ASYMPTOTICS OF GREEN-FUNCTIONS ON A CLASS OF SOLVABLE LIE-GROUPS

We study the Green kernel at infinity for random walks and diffusions on the solvable Lie groups which are semi-direct extensions of simply connected nilpotent groups by an abelian group isomorphic to R-d. We n otice that Markov processes on Hadamard homogeneous Riemannian manifol ds can be seen as random walks of the above type if their transition k ernel commutes with isometries (e.g. Brownian Motion). This leads to a description of the Martin topology on the Poisson boundary and, in th e case of Riemannian symmetric spaces, to precise asymptotics for the Green kernel and the Martin kernel in the regular directions for 'disc rete brownian motions'.