Hamilton-Jacobi equations from mean-field spin glasses

TL;DR

This paper establishes the well-posedness of Hamilton-Jacobi equations derived from mean-field spin glass models in the viscosity sense, connecting finite-dimensional approximations with infinite-dimensional solutions.

Contribution

It introduces a viscosity solution framework for these equations, proving comparison principles and convergence of approximations, and relates existing solution notions to viscosity solutions.

Findings

Proved comparison principle for the equations.

Established convergence of finite-dimensional approximations.

Connected existing solution concepts to viscosity solutions.

Abstract

We give a meaning to the Hamilton--Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness. Originally defined on the set of monotone probability measures, these equations can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with an empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existence of solutions. Under additional convexity conditions, we show that the solution can be represented by a version of the Hopf--Lax formula, or the Hopf formula on cones. Previously, two notions of solutions were considered, one defined directly as the Hopf--Lax formula, and another as limits of finite-dimensional approximations. They have been proven to describe the limit free energy in a wide class of mean-field spin glass models. This work shows that these two kinds of solutions are viscosity solutions.