  We perform a systematic study of optimization problems in the Wasserstein
spaces that are analogs of infinite horizon, deterministic control problems. We
derive necessary conditions on action minimizing paths and present a sufficient
condition for their existence. We also verify that the corresponding
generalized value functions are a type of viscosity solution of a time
independent, Hamilton-Jacobi equation in the space of probability measures.
Finally, we prove a special case of a conjecture involving the subdifferential
of generalized value functions and their relation to action minimizing paths.
