Powers of Brownian Green Potentials

Abstract

In this article we study stability properties of $g_O$, the standard Green kernel for $O$ an open regular set in $R^d$. In $d\ge 3$ we show that $g_O^β$ is again a Green kernel of a Markov Feller process, for any power $β\in [1,d/(d-2))$. In dimension $d=2$, if $O$ is an open Greenian regular set, we show the same result for $g_O^β$, for any $β\ge 1$ and for the kernel $\exp(αg_O)$, when $α\in (0,2π)$.