An entropy generation formula on $RCD(K,\infty)$ spaces

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.