Index 369 kinetic energy, 238 kinetic equations, 223 Knott-Smith criterion, 66, 70 Krein-Milman theorem, 15 Kullback information, 278 L Lagrangian point of view, 113, 165 p-Laplace equation, 98, 266 layer cake representation, 45 Lebesgue density point, 132 Legendre-Fenchel transform, 23, 55 c-transform, 86 L´ evy distance, 44 Lieb’s theorem, 195 linear programming, 23 linearization of Gaussian isoperimetric inequality, 284 of logarithmic Sobolev inequality, 283 of Monge-Amp` ere equation, 127 of quadratic Wasserstein distance, 233 local compactness, 18 logarithmic Sobolev inequality, 279, 287 Stam-Gross, 282 Lusin’s theorem, 71 M marginals, 2 McCann’s interpolation, 148 McCann’s theorem (optimal transportation on Riemannian manifolds), 96 McKean-Vlasov equation, 252, 304, 341 mean-field equation, 254, 340 measure-preserving maps, 108 minimax principle, 21 Minkowski sum, 184 Monge’s problem, 4, 97 Monge-Amp` ere equation, 125, 318 Aleksandrov solution, 129, 136 almost everywhere, 133 Brenier solution, 130 linearization, 127 viscosity solution, 130 monotone rearrangement, 75 O optimal matching, 6 Otto’s interpretation, 245 P P´ olya-Szeg¨ o principle, 203 particle systems, 283, 336, 340 Poincar´ e inequality, 283 polar factorization of matrices, 123 of vector fields, 109 Polish spaces, 18, 205 porous medium equation, 252, 270, 302 potential (of transportation), 98 Pr´ ekopa-Leindler inequality, 186, 196 for Gaussians, 189, 196 pressure incompressible Euler equation, 111 thermodynamical, 154, 272 probabilistic Kantorovich problem, 3 projection onto S(Ω), 110 Prokhorov distance, 44 Prokhorov’s theorem, 25 push-forward, 4 Q quantum field theory, 283 R Rademacher’s theorem, 52 rearrangement, 107 radially symmetric monotone, 109 rearrangement inequality, 164 regularity theory (Caffarelli, Urbas), 137 relaxation, 9 reparametrization techniques, 183 rescaled convolution, 220 Ricci curvature, 163, 178, 285 Riemannian manifolds, 94, 163, 285 Riesz’ rearrangement inequality, 164 Riesz’ theorem, 18 rigid body motion, 312 Rockafellar’s theorem, 82, 87 R¨ uschendorf’s theorem, 87 S Sanov’s theorem, 344 selection principle, 100 semi-geostrophic system, 14, 266, 323 Shannon-Stam inequality, 194, 285 shipper’s problem, 20 shocks, 166, 178 Skorohod’s representation theorem, 219 small sets, 52 Sobolev inequality, 200 optimal, 201, 282, 302 spectral gap, 283 stability of optimal mapping, 71 sticky particles, 170 Strassen’s theorem, 44 subdifferential, 53 c-superdifferential, 86, 90

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