Brownian motion on treebolic space: escape to infinity
| Autor: | Laurent Saloff-Coste, Alexander Bendikov, Wolfgang Woess, Maura Salvatori |
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| Rok vydání: | 2015 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Group (mathematics) General Mathematics Hyperbolic geometry Probability (math.PR) 010102 general mathematics 60J25 60J65 53C23 20F65 05C05 Boundary (topology) Group Theory (math.GR) Locally compact group Lattice (discrete subgroup) 01 natural sciences 010104 statistics & probability Differential Geometry (math.DG) FOS: Mathematics Upper half-plane 0101 mathematics Busemann function Mathematics - Group Theory Laplace operator Mathematics - Probability Mathematics |
| Zdroj: | Revista Matemática Iberoamericana. 31:935-976 |
| ISSN: | 0213-2230 |
| DOI: | 10.4171/rmi/859 |
| Popis: | 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. Revista Matematica Iberoamericana, to appear |
| Databáze: | OpenAIRE |
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