We establish a new and intriguing connection between the Fokker-Planck equation with gradient drift term and an associated free energy functional. Namely, we demonstrate that such a Fokker-Planck equation may be interpreted as a gradient flux, or a steepest descent, of a free energy functional with respect to a certain metric. This is accomplished through the construction of a time-discrete iterative variational scheme whose solutions converge to the solution of the Fokker-Planck equation. The time step in this scheme is governed by the Wasserstein metric on probability measures.
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Jordan et al. (1997) studied this question.
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