Brownian Motion and Harmonic Functions on Sol(p,q)

The Lie group Sol(p,q) is the semidirect product induced by the action of $\mathbb R$ on $\mathbb R^2$ which is given by (x,y)↦(e^pzx,e^−qzy), $z\in\mathbb R$. Viewing Sol(p,q) as a three-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 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.