| Preface to the Second Edition | p. ix |
| Introduction | p. 1 |
| Notation | p. 18 |
| Gradient Flow in Metric Spaces | p. 21 |
| Curves and Gradients in Metric Spaces | p. 23 |
| Absolutely continuous curves and metric derivative | p. 23 |
| Upper gradients | p. 26 |
| Curves of maximal slope | p. 30 |
| Curves of maximal slope in Hilbert and Banach spaces | p. 32 |
| Existence of Curves of Maximal Slope | p. 39 |
| Main topological assumptions | p. 42 |
| Solvability of the discrete problem and compactness of discrete trajectories | p. 44 |
| Generalized minimizing movements and curves of maximal slope | p. 45 |
| The (geodesically) convex case | p. 49 |
| Proofs of the Convergence Theorems | p. 59 |
| Moreau-Yosida approximation | p. 59 |
| A priori estimates for the discrete Solutions | p. 66 |
| A compactness argument | p. 69 |
| Conclusion of the proofs of the convergence theorems | p. 71 |
| Generation of Contraction Semigroups | p. 75 |
| Cauchy-type estimates for discrete Solutions | p. 82 |
| Discrete variational inequalities | p. 82 |
| Piecewise affine Interpolation and comparison results | p. 84 |
| Convergence of discrete Solutions | p. 89 |
| Convergence when the initial datum u0 ∈ <$>D(phi)<$> | p. 89 |
| Convergence when the initial datum u0 ∈ <$>overline{D(phi)}<$> | p. 92 |
| Regularizing effect, uniqueness and the semigroup property | p. 93 |
| Optimal error estimates | p. 97 |
| The case = 0 | p. 97 |
| The case &neq; 0 | p. 99 |
| Gradient Flow in the Space of Probability Measures | p. 103 |
| Preliminary Results on Measure Theory | p. 105 |
| Narrow convergence, tightness, and uniform integrability | p. 106 |
| Unbounded and l.s.c. integrands | p. 109 |
| Hilbert spaces and weak topologies | p. 113 |
| Transport of measures | p. 118 |
| Measure-valued maps and disintegration theorem | p. 121 |
| Convergence of plans and convergence of maps | p. 124 |
| Approximate differentiability and area formula in Euclidean spaces | p. 128 |
| The Optimal Transportation Problem | p. 133 |
| Optimality conditions | p. 135 |
| Optimal transport maps and their regularity | p. 139 |
| Approximate differentiability of the optimal transport map | p. 142 |
| The infinite dimensional case | p. 147 |
| The quadratic case p = 2 | p. 149 |
| The Wasserstein Distance and its Behaviour along Geodesics | p. 151 |
| The Wasserstein distance | p. 151 |
| Interpolation and geodesics | p. 158 |
| The curvature properties of <$>{cal P}_2(X)<$> | p. 160 |
| A.C. Curves in <$>{cal P}_p(X)<$> and the Continuity Equation | p. 167 |
| The continuity equation in <$>{op R}^d<$> | p. 169 |
| A probabilistic representation of Solutions of the continuity equation | p. 178 |
| Absolutely continuous curves in <$>{cal P}_p(X)<$> | p. 182 |
| The tangent bundle to <$>{cal P}_p(X)<$> | p. 189 |
| Tangent space and optimal maps | p. 194 |
| Convex Functionals in <$>{cal P}_p(X)<$> | p. 201 |
| -geodesically convex functionals in <$>{cal P}_p(X)<$> | p. 202 |
| Convexity along generalized geodesics | p. 205 |
| Examples of convex functionals in <$>{cal P}_p(X)<$> | p. 209 |
| Relative entropy and convex functionals of measures | p. 215 |
| Log-concavity and displacement convexity | p. 220 |
| Metric Slope and Subdifferential Calculus in <$>{cal P}_p(X)<$> | p. 227 |
| Subdifferential calculus in <$>{cal P}_2^r(X)<$>: the regular case | p. 229 |
| The case of -convex functionals along geodesics | p. 231 |
| Regular functionals | p. 232 |
| Differentiability properties of the p-Wasserstein distance | p. 234 |
| Subdifferential calculus in <$>{cal P}_p(X)<$>: the general case | p. 240 |
| The case of -convex functionals along geodesics | p. 244 |
| Regular functionals | p. 246 |
| Example of subdifferentials | p. 254 |
| Variational integrals: the smooth case | p. 254 |
| The potential energy | p. 255 |
| The internal energy | p. 257 |
| The relative internal energy | p. 265 |
| The interaction energy | p. 267 |
| The opposite Wasserstein distance | p. 269 |
| The sum of internal, potential and interaction energy | p. 272 |
| Relative entropy and Fisher Information in infinite dimensions | p. 276 |
| Gradient Flows and Curves of Maximal Slope in <$>{cal P}_p(X)<$> | p. 279 |
| The gradient flow equation and its metric formulations | p. 280 |
| Gradient flows and curves of maximal slope | p. 283 |
| Gradient flows for -convex functionals | p. 284 |
| The convergence of the "Minimizing Movement" scheme | p. 286 |
| Gradient flows for -convex functionals along generalized geodesics | p. 295 |
| Applications to Evolution PDE's | p. 298 |
| Gradient flows in <$>{cal P}_p(X)<$> for regular functionals | p. 304 |
| Appendix | p. 307 |
| Carathéodory and normal integrands | p. 307 |
| Weak convergence of plans and disintegrations | p. 308 |
| PC metric spaces and their geometric tangent cone | p. 310 |
| The geometric tangent spaces in <$>{cal P}_2(X)<$> | p. 314 |
| Bibliography | p. 331 |
| Index | p. 333 |
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