# Constrained deepest descent in the \$2\$-Wasserstein metric | Annals of Mathematics

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 F0 on ℝd and a time-step h>0, we seek to minimize I(F)=hS(F)+W22(F0,F) over all of the probability densities F that have the same mean and variance as F0, 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.

KEYWORDS

SHARE & LIKE

### 数学年刊（Annals of Mathematics）

0 Following 0 Fans 0 Projects 674 Articles

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Abstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Abstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Abstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Abstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Abstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Abstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge