Hamilton--Jacobi equations for controlled gradient flows: the comparison principle

TL;DR

This paper proves a comparison principle for Hamilton--Jacobi equations linked to controlled gradient flows on metric spaces, advancing the theoretical understanding of such equations in various mathematical contexts.

Contribution

It establishes a comparison principle for Hamilton--Jacobi equations associated with controlled gradient flows, extending the theory to new classes of metric space gradient flows.

Findings

Proves comparison principle for Hamilton--Jacobi equations in metric spaces.

Applies to gradient flows on Hilbert and Wasserstein spaces.

Utilizes EVI formulation and Tataru's distance for key estimates.

Abstract

Motivated by recent developments in the fields of large deviations for interacting particle system and mean field control, we establish a comparison principle for the Hamilton--Jacobi equation corresponding to linearly controlled gradient flows of an energy function $\cE$ defined on a metric space $(E,d)$. Our analysis is based on a systematic use of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional $\cE$ satisfying McCann's condition.