Powers of Brownian Green Potentials

Abstract

In this article we study stability properties of \(g_{_{\mathcal {O}}}\!\), the standard Green kernel for \(\mathcal {O}\) an open regular set in \(\mathbb {R}^{d}\). In d ≥ 3 we show that \( g^{\beta }_{_{\mathcal {O}}}\) is again a Green kernel of a Markov Feller process, for any power β ∈ [1,d/(d − 2)). In dimension d = 2, we show the same result for \( g^{\beta }_{_{\mathcal {O}}}\), for any β ≥ 1 and for kernels \(\exp (\alpha g_{_{\mathcal {O}}}\!), \exp (\alpha g_{_{\mathcal {O}}}\!)-1\), for α ∈ (0,2π), when \(\mathcal {O}\) is an open Greenian regular set whose complement contains a ball.