Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations

Abstract

This talk focuses on the vanishing discount problem for first-order Hamilton–Jacobi (HJ) equations: as the discount parameter tends to zero, do discounted solutions converge and—when they do—which critical solution is selected? The question is closely linked to the ergodic approximation introduced by Lions, Papanicolaou, and Varadhan in their seminal preprint on the periodic homogenization of Hamilton–Jacobi equations. I will first review a selection result (joint with A. Fathi, R. Iturriaga, and M. Zavidovique; Invent. Math. (2016)) that establishes convergence and identifies a canonical limit in the convex, coercive setting, drawing on weak KAM ideas, variational and representation principles, and Mather measures to describe the structure of the limit. I will then discuss more recent results for Hamilton–Jacobi equations with nonlinear dependence on the solution itself. In this framework, the usual strict monotonicity in the solution variable that underpins comparison principles may be weakened or fail on parts of the domain. I will explain what this loss of monotonicity implies for the uniqueness of discounted solutions and for the vanishing-discount limit—highlighting conditions that still ensure convergence and selection, mechanisms that produce divergent families of discounted solutions, and illustrative examples (from recent joint work with P. Ni, J. Yan, and M. Zavidovique).