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Asymptotics of Green Functions on a Class of Solvable Lie Groups

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Abstract

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 Rd. We notice that Markov processes on Hadamard homogeneous Riemannian manifolds can be seen as random walks of the above type if their transition kernel commutes with isometries (e.g. Brownian Motion). This leads to a description of the Martin topology on the Poisson boundary and, in the case of Riemannian symmetric spaces, to precise asymptotics for the Green kernel and the Martin kernel in the regular directions for ‘discrete brownian motions’.

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  1. Laboratoire de Probabilités, URA 224 CNRS, Université Pierre et Marie Curie, Tour 56 3éme étage, 4 Place Jussieu, 75252, Paris Cedex 05, france

    Martine Babillot

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Babillot, M. Asymptotics of Green Functions on a Class of Solvable Lie Groups. Potential Analysis 8, 69–100 (1998). https://doi.org/10.1023/A:1017991923947

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