Minimal thinness for subordinate Brownian motion in half-space
[L’effilement minimal pour le mouvement brownien subordonné dans un demi-espace]
Kim, Panki ; Song, Renming ; Vondraček, Zoran
Annales de l'Institut Fourier, Tome 62 (2012), p. 1045-1080 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

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.

Publié le : 2012-01-01
DOI : https://doi.org/10.5802/aif.2716
Classification:  60J50,  31C40,  31C35,  60J45,  60J75
Mots clés: effilement minimal, mouvement brownien subordonné, principe de Harnack à la frontiére, fonction de Green, noyau de Martin 
@article{AIF_2012__62_3_1045_0,
     author = {Kim, Panki and Song, Renming and Vondra\v cek, Zoran},
     title = {Minimal thinness for subordinate Brownian motion in half-space},
     journal = {Annales de l'Institut Fourier},
     volume = {62},
     year = {2012},
     pages = {1045-1080},
     doi = {10.5802/aif.2716},
     zbl = {1273.60096},
     mrnumber = {3013816},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 1045-1080. doi : 10.5802/aif.2716. http://gdmltest.u-ga.fr/item/AIF_2012__62_3_1045_0/

[1] Armitage, D. H.; Gardiner, S. J. Classical potential theory, Springer-Verlag London Ltd., London, Springer Monographs in Mathematics (2001) | MR 1801253

[2] Bertoin, J. Lévy processes, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 121 (1996) | MR 1406564 | Zbl 0861.60003

[3] Beurling, A. A minimum principle for positive harmonic functions, Ann. Acad. Sci. Fenn. Ser. A I No., Tome 372 (1965), pp. 7 | MR 188466 | Zbl 0139.06402

[4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications, Tome 27 (1987) | MR 898871 | Zbl 0667.26003

[5] Bliedtner, J.; Hansen, W. Potential theory. An analytic and probabilistic approach to balayage, Springer-Verlag, Berlin, Universitext (1986) | MR 850715 | Zbl 0706.31001

[6] Bogdan, K. The boundary Harnack principle for the fractional Laplacian, Studia Math., Tome 123 (1997) no. 1, pp. 43-80 | MR 1438304 | Zbl 0870.31009

[7] Bogdan, K. Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J., Tome 29 (1999) no. 2, pp. 227-243 http://projecteuclid.org/getRecord?id=euclid.hmj/1206125005 | MR 1704245 | Zbl 0936.31008

[8] Bogdan, K.; Byczkowski, T.; Kulczycki, T.; Ryznar, M.; Song, R.; Vondraček, Z. Potential analysis of stable processes and its extensions, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1980 (2009) (Edited by Piotr Graczyk and Andrzej Stos) | Article | MR 2569321

[9] Burdzy, K. Brownian excursions and minimal thinness. I, Ann. Probab., Tome 15 (1987) no. 2, pp. 676-689 http://www.jstor.org/stable/2244068 | Article | MR 885137 | Zbl 0656.60051

[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 2775111

[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 2912450

[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., Tome 159 (1998) no. 1, pp. 267-294 | Article | MR 1654115 | Zbl 0954.60003

[14] Dahlberg, B. A minimum principle for positive harmonic functions, Proc. London Math. Soc. (3), Tome 33 (1976) no. 2, pp. 238-250 | Article | MR 409847 | Zbl 0342.31004

[15] Doob, J. L. Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 262 (1984) | MR 731258 | Zbl 0549.31001

[16] Föllmer, H. Feine Topologie am Martinrand eines Standardprozesses, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Tome 12 (1969), pp. 127-144 | Article | MR 245092 | Zbl 0172.21601

[17] Gardiner, S. J. A short proof of Burdzy’s theorem on the angular derivative, Bull. London Math. Soc., Tome 23 (1991) no. 6, pp. 575-579 | Article | MR 1135189 | Zbl 0718.31005

[18] Kim, P.; Song, R. Boundary behavior of harmonic functions for truncated stable processes, J. Theoret. Probab., Tome 21 (2008) no. 2, pp. 287-321 | Article | MR 2391246

[19] Kim, P.; Song, R.; Vondraček, Z. Boundary Harnack principle for subordinate Brownian motions, Stochastic Process. Appl., Tome 119 (2009) no. 5, pp. 1601-1631 | Article | MR 2513121

[20] Kim, P.; Song, R.; Vondraček, Z. On the potential theory of one-dimensional subordinate Brownian motions with continuous components, Potential Anal., Tome 33 (2010) no. 2, pp. 153-173 | Article | MR 2658980

[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) | Article

[24] Kunita, H.; Watanabe, T. Markov processes and Martin boundaries. I, Illinois J. Math., Tome 9 (1965), pp. 485-526 | MR 181010 | Zbl 0147.16505

[25] Naïm, L. Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, Grenoble, Tome 7 (1957), pp. 183-281 | Article | Numdam | MR 100174 | Zbl 0086.30603

[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 492329 | Zbl 0413.60067

[27] Rao, M.; Song, R.; Vondraček, Z. Green function estimates and Harnack inequality for subordinate Brownian motions, Potential Anal., Tome 25 (2006) no. 1, pp. 1-27 | Article | MR 2238934

[28] Schilling, R. L.; Song, R.; Vondraček, Z. Bernstein functions, Walter de Gruyter & Co., Berlin, de Gruyter Studies in Mathematics, Tome 37 (2010) (Theory and applications) | MR 2598208

[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 477597 | Zbl 0317.42019

[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 588344 | Zbl 0339.31005