Dynamical Systems and Hamilton–Jacobi–Bellman Equations on the Wasserstein Space and their L2 Representations

Several optimal control problems in \BbbR d, like systems with uncertainty, control of flock dynamics, or control of multiagent systems, can be naturally formulated in the space of probability measures in \BbbR d . This leads to the study of dynamics and viscosity solutions to the Hamilton-- Jacobi--Bellman equation satisfied by the value functions of those control problems, both stated in the Wasserstein space of probability measures. Since this space can be also viewed as the set of the laws of random variables in a suitable L2 space, the main aim of the paper is to study such control systems in the Wasserstein space and to investigate the relations between dynamical systems in Wasserstein space and their representations by dynamical systems in L2, both from the points of view of trajectories and of (first-order) Hamilton--Jacobi--Bellman equations.