Recent progress in generalized Hamiltonian gradient flow: Singularities

TL;DR

This survey explores the generalized Hamiltonian gradient flow framework for Hamilton-Jacobi equations, highlighting new variational constructions, dynamical characterizations of measures, and open problems in singular dynamics and optimal transport.

Contribution

It introduces a variational scheme for generalized characteristics, characterizes Mather measures via semi-flow dynamics, and discusses open problems connecting singular dynamics with ergodic theory and optimal transport.

Findings

Constructed generalized characteristics via a variational minimizing movement scheme.

Proved invariant measures of the GHGF semi-flow are precisely Mather measures.

Identified open problems related to uniqueness, stability, and extensions of the framework.

Abstract

This paper is a survey of the generalized Hamiltonian gradient flow (GHGF) framework for Hamilton-Jacobi equations, with an emphasis on the propagation of singularities and its connections to weak KAM theory, optimal transport and mean field control. In addition to reviewing the main ideas and known results, we present two new contributions. First, we provide a variational construction of generalized characteristics via a minimizing movement scheme; by taking the weak limit of approximate solutions and using Young measure compactness, we show that the limiting curve satisfies the generalized characteristic differential inclusion. Second, we lift the forward dynamics of strict singular characteristics to a semi-flow on the space of trajectories and study its invariant probability measures. We prove that the only invariant measures of the GHGF semi-flow that attain the critical value \(c[H]\) are precisely the projected Mather measures, thereby giving a new dynamical characterization of Mather's minimal measures as well as Ma\~n\'e's critical value. Finally, we discuss a number of open problems that arise from the GHGF perspective, including questions on uniqueness of strict singular characteristics, rectifiability of cut loci, stability under perturbations, contact Hamiltonian systems, vanishing noise limits, and extensions to non-convex or low-regularity Hamiltonians. These problems highlight the deeper connections between singular dynamics, ergodic theory, optimal transport, and geometric analysis, and indicate directions for future research.