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Zbl 0721.60005
Babillot, M.
Comportement asymptotique du mouvement brownien sur une variété homogène à courbure négative ou nulle. (Asymptotic behaviour of Brownian manifolds with non positive curvature). (French)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 27, No.1, 61-90 (1991). ISSN 0246-0203

Author's summary: A homogeneous Riemannian manifold M with nonpositive sectional curvature is a solvable Lie group which is the skew product of an Abelian group A isomorphic to $R\sp d$ and a nilpotent Lie group N. Under the assumption of no Euclidean factor in the De Rham decomposition of M, we study the geometrical configuration of the weights of the adjoint representation of A on N, from which we deduce that the sum of the weights belongs to the interior of some Weyl chamber, like in the symmetric case. This result, with classical arguments, gives the asymptotic behavior of Brownian motion on M and the fact that the exit boundary can be identified with N.
[A.Mukherjea (Tampa)]

MSC 2000:: *60B15 Probability measures on groups
53C30 Homogeneous manifolds

Keywords: Riemannian manifold; nilpotent Lie group; De Rham decomposition; Weyl chamber; Brownian motion

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