Well-posedness for Hamilton–Jacobi equations for Controlled EVI Flows on the Wasserstein-2 Space

This talk will be at UNC; see their website for details. Hamilton-Jacobi (HJ) equations encode the infinitesimal evolution of value functions in control problems and are, for example, central in dynamic large deviations. In infinite-dimensional settings, such as the Wasserstein-2 space in Dawson–Gärtner type results, well-posedness of the associated HJ equation is delicate and remains only partially understood. We introduce a formulation based on two-sided Hamiltonian bounds via functional inequalities, tailored to infinitesimal Riemannian geodesic spaces. For dynamics driven by lambda-convex energies in the sense of EVI gradient flows, we prove a comparison/uniqueness result. In the Wasserstein-2 space we further establish existence, yielding well-posedness for Hamilton–Jacobi equations associated with linearly controlled EVI gradient flows.