Potential theory for Lévy stable processes

• Autorzy: Bogdan K. , Stós A. , Sztonyk P.
• Języki publikacji: EN

Abstrakty

EN

We investigate properties of functions that are harmonic with respect to symmetric stable processes which are not necessarily rotation invariant. We prove the Harnack inequality and the boundary Harnack principle for a Lipschitz domain. To obtain this we use Poisson kernel estimates. We also give an estimate concerning the rate of decay of harmonic functions near the boundary of the domain.

Słowa kluczowe

EN

boundary Harnack principle; Harnack inequality; stable processes

PL

nierówność Harnacka; proces stabilny; zasada ograniczenia Harnacka

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2002
• Tom: Vol. 50, no 3
• Strony: 361--372
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Bogdan K.

• Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

autor

Stós A.

• Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

autor

Sztonyk P.

• Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

• [1] J. Bertoin, Lévy processes, Cambridge University Press, Cambridge 1996.
• [2] K. Bogdan, T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Stud. Math., 133 (1) (1999) 53-92.
• [3] K. Bogdan, T. Byczkowski, Potential theory of the Schrödinger operator based on the fractional Laplacian, Prob. Math. Stat., 20 (2000) 293-335.
• [4] R. M. Blumenthal, R. K. Getoor, Markov Processes and Potential Theory, Pure Appl. Math., Academic Press Inc., New York 1968.
• [5] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1) (1997) 43-80.
• [6] K. L. Chung, Z. Zhao, From Brownian motion to Schrödinger’s equation, Springer-Verlag, New York 1995.
• [7] P. Głowacki, W. Hebisch, Pointwise estimates for densities of stable semigroups of measures, Studia Math., 104 (1993) 243-258.
• [8] N. Ikeda, S. Watanabe, 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.
• [9] W. E. Pruitt, S. J. Taylor, The potential kernel and hitting probabilities for the general stable process in Rd, Trans. Amer. Math. Soc., 146 (1969) 299-321.
• [10] R. Song, J.-M. Wu, Boundary Harnack principle for symmetric stable processes, J. Funct. Anal., 168 (2) (1999) 403-427.
• [11] P. Sztonyk, On harmonic measures for Lévy processes, Prob. Math. Statist., 20 (2) (2000) 383-390.
• [12] S. J. Taylor, Sample path properties of a transient stable process, J. Math. Mech., 16 (1967) 1229-1246.