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1 LM-Orsay - Laboratoire de Mathématiques d-Orsay 2 Dipartimento de Matematica Milano 3 TU Graz - Institut für Mathematische Strukturtheorie Math C

Abstract : The Lie group Solp,q is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by x,y -> exp{p z} x, exp{-q z} y, where z is in R. Viewing Solp,q as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and compute the rate of escape. Also, we introduce the natural geometric compactification of Solp,q and explain how Brownian motion converges almost surely to the boundary in the resulting topology. We also study all positive harmonic functions for the Laplacian with drift, and determine explicitly all minimal harmonic functions. All this is carried out with a strong emphasis on understanding and using the geometric features of Solp,q, and in particular the fact that it can be described as the horocyclic product of two hyperbolic planes with curvatures -p^2 and -q^2, respectively.

Keywords : Sol-group hyperbolic plane horocyclic product Laplacian Brownian motion central limit theorem rate of escape boundary positive harmonic functions

Author: Sara Brofferio - Maura Salvatori - Wolfgang Woess -



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