Hamilton–Jacobi Equations in Spaces of Measures

This thesis investigates first-order Hamilton–Jacobi equations in spaces of measures and their connections with optimal control problems. The main objective is to develop a viscositysolution theory compatible with the intrinsic geometry of these spaces, and to establish existence, comparison, stability, and representation results for the associated equations. We begin with Hamilton–Jacobi equations on the Wasserstein space P2(R d ). We develop a unified viscosity-solution framework covering both first-order Hamilton–Jacobi equations and semilinear Hamilton–Jacobi equations driven by an idiosyncratic operator defined on the Wasserstein space. Within this setting, we establish a vanishing-viscosity limit extending beyond the classical control-theoretic framework: solutions of semilinear Hamilton–Jacobi equations converge to the corresponding first-order equation as the idiosyncratic noise vanishes, and we obtain an optimal convergence rate. We also present several results of independent interest, including existence results for the first-order equation obtained through a suitable Hopf–Lax representation formula, and a characterization of the action of the idiosyncratic operator on geodesically convex functions. We then turn to Hamilton–Jacobi equations on Riemannian manifolds. Combining tools from viscosity-solution theory, Weak KAM theory, and superdifferential calculus, we establish comparison principles for equations with both convex and non-convex Hamiltonians using global penalization arguments based on the geodesic distance. These results are subsequently extended to p-Wasserstein spaces of probability measures over a Riemannian manifold, where analogous comparison principles are proved. In this framework, we also study finite-horizon optimal control problems in P2, establishing local regularity properties of the value function, the Dynamic Programming Principle, and the consistency between the value function and the Hamilton–Jacobi equation in the viscosity sense. Finally, we investigate Hamilton–Jacobi equations in the Hellinger space of finite nonnegative measures. We introduce the geometric structure required to perform differential calculus in this setting, including geodesics, parallel transport, and subdifferential notions compatible with the flat calculus on measures. Within this framework, we establish a comparison principle and prove that the value function of a finite–horizon control problem is a viscosity solution of the associated Hamilton–Jacobi equation. Overall, the thesis develops a unified geometric approach to viscosity solutions and optimal control in spaces of measures, extending classical Hamilton–Jacobi theory to new infinite-dimensional settings.