Minimal thinness for subordinate Brownian motion in half-space
Let \(H\) denote the upper half-space in \(\mathbb{R} ^ {d}\), for \(d \geq 2\). The goal of this paper is to show the following two results. Theorem 1. Let \(A\) be a Borel subset of \(H\) and assume that \[ \int_ {A\cap B(0, 1)} | x |^ {-d}\, d x = \infty. \] Then \(A\) is not minimally thin in \(H\) with respect to Brownian motion at \(z=0\). Theorem 2. Let \(f:\mathbb{R}^ {d-1} \rightarrow [0, +\infty )\) be a Lipschitz function. The set \[ A:= \{ x=(\tilde x , x_d) \in H : 0 < x_d \leq f(\tilde x) \} \] is minimally thin in \(H\) with respect to the Brownian motion at \(z=0\) if and only if \[ \int _ {\{|\tilde x|< 1\}} f(\tilde x) | \tilde x |^ {-d} d x < \infty \] are still valid in exactly the same form when the Brownian motion is replaced with a wide class of rotationally invariant Lévy processes (including rotationally invariant \(\alpha \)-stable processes, \(\alpha \in (0, 2)\)). The proofs rely on recent results of the authors, namely, a boundary Harnack principle and a sharp estimate of the Green function for certain subordinate Brownian motions (see [Stochastic Processes Appl. 119, No. 5, 1601--1631 (2009; Zbl 1166.60046)]).
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