A combinatorial representation for the invariant measure of diffusion processes on metric graphs

We give a generalization to a continuous setting of the classic Markov chain tree Theorem. In particular, we consider an irreducible diffusion process on a metric graph. The unique invariant measure has an atomic component on the vertices and an absolutely continuous part on the edges. We show that the corresponding density at $x$ can be represented by a normalized superposition of the weights associated to metric arborescences oriented toward the point $x$. The weight of each oriented metric arborescence is obtained by the exponential of integrals of the form $\int\frac{b}{\sigma^2}$ along the oriented edges time a weight for each node determined by the local orientation of the arborescence around the node time the inverse of the diffusion coefficient at $x$. The metric arborescences are obtained cutting the original metric graph along some edges.