Un théorème de Choquet-Deny pour les groupes moyennables. (A theorem of Choquet-Deny for amenable groups) (Q1092497)

Let G be a connected locally compact separable amenable group. Let \(\sigma\) be a positive measure on the Borel \(\sigma\)-field of G. We study the positive Borel functions h on G which satisfy: \[ \forall g\in G,\int _{G}h(gx)\sigma (dx)=\int _{G}h(xg)\sigma (dx)=h(g). \] Under ``smooth'' assumptions on \(\sigma\), we establish an integral representation of these functions in term of exponentials.