no. 3
Minimal thinness for subordinate Brownian motion in half-space
Kim, Panki1; Song, Renming2; Vondraček, Zoran3
1 Department of Mathematics, Seoul National University, Building 27, 1 Gwanak-ro, Gwanak-gu, Seoul 151-747, Republic of Korea
2 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
3 Department of Mathematics, University of Zagreb, Zagreb, Croatia
Annales de l'Institut Fourier, Volume 62 (2012) no. 3, pp. 1045-1080.

We study minimal thinness in the half-space H:={x=(x ˜,x d ):x ˜∈ℝ d-1 ,x d >0} for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of H below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.

Nous étudions l’effilement minimal dans le demi-espace H:={x=(x ˜,x d ):x ˜∈ℝ d-1 ,x d >0} pour une classe grande de mouvements brownien subordonnés. Nous montrons que le même test pour l’effilement minimal d’un sous-ensemble sous le graphe d’une fonction non-négative lipschitzienne est valable pour tous les processus dans la classe considérée. Dans le cas classique du mouvement brownien ce test a été démontré par Burdzy.

DOI: 10.5802/aif.2716
Classification: 60J50, 31C40, 31C35, 60J45, 60J75
Keywords: Minimal thinness, subordinate Brownian motion, boundary Harnack principle, Green function, Martin kernel
Kim, Panki 1; Song, Renming 2; Vondraček, Zoran 3

1 Department of Mathematics, Seoul National University, Building 27, 1 Gwanak-ro, Gwanak-gu, Seoul 151-747, Republic of Korea
2 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
3 Department of Mathematics, University of Zagreb, Zagreb, Croatia 
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Kim, Panki; Song, Renming; Vondraček, Zoran. Minimal thinness for subordinate Brownian motion in half-space. Annales de l'Institut Fourier, Volume 62 (2012) no. 3, pp. 1045-1080. doi : 10.5802/aif.2716. https://aif.centre-mersenne.org/articles/10.5802/aif.2716/

[1] Armitage, D. H.; Gardiner, S. J. Classical potential theory, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 2001 | MR

[2] Bertoin, J. Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996 | MR | Zbl

[3] Beurling, A. A minimum principle for positive harmonic functions, Ann. Acad. Sci. Fenn. Ser. A I No., Volume 372 (1965), pp. 7 | MR | Zbl

[4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987 | MR | Zbl

[5] Bliedtner, J.; Hansen, W. Potential theory. An analytic and probabilistic approach to balayage, Universitext, Springer-Verlag, Berlin, 1986 | MR | Zbl

[6] Bogdan, K. The boundary Harnack principle for the fractional Laplacian, Studia Math., Volume 123 (1997) no. 1, pp. 43-80 | MR | Zbl

[7] Bogdan, K. Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J., Volume 29 (1999) no. 2, pp. 227-243 http://projecteuclid.org/getRecord?id=euclid.hmj/1206125005 | MR | Zbl

[8] Bogdan, K.; Byczkowski, T.; Kulczycki, T.; Ryznar, M.; Song, R.; Vondraček, Z. Potential analysis of stable processes and its extensions, Lecture Notes in Mathematics, 1980, Springer-Verlag, Berlin, 2009 (Edited by Piotr Graczyk and Andrzej Stos) | DOI | MR

[9] Burdzy, K. Brownian excursions and minimal thinness. I, Ann. Probab., Volume 15 (1987) no. 2, pp. 676-689 http://www.jstor.org/stable/2244068 | DOI | MR | Zbl

[10] Chen, Z.-Q.; Kim, P.; Song, R. Global heat kernel estimate for Δ+Δ α/2 in half space like open sets, Preprint (2011) | MR

[11] Chen, Z.-Q.; Kim, P.; Song, R.; Vondraček, Z. Boundary Harnack principle Δ+Δ α/2 , To appear in Trans. Amer. Math. Soc. (2011) | MR

