Interpolation results for pathwise Hamilton-Jacobi equations

TL;DR

This paper investigates how the regularity of paths and Hamiltonians affects the solutions of pathwise Hamilton-Jacobi equations, using interpolation methods and exploring convex function representations.

Contribution

It introduces new criteria linking path and Hamiltonian regularity, and characterizes the largest class of Hamiltonians ensuring well-posedness for all continuous paths.

Findings

Criteria for Hamiltonian regularity based on Sobolev, Besov, Hölder, and variation norms.

Identification of the largest space of Hamiltonians for well-posedness.

Analysis of functions representable as differences of convex functions.

Abstract

We study the interplay between the regularity of paths and Hamiltonians in the theory of pathwise Hamilton-Jacobi equations with the use of interpolation methods. The regularity of the paths is measured with respect to Sobolev, Besov, H\"older, and variation norms, and criteria for the Hamiltonians are presented in terms of both regularity and structure. We also explore various properties of functions that are representable as the difference of convex functions, the largest space of Hamiltonians for which the equation is well-posed for all continuous paths. Finally, we discuss some open problems and conjectures.