The vanishing discount problem for nonlocal Hamilton-Jacobi equations

TL;DR

This work extends vanishing discount analysis to nonlocal Hamilton–Jacobi equations on the torus by incorporating a general integro-differential operator with jumps. It develops a duality framework that defines nonlocal Mather and discounted Mather measures via Hahn–Banach separation, and proves that discounted solutions $u_\lambda$ converge uniformly to a distinguished critical solution $u_0$ as $\lambda\to0^+$. The analysis hinges on a localization argument, a Fenchel-transform-based Lagrangian $L$, and the convergence of discounted measures to (non-discounted) Mather measures, yielding a variational characterization of the limit. Applications cover convex, superlinear Hamiltonians in a broad class $\mathscr{H}_{per}$, where the critical constant exists and ergodic convergence holds, with smoothing and closed-measure minimization tools enabling broader applicability and a robust variational interpretation of the asymptotics.

Abstract

We establish a convergence result for the vanishing discount problem in the context of nonlocal HJ equations. We consider a fairly general class of discounted first-order and convex HJ equations which incorporate an integro-differential operator posed on the $d$-dimensional torus, and we show that the solutions converge to a specific critical solution as the discount factor tends to zero. Our approach relies on duality techniques for nonlocal convex HJ equations, building upon Hahn-Banach separation theorems to develop a generalized notion of Mather measure. The results are applied to a specific class of convex and superlinear Hamiltonians.

Theorems & Definitions (88)

• Theorem 1
• Theorem 1.1: Aleksandrov's Theorem
• Lemma 1.2
• proof
• Definition 1.3
• Proposition 1.4
• Theorem 2.1
• proof
• Theorem 2.2
• proof