2020/02/03 by Michele Aleandri, Aleandri, Michele, Matteo Colangeli +3 · 2 citations
Computer Science · Mathematics · #FOS: Mathematics #Graph theory and applications #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2002.00654
openalex publication_date 2020/02/03 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We give a generalization to a continuous setting of the classic Markov chain\ntree Theorem. In particular, we consider an irreducible diffusion process on a\nmetric graph. The unique invariant measure has an atomic component on the\nvertices and an absolutely continuous part on the edges. We show that the\ncorresponding density at x can be represented by a normalized superposition\nof the weights associated to metric arborescences oriented toward the point\nx. The weight of each oriented metric arborescence is obtained by the\nexponential of integrals of the form \∫\(b)/(\σ2) along the\noriented edges time a weight for each node determined by the local orientation\nof the arborescence around the node time the inverse of the diffusion\ncoefficient at x. The metric arborescences are obtained cutting the original\nmetric graph along some edges.\n