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 parameter 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 polyhedron of one-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.