The heat semigroup and Brownian motion on strip complexes

Abstract

We introduce the notion of strip complex. A strip complex is a special type of complex obtained by gluing "strips" along their natural boundaries according to a given graph structure. The most familiar example is the one dimensional complex classically associated with a graph, in which case the strips are simply copies of the unit interval (our setup actually allows for variable edge length). A leading key example is treebolic space, a geometric object studied in a number of recent articles, which arises as a horocyclic product of a metric tree with the hyperbolic plane. In this case, the graph is a regular tree, the strips are [0 , 1] × R, and each strip is equipped with the hyperbolic geometry of a specific strip in upper half plane. We consider natural families of Dirichlet forms on a general strip complex and show that the associated heat kernels and harmonic functions have very strong smoothness properties. We study questions such as essential selfadjointness of the underlying differential operator acting on a suitable space of smooth functions satisfying a Kirchoff type condition at points where the strip complex bifurcates. Compatibility with projections that arise from proper group actions is also considered.

Brownian motion on treebolic space: positive harmonic functions

“…We also write ∆ α,β , Dom(∆ α,β ) for its unique self-adjoint extension. Basic properties of this Laplacian and the associated heat semigroup are derived in [2]. In particular, there is the positive, continuous, symmetric heat kernel h α,β (t, w, z) on (0, ∞) × HT × HT such that for all f ∈ C c (HT),…”

“…Thus, in treebolic space HT(q, p), infinitely many copies of the strips S k are glued together as follows: to each vertex v of T there corresponds the (1) This figure also appears in [2]. bifurcation line…”

“…Brownian motion is induced by a Laplace operator ∆ α,β , where the parameter α is the coefficient of a vertical drift term in the interior of each strip, while β is responsible for (again vertical) Kirchhoff type bifurcation conditions along the lines L v . The serious task of rigorously constructing ∆ α,β as an essentially self-adjoint diffusion operator was undertaken in the general setting of strip complexes in [2]. For the specific case of HT, it is explained in detail in [3].…”

Brownian motion on treebolic space: escape to infinity

“…When q = p, that group contains the amenable Baumslag-Solitar group BS(p) as a co-compact lattice, while when q = 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 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.…”

“…When q = p, that group contains the amenable Baumslag-Solitar group BS(p) as a co-compact lattice, while when q = p, it is amenable, but non-unimodular. HT(q, p) is a key example of a strip complex in the sense of [4].…”