Brownian motion in riemannian admissible complex

Abstract

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