A cost on paths of measures which induces the Fokker–Planck equation

In [12], Feng and Nguyen (2012) define a cost on curves of measures which is finite exactly on the curves which solve a Fokker–Planck equation with L2L2 drift. In this paper, using ideas of D. Gomes and E. Valdinoci, we give a different construction of the cost of Feng and Nguyen (2012).