Projects per year
Abstract
The Lie group Sol(p,q) is the semidirect product induced by the action of formula on formula which is given by (x,y)↦(epzx,e−qzy), formula. Viewing Sol(p,q) as a threedimensional manifold, it carries a natural Riemannian metric and Laplace–Beltrami operator. We add a linear drift term in the zvariable 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 these are 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 −p2 and −q2, respectively.
Original language  English 

Pages (fromto)  51825218 
Journal  International Mathematics Research Notices 
Volume  2012 
Issue number  22 
DOIs  
Publication status  Published  2012 
Fields of Expertise
 Information, Communication & Computing
Treatment code (Nähere Zuordnung)
 Basic  Fundamental (Grundlagenforschung)
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Dive into the research topics of 'Brownian motion and harmonic functions on Sol(p,q)'. Together they form a unique fingerprint.Projects
 2 Finished

FWF  Hyperbolic Structures  Hyperbolic Structures in Stochastics, Graph Theory, and Topology
15/05/12 → 14/11/17
Project: Research project

FWF  Horozyklische Produkte  Random walks, random configurations, and horocyclic products
Huss, W., Sobieczky, F., Woess, W. & Parkinson, J.
1/10/06 → 30/09/09
Project: Research project