Envelope representation of Hamilton-Jacobi equations from spin glasses

TL;DR

This paper develops an envelope representation for the unique viscosity solution of a Hamilton-Jacobi equation related to spin glass models, using PDE techniques to express the solution as an average along characteristic lines.

Contribution

It introduces a novel envelope-type formula for the PDE solution, extending Evans' representation to infinite-dimensional, cone-restricted settings in spin glass theory.

Findings

Derived an envelope representation formula for the PDE solution.

Expressed the solution as an average along characteristic lines.

Addressed technical challenges of infinite-dimensionality and cone domain.

Abstract

Recently, [arXiv:2311.08980] demonstrated that, if it exists, the limit free energy of possibly non-convex spin glass models must be determined by a characteristic of the associated infinite-dimensional non-convex Hamilton-Jacobi equation. In this work, we investigate a similar theme purely from the perspective of PDEs. Specifically, we study the unique viscosity solution of the aforementioned equation and derive an envelope-type representation formula for the solution, in the form proposed by Evans in [doi:10.1007/s00526-013-0635-3]. The value of the solution is expressed as an average of the values along characteristic lines, weighted by a non-explicit probability measure. The technical challenges arise not only from the infinite dimensionality but also from the fact that the equation is defined on a closed convex cone with an empty interior, rather than on the entire space. In the introduction, we provide a description of the motivation from spin glass theory and present the corresponding results for comparison with the PDE results.