Giulia Pallotta | Searcharxiv

arXiv2026

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.

math.PR↗

arXiv2025

Freidlin-Wentzell solutions of discrete Hamilton Jacobi equations

We consider a sequence of finite irreducible Markov chains with exponentially small transition rates: the transition graph is a fixed, finite, strongly connected directed graph; the transition rates decay exponentially on a paramenter N with a given rate that varies from edge to edge. The stationary equation uniquely identifies the invariant measure for each N, but at exponential scale in the limit as N goes to infinity reduces to a discrete equation for the large deviation rate functional of the invariant measure, that in general has not an unique solution. In analogy with the continuous case of diffusions, we call such equation a discrete Hamilton-Jacobi equation. Likewise in the continuous case we introduce a notion of viscosity supersolutions and viscosity subsolutions and give a detailed geometric characterization of the solutions in terms of special faces of the polyedron of Lipschitz functions on the transition graph. This parallels the weak KAM theory in a purely discrete setting. We identify also a special vanishing viscosity solution obtained in the limit from the combinatorial representation of the invariant measure given by the matrix tree theorem. The result gives a selection principle on the set of solutions to the discrete Hamilton-Jacobi equation obtained by the Freidlin and Wentzell minimal arborescences construction; this enlightens and parallels what happens in the continuous setting.

math.PR↗