Harmonic functions of Brownian motions on metric graphs

We consider diffusions on locally finite, connected graphs, specifically, a generalization of Walsh's Brownian motion in R-2. In this generalized setting, we classify harmonic functions and introduce an embedded Markov chain associated to such processes. In exploring the relationship between the Brownian motion on a graph and its associated Markov chain, we examine conditions under which the process is reversible and derive the Dirichlet form for the reversible process. We end with a derivation of the Laplace transform of passage times for Brownian motion on a graph. (C) 2015 AIP Publishing LLC.