Constrained deepest descent in the $2$-Wasserstein metric

Abstract

We study several constrained variational problems in the $2$-Wasserstein metric for which the set of probability densities satisfying the constraint is not closed. For example, given a probability density $F_0$ on $\mathbb{R}^d$ and a time-step $h>0$, we seek to minimize $I(F) = hS(F) + W_2^2(F_0,F)$ over all of the probability densities $F$ that have the same mean and variance as $F_0$, where $S(F)$ is the entropy of $F$. We prove existence of minimizers. We also analyze the induced geometry of the set of densities satisfying the constraint on the variance and means, and we determine all of the geodesics on it. From this, we determine a criterion for convexity of functionals in the induced geometry. It turns out, for example, that the entropy is uniformly strictly convex on the constrained manifold, though not uniformly convex without the constraint. The problems solved here arose in a study of a variational approach to constructing and studying solutions of the nonlinear kinetic Fokker-Planck equation, which is briefly described here and fully developed in a companion paper.