Brownian motion on treebolic space: escape to infinity

Treebolic space is an analog of the Sol geometry, namely, it is the horocylic product of the hyperbolic upper half plane H and the homogeneous tree T with degree p+1 > 2, the latter seen as a one-complex. Let h be the Busemann function of T with respect to a fixed boundary point. Then for real q > 1 and integer p > 1, treebolic space HT(q,p) consists of all pairs (z=x+i y,w) in H x T with h(w) = log_{q} y. It can also be obtained by glueing together horziontal strips of H in a tree-like fashion. We explain the geometry and metric of HT and exhibit a locally compact group of isometries (a horocyclic product of affine groups) that acts with compact quotient. When q=p, that group contains the amenable Baumslag-Solitar group BS(p)$ as a co-compact lattice, while when q and p are distinct, it is amenable, but non-unimodular. HT(q,p) is a key example of a strip complex in the sense of our previous paper in Advances in Mathematics 226 (2011) 992-1055. Relying on the analysis of strip complexes developed in that paper, we consider a family of natural Laplacians with "vertical drift" and describe the associated Brownian motion. The main difficulties come from the singularites which treebolic space (as any strip complex) has along its bifurcation lines. In this first part, we obtain the rate of escape and a central limit theorem, and describe how Brownian motion converges to the natural geometric boundary at infinity. Forthcoming work will be dedicated to positive harmonic functions.Comment: Revista Matematica Iberoamericana, to appea

Brownian motion on treebolic space: positive harmonic functions

Treebolic space HT(q,p) is a key example of a strip complex in the sense of Bendikov, Saloff-Coste, Salvatori, and Woess [Adv. Math. 226 (2011), 992-1055]. It is an analog of the Sol geometry, namely, it is a horocylic product of the hyperbolic upper half plane with a "stretching" parameter q and the homogeneous tree T with vertex degree p+1 < 2, the latter seen as a one-complex. In a previous paper [arXiv:1212.6151, Rev. Mat. Iberoamericana, in print] we have explored the metric structure and isometry group of that space. Relying on the analysis on strip complexes, a family of natural Laplacians with "vertical drift" and the escape to infinity of the associated Brownian motion were considered. Here, we undertake a potential theoretic study, investigating the positive harmonic functions associated with those Laplacians. The methodological subtleties stem from the singularites of treebolic space at its bifurcation lines. We first study harmonic functions on simply connected sets with "rectangular" shape that are unions of strips. We derive a Poisson representation and obtain a solution of the Dirichlet problem on sets of that type. This provides properties of the density of the induced random walk on the collection of all bifuraction lines. Subsequently, we prove that each positive harmonic function with respect to that random walk has a unique extension which is harmonic with respect to the Laplacian on treebolic space. Finally, we derive a decomposition theorem for positive harmonic functions on the entire space that leads to a characterisation of the weak Liouville property. We determine all minimal harmonic functions in those cases where our Laplacian arises from lifting a (smooth) hyperbolic Laplacian with drift from the hyperbolic plane to treebolic space

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

On some spaces of holomorphic functions of exponential growth on a half-plane

In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on [0, +∞). We obtain a Paley–Wiener theorem for M2ω, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that Mpω contains functions of order 1. Moreover, we prove that the orthogonal projection from Lp(R,dω) into Mpω is unbounded for p ≠ 2. Furthermore, we compare the spaces Mpω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently