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 = T-p, with degree p + 1 >= 3, 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 >= 2, treebolic space HT(q, p) consists of all pairs (z = x + iy, w) is an element of H x T with h(w) = log(q) y. It can also be obtained by glueing together horizontal strips of Elf 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 not equal p, it is amenable, but non-unimodular. HT(q, p) is a key example of a strip complex in the sense of [4]. Relying on the analysis of strip complexes developed by the same authors in [4], we consider a family of natural Laplacians with "vertical drift" and describe the associated Brownian motion. The main difficulties come from the singularities 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.

Brownian motion on treebolic space: escape to infinity / A. Bendikov, L. Saloff-Coste, M. Salvatori, W. Woess. - In: REVISTA MATEMATICA IBEROAMERICANA. - ISSN 0213-2230. - 31:3(2015), pp. 935-976.

Brownian motion on treebolic space: escape to infinity

M. Salvatori
Penultimo
;
2015

Abstract

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 = T-p, with degree p + 1 >= 3, 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 >= 2, treebolic space HT(q, p) consists of all pairs (z = x + iy, w) is an element of H x T with h(w) = log(q) y. It can also be obtained by glueing together horizontal strips of Elf 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 not equal p, it is amenable, but non-unimodular. HT(q, p) is a key example of a strip complex in the sense of [4]. Relying on the analysis of strip complexes developed by the same authors in [4], we consider a family of natural Laplacians with "vertical drift" and describe the associated Brownian motion. The main difficulties come from the singularities 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.
tree; hyperbolic plane; horocyclic product; Laplacian; Brownian motion; rate of escape; central limit theorem; boundary convergence
Settore MAT/05 - Analisi Matematica
Settore MAT/06 - Probabilita' e Statistica Matematica
2015
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/325467
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