The heat semigroup and Brownian motion on strip complexes

TL;DR

This paper introduces the concept of strip complexes, explores their geometric and analytic properties, and studies heat kernels and harmonic functions, with applications to structures like treebolic space.

Contribution

It defines strip complexes, analyzes their heat kernels and harmonic functions, and examines selfadjointness and symmetry properties in this new geometric setting.

Findings

Heat kernels on strip complexes exhibit strong smoothness properties.

The differential operator is essentially selfadjoint under certain conditions.

Compatibility with group actions preserves key analytic features.

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 the closed unit interval times the real line, 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.