Friedlin-Wentzell solutions of discrete Hamilton Jacobi equations

TL;DR

The paper develops a discrete Freidlin–Wentzell framework for finite directed graphs by deriving a stationary discrete Hamilton–Jacobi equation from large deviations of invariant measures and characterizing its entire solution set through viscosity theory and graph geometry. It identifies a canonical Friedlin–Wentzell solution via a combinatorial matrix-tree representation and shows this special solution arises as the vanishing-viscosity limit, mirroring the continuous theory. The work builds a discrete weak KAM-like structure, connecting in-depth geometric descriptions (Lip1 faces, unicyclic components, and quasipotentials) with arborescence-based optimal costs, and demonstrates how these ideas relate to shortest-path algorithms and metric-graph theory. Overall, it provides a tractable, graph-based paradigm for metastability and large deviations in finite networks, bridging stochastic processes, variational principles, and discrete geometry.

Abstract

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. This gives a selection principle on the set of solutions to the discrete Hamilton Jacobi equation obtained by the Friedlin and Wentzell minimal arborescences construction; this enlights and parallels what happens in the continuous setting.

Figures (3)

• Figure 2: An in-unicyclic component on the left and an out-unicyclic component on the right
• Figure 3: The switch of orientation from an arborescence oriented toward $x$ to an arborescence oriented toward $y$.
• Figure 4: An example of the sunshine transformation for the case of a continuous function $S$ (in black) with $\int_0^1f(y)dy >0$. The light is assumed to arrive horizontally from the left; parts of the graph below the horizontal red line are in shadow, while the remaining parts are illuminated. The sunshine transformation retains the shadowed parts of the graph and replace the illuminated parts with horizontal segments, shifting them vertically to ensure that the final transformed graph remains continuous. An example of this transformation is illustrated on a portion of the graph inside the rectangle at the upper left. The transformed graph is periodic of period one. In the case $\int_0^1f(y)dy< 0$, the construction is performed with the light arriving horizontally from the right. For $\int_0^1f(y)dy=0$, the graph is already periodic, every point of the graph is in shadow, and the transformation does not modify the graph.

Theorems & Definitions (44)

• Lemma 2.2: HJ equation
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Definition 4.1
• Definition 4.2
• Definition 4.3
• Definition 4.4
• Lemma 4.5