Minimal thinness for subordinate Brownian motion in half-space
[L’effilement minimal pour le mouvement brownien subordonné dans un demi-espace]
Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1045-1080.

Nous étudions l’effilement minimal dans le demi-espace H:={x=(x ˜,x d ):x ˜ 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.

We study minimal thinness in the half-space H:={x=(x ˜,x d ):x ˜ 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.

DOI : 10.5802/aif.2716
Classification : 60J50, 31C40, 31C35, 60J45, 60J75
Keywords: Minimal thinness, subordinate Brownian motion, boundary Harnack principle, Green function, Martin kernel
Mot clés : effilement minimal, mouvement brownien subordonné, principe de Harnack à la frontiére, fonction de Green, noyau de Martin
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
@article{AIF_2012__62_3_1045_0,
     author = {Kim, Panki and Song, Renming and Vondra\v{c}ek, Zoran},
     title = {Minimal thinness for subordinate {Brownian} motion in half-space},
     journal = {Annales de l'Institut Fourier},
     pages = {1045--1080},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {3},
     year = {2012},
     doi = {10.5802/aif.2716},
     mrnumber = {3013816},
     zbl = {1273.60096},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2716/}
}
TY  - JOUR
AU  - Kim, Panki
AU  - Song, Renming
AU  - Vondraček, Zoran
TI  - Minimal thinness for subordinate Brownian motion in half-space
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 1045
EP  - 1080
VL  - 62
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2716/
DO  - 10.5802/aif.2716
LA  - en
ID  - AIF_2012__62_3_1045_0
ER  - 
%0 Journal Article
%A Kim, Panki
%A Song, Renming
%A Vondraček, Zoran
%T Minimal thinness for subordinate Brownian motion in half-space
%J Annales de l'Institut Fourier
%D 2012
%P 1045-1080
%V 62
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2716/
%R 10.5802/aif.2716
%G en
%F AIF_2012__62_3_1045_0
Kim, Panki; Song, Renming; Vondraček, Zoran. Minimal thinness for subordinate Brownian motion in half-space. Annales de l'Institut Fourier, Tome 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

Cité par Sources :