A comparison principle for semilinear Hamilton-Jacobi-Bellman equations in the Wasserstein space

TL;DR

This paper establishes a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the Wasserstein space, ensuring uniqueness and stability without relying on mean field control formulations.

Contribution

It introduces a novel method leveraging Wasserstein distance differentiability and entropy penalization to prove comparison principles for a broad class of equations.

Findings

Proves uniqueness of viscosity solutions in Wasserstein space.

Applies to equations with nonconvex Hamiltonians and measure-dependent volatility.

Shows the value function from a mean-field control problem is the unique solution for convex Hamiltonians.

Abstract

The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the $2$-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the $1$-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.