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 ≥ 3 we show that g^β__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^β__𝒪 , for any β ≥ 1 and for kernels exp (α g__𝒪), exp (α g__𝒪)-1 , for α ∈ (0,2 π ), when 𝒪 is an open Greenian regular set whose complement contains a ball.