  J. Feng and T. Nguyen have shown that the solutions of the Fokker-Planck
equation in $\R^d$ satisfy an entropy generation formula. We prove that, in
compact metric measure spaces with the $RCD(K,\infty)$ property, a similar
result holds for curves of measures whose density is bounded away from zero and
infinity. We use this fact to show the existence of minimal characteristics for
the stochastic value function.
