The Invariant Measure of Multiscale Markov Chains via Fast Arborescence Factorization

We consider a family of continuous-time Markov chains with finite strongly connected transition graph and rates $\left(r_N\right)_{N>0}$ depending on a parameter $N$, so that, when $N$ is large, transitions may happen on different time scales. Under suitable general assumptions on the asymptotic behavior of the rates, we give a recursive characterization of the limiting invariant measure. The recursion is encoded in a forest structure equivalent to the one recently developed in the analysis of dynamical aspects of metastability \cite{BL,LX}. Our proof is based on a combinatorial representation of the invariant measure, given by the Markov chain tree theorem. Basic steps are the reduction of the chain by a trace process, the introduction of an effective dynamics, and a careful analysis of the set of relevant arborescences in the expansion. In particular we use a factorization of fast arborescences. As a byproduct we obtain properties of the arborescences of generalized star-delta reductions of weighted digraphs.