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

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