Powers of Brownian Green Potentials

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 $g_O^{\beta }$ is again a Green kernel of a Markov Feller process, for any power $\beta \in [1,d\mathrm{/}(d-2))$. In dimension $d=2$, if $O$ is an open Greenian regular set, we show the same result for $g_O^{\beta }$, for any $\beta \ge 1$ and for the kernel $\exp (\alpha g_O)$, when $\alpha \in (0,2\pi )$.