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Brownian motion and Harmonic functions on Sol(p,q)
Brofferio S. et al
http://hal.archives-ouvertes.fr/hal-00710645
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Sara Brofferio ()1, Maura Salvatori2, Wolfgang Woess ()3
1:  LM-Orsay - Laboratoire de Mathématiques d'Orsay
http://www.math.u-psud.fr
CNRS : UMR8628 – Université Paris XI - Paris Sud
France
2:  Dipartimento de Matematica [Milano]
Università degli studi di Milano
Via Saldini 50 20133 Milano
Italy
3:  TU Graz - Institut für Mathematische Strukturtheorie (Math C)
http://www.math.tugraz.at
Technische Universität, Graz
Steyrergasse 30 8010 Graz
Austria
Mathematics/Probability
Mathematics/Differential Geometry
Brownian motion and Harmonic functions on Sol(p,q)
The Lie group Sol(p,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 Sol(p,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 Sol(p,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 Sol(p,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.
English
2012-10-11

Sol-group – hyperbolic plane – horocyclic product – Laplacian – Brownian motion – central limit theorem – rate of escape – boundary – positive harmonic functions
58J65, 31C12, 60J50

MAURA SALVATORI, WOLFGANG WOESS