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.