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We introduce a stochastic version of the optimal transport problem. Weprovide an analysis by means of the study of the associatedHamilton-Jacobi-Bellman equation, which is set on the set of probabilitymeasures. We introduce a new definition of viscosity solutions of thisequation, which yields general comparison principles, in particular for casesinvolving terms modeling stochasticity in the optimal control problem. We arethen able to establish results of existence and uniqueness of viscositysolutions of the Hamilton-Jacobi-Bellman equation. These results rely oncontrollability results for stochastic optimal transport that we alsoestablish.

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