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

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.