Stochastic optimal transport and Hamilton–Jacobi–Bellman equations on the set of probability measures

We introduce a stochastic version of the optimal transport problem. We provide an analysis by means of the study of the associated Hamilton–Jacobi–Bellman equation, which is set on the set of probability measures. We introduce a new definition of viscosity solutions of this equation, which yields general comparison principles, in particular for cases involving terms modeling stochasticity in the optimal control problem. We are then able to establish results of existence and uniqueness of viscosity solutions of the Hamilton–Jacobi–Bellman equation. These results rely on controllability results for stochastic optimal transport that we also establish.