368 Index D differentiability of Wasserstein distance, 262 differential cost, 144 Dirichlet boundary condition, 127 Dirichlet form, 278, 283 disintegration of measure, 209 displacement convexity, 150 displacement interpolation, 148, 172 distribution function (cumulative), 73, 218 inverse, 73 divergence, 167 divergence formulation of the Kantorovich- Rubinstein problem, 38 double convexification trick, 51, 60 duality for c-concave functions, 33, 87 Kantorovich, 19 Legendre-Fenchel-Rockafellar, 24, 56, 87 E elliptic equation linear, 126 nonlinear, 127 empirical measure, 327, 344 energy functional interaction, 152, 159 internal, 152, 155 potential, 152, 158 entropy, 226, 270 and entropy production, 271, 288 metric, 316 entropy power functional, 194, 281 Euler equation incompressible, 111 pressureless, 170 two-dimensional incompressible, 254 Eulerian point of view, 113, 165, 173 exponential map (Riemannian), 95 extremal points (of a convex set), 5 F Fenchel-Rockafellar theorem (duality), 24 Fisher information, 278, 279 FKG inequalities, 331 Fokker-Planck equation, 252, 274 free energy, 270 free probability theory, 278 G Gagliardo-Nirenberg inequality, 302 Γ2 formalism, 275 Gangbo-McCann theorem concave cost, 93 convex cost, 92 Gauss curvature (prescribed), 126 Gaussian, 193, 291 gaussian, 281 gluing lemma, 208 Grad’s number, 311 gradient (for Wasserstein metric), 251 gradient flow, 251, 256 for Wasserstein metric, 252, 259, 268 granular flow, 269, 304, 320 H H theorem, 226 H−1 norm, 234 Hahn-Banach theorem, 24 Hamilton-Jacobi equation, 172, 174 and Kantorovich duality, 177 viscous, 296 Hamiltonian system, 251 for Wasserstein metric, 252 Hardy-Littlewood-Sobolev inequality, 286 Hausdorff-Young inequality, 194 heat equation, 252 Helmholtz decomposition, 122, 124 Henstock-McBeath inequalities, 189 Herbst’s theorem, 280 Hessian measure, 129 Hessians (for Wasserstein metric), 271 Hoeffding-Fr´ echet theorem, 75 Hopf-Lax formula, 175, 287 HWI inequality, 297 hypercontractivity, 286 I image measure, 4 implicit function theorem (for convex func- tions), 85 incompressibility, 111, 169 indicator function (in convex analysis), 22 information theory, 285 interpolation inequality (HWI), 297 isoperimetry Euclidean, 184, 283 Gaussian, 284 K Kantorovich duality, 19, 89 in terms of Hamilton-Jacobi equations, 177 Kantorovich norm, 213 Kantorovich’s problem, 2 Kantorovich-Rubinstein distance, 207 Kantorovich-Rubinstein problem (transship- ment), 37 Kantorovich-Rubinstein theorem, 34

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