tailieunhanh - Đề tài " Constrained steepest descent in the 2-Wasserstein metric "

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 2 h 0, we seek to minimize I(F ) = hS(F )+W2 (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. | Annals of Mathematics Constrained steepest descent in the 2-Wasserstein metric By E. A. Carlen and W. Gangbo Annals of Mathematics 157 2003 807 846 Constrained steepest descent in the 2-Wasserstein metric By E. A. Carlen and W. Gangbo 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 W2 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. Contents 1. Introduction 2. Riemannian geometry of the 2-Wasserstein metric 3. Geometry of the constraint manifold 4. The Euler-Lagrange equation 5. Existence of minimizers References The work of the first named author was partially supported by . . grant DMS-00-70589. The work of the second named author was partially supported by . . grants DMS-99-70520 and DMS-00-74037. 808 E. A. CARLEN AND W. GANGBO 1. Introduction Recently there has been considerable progress in understanding a wide range of dissipative evolution equations in terms of variational problems involving the Wasserstein metric. In particular Jordan Kinderlehrer and Otto have shown in 12 that the heat equation is gradient .