On Viscosity Solutions of Hamilton-Jacobi Equations in the Wasserstein space and the Vanishing Viscosity Limit

TL;DR

This paper develops a unified framework for viscosity solutions of Hamilton-Jacobi equations in Wasserstein space and proves their convergence in the vanishing viscosity limit, extending classical control theory results.

Contribution

It introduces a novel unified approach for viscosity solutions in Wasserstein space and establishes a vanishing-viscosity limit with optimal convergence rate.

Findings

Established existence theorems for first-order equations.

Proved convergence of semilinear to first-order solutions as noise vanishes.

Provided new insights into the action of the operator on convex functions.

Abstract

The aim of this article is twofold. First, we develop a unified framework for viscosity solutions to both first-order Hamilton-Jacobi equations and semilinear Hamilton-Jacobi equations driven by the idiosyncratic operator, defined on the Wasserstein Space. Second, we establish a vanishing-viscosity limit-extending beyond the classical control-theoretic setting-for solutions of semilinear Hamilton-Jacobi equations, proving their convergence to the corresponding first-order solution as the idiosyncratic noise vanishes. Our approach provides an optimal convergence rate. We also present some results of independent interest. These include existence theorems for the first-order equation, obtained through an appropriate Hopf-Lax representation, and a useful description of the action of the idiosyncratic operator on geodesically convex functions.