Brownian motion in Riemannian admissible complexes

Abstract

The purpose of this work is to construct a Brownian motion with values in simplicial complexes with piecewise differential structure. In order to state and prove the existence of such a Brownian motion, we define a family of continuous Markov processes with values in an admissible complex. We call a process in this family an isotropic transport process. We first show that the family of isotropic processes contains a subsequence which converges weakly to a measure which we call the Wiener measure. Then, using the finite dimensional distributions of this Wiener measure, we construct a new admissible complex valued continuous Markov process, the Brownian motion. We conclude with a geometric analysis of this Brownian motion and determine the recurrent or transient behavior of such a process.