Brownian motion and Harmonic functions on Sol(p,q)

TL;DR

This paper investigates Brownian motion with drift on Sol(p,q), deriving limit theorems, analyzing harmonic functions, and exploring geometric compactification, emphasizing the structure as a horocyclic product of hyperbolic planes.

Contribution

It provides a detailed analysis of Brownian motion with drift on Sol(p,q), including limit theorems, harmonic functions, and geometric boundary behavior, with explicit characterizations.

Findings

Derived a central limit theorem for Brownian motion with drift on Sol(p,q)

Computed the rate of escape for the process

Explicitly characterized all minimal positive harmonic functions

Abstract

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.