Minimal thinness for subordinate Brownian motion in half-space

Kim, Panki; Song, Renming; Vondraček, Zoran

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1045-1080.

Nous étudions l’effilement minimal dans le demi-espace $H:=\{x=(\tilde{x},x_d):\tilde{x}\in {ℝ}^{d-1},x_d>0\}$ 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 $H:=\{x=(\tilde{x},x_d):\tilde{x}\in {ℝ}^{d-1},x_d>0\}$ 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.

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

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