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We study analogs of value functions arising in classical mechanics in thespace of probability measures endowed with the Wasserstein metric $W_p$, for$1<p<\infty$. Our main result is that each of these generalized value functionsis a type of viscosity solution of an appropriate Hamilton-Jacobi equation,completing a program initiated by Gangbo, Tudorascu, and Nguyen. Of particularinterest is a formula we derive for a generalized value function when theassociated potential energy is of the form ${\calV}(\mu)=\int_{\mathbb{R}^d}V(x)d\mu(x)$. This formula allows us to makerigorous a well known heuristic connection between Euler-Poisson equations andclassical Hamilton-Jacobi equations. Further results are presented whichsuggest there is a rich theory to be developed of deterministic control in theWasserstein spaces.

Value functions in the Wasserstein spaces: finite time horizons