16. Kim P., and Song R. Potential Theory of truncated stable processes. Math. Z. 256 (2007) 139-173
17. Kim P., and Song R. Boundary behavior of harmonic functions for truncated stable processes. J. Theoret. Probab. 21 (2008) 287-321
18. Kunita H., and Watanabe T. Markov processes and Martin boundaries. Illinois J. Math. 9 (1965) 485-526
19. Kyprianou A.E. Introductory Lectures on Fluctuations of Lévy Processes with Applications (2006), Springer, Berlin
20. Martio O., and Vuorinen M. Whitney cubes, p-capacity, and Minkowski content. Exposition. Math. 5 1 (1987) 17-40
21. Rao M., Song R., and Vondraček Z. Green function estimates and Harnack inequality for subordinate Brownian motions. Potential Anal. 25 1 (2006) 1-27
22. Ryznar M. Estimates of Green function for relativistic α-stable process. Potential Anal. 17 (2002) 1-23
23. Schilling R.L. Subordination in the sense of Bochner and a related functional calculus. J. Austral. Math. Soc. Ser. A 64 (1998) 368-396
24. Song R., and Vondraček Z. Harnack inequalities for some classes of Markov processes. Math. Z. 246 (2004) 177-202
25. Song R., and Vondraček Z. Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Probab. 19 (2006) 817-847
26. R. Song, Z. Vondraček, Potential theory of subordinate Brownian motions, preprint, 2007
27. Song R., and Wu J. Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168 2 (1999) 403-427
28. Sztonyk P. On harmonic measure for Lévy processes. Probab. Math. Statist. 20 (2000) 383-390
29. Sztonyk P. Boundary potential theory for stable Lévy processes. Colloq. Math. 95 2 (2003) 191-206