We consider diffusions on locally finite, connected graphs, specifically, a generalization of Walsh’s Brownian motion in ℝ2. 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.
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2015
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