[12] Chen, Z.-Q.; Kim, P.; Song, R.; Vondraček, Z. Sharp Green function estimates for Δ+Δ α/2 in C 1,1 open sets and their applications, To appear in Illinois J. Math. (2011)

[13] Chen, Z.-Q.; Song, R. Martin boundary and integral representation for harmonic functions of symmetric stable processes, J. Funct. Anal., Volume 159 (1998) no. 1, pp. 267-294 | DOI | MR | Zbl

[14] Dahlberg, B. A minimum principle for positive harmonic functions, Proc. London Math. Soc. (3), Volume 33 (1976) no. 2, pp. 238-250 | DOI | MR | Zbl

[15] Doob, J. L. Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 262, Springer-Verlag, New York, 1984 | MR | Zbl

[16] Föllmer, H. Feine Topologie am Martinrand eines Standardprozesses, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 12 (1969), pp. 127-144 | DOI | MR | Zbl

[17] Gardiner, S. J. A short proof of Burdzy’s theorem on the angular derivative, Bull. London Math. Soc., Volume 23 (1991) no. 6, pp. 575-579 | DOI | MR | Zbl

[18] Kim, P.; Song, R. Boundary behavior of harmonic functions for truncated stable processes, J. Theoret. Probab., Volume 21 (2008) no. 2, pp. 287-321 | DOI | MR

[19] Kim, P.; Song, R.; Vondraček, Z. Boundary Harnack principle for subordinate Brownian motions, Stochastic Process. Appl., Volume 119 (2009) no. 5, pp. 1601-1631 | DOI | MR

[20] Kim, P.; Song, R.; Vondraček, Z. On the potential theory of one-dimensional subordinate Brownian motions with continuous components, Potential Anal., Volume 33 (2010) no. 2, pp. 153-173 | DOI | MR

[21] Kim, P.; Song, R.; Vondraček, Z. Potential theory of subordinate Brownian motions revisited, To appear in a volume in honor of Prof. Jiaan Yan (2011)

[22] Kim, P.; Song, R.; Vondraček, Z. Potential theory of subordinate Brownian motions with Gaussian components, Preprint (2011)

[23] Kim, P.; Song, R.; Vondraček, Z. Two-sided Green function estimates for killed subordinate Brownian motions, To appear in Proc. London Math. Soc. (2012) | DOI

[24] Kunita, H.; Watanabe, T. Markov processes and Martin boundaries. I, Illinois J. Math., Volume 9 (1965), pp. 485-526 | MR | Zbl

[25] Naïm, L. Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, Grenoble, Volume 7 (1957), pp. 183-281 | DOI | Numdam | MR | Zbl

[26] Port, S. C.; Stone, C. J. Brownian motion and classical potential theory, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978 (Probability and Mathematical Statistics) | MR | Zbl

[27] Rao, M.; Song, R.; Vondraček, Z. Green function estimates and Harnack inequality for subordinate Brownian motions, Potential Anal., Volume 25 (2006) no. 1, pp. 1-27 | DOI | MR

[28] Schilling, R. L.; Song, R.; Vondraček, Z. Bernstein functions, de Gruyter Studies in Mathematics, 37, Walter de Gruyter & Co., Berlin, 2010 (Theory and applications) | MR

[29] Sjögren, P. La convolution dans L 1 faible de R n , Séminaire Choquet, 13e année (1973/74), Initiation à l’analyse, Exp. No. 14, Secrétariat Mathématique, Paris, 1975, pp. 10 | Numdam | MR | Zbl

[30] Sjögren, P. Une propriété des fonctions harmoniques positives, d’après Dahlberg, Séminaire de Théorie du Potentiel de Paris, No. 2 (Univ. Paris, Paris, 1975–1976), Springer, Berlin, 1976, p. 275-282. Lecture Notes in Math., Vol. 563 | MR | Zbl

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