| Introduction |
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1 | (17) |
| Notation |
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18 | (3) |
| I Gradient Flow in Metric Spaces |
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21 | (82) |
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1 Curves and Gradients in Metric Spaces |
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23 | (16) |
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1.1 Absolutely continuous curves and metric derivative |
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23 | (3) |
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26 | (4) |
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1.3 Curves of maximal slope |
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30 | (2) |
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1.4 Curves of maximal slope in Hilbert and Banach spaces |
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32 | (7) |
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2 Existence of Curves of Maximal Slope and their Variational Approximation |
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39 | (20) |
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2.1 Main topological assumptions |
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42 | (2) |
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2.2 Solvability of the discrete problem and compactness of discrete trajectories |
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44 | (1) |
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2.3 Generalized minimizing movements and curves of maximal slope |
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45 | (4) |
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2.4 The (geodesically) convex case |
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49 | (10) |
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3 Proofs of the Convergence Theorems |
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59 | (16) |
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3.1 Moreau-Yosida approximation |
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59 | (7) |
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3.2 A priori estimates for the discrete solutions |
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66 | (3) |
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3.3 A compactness argument |
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69 | (2) |
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3.4 Conclusion of the proofs of the convergence theorems |
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71 | (4) |
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4 Uniqueness, Generation of Contraction Semigroups, Error Estimates |
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75 | (28) |
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4.1 Cauchy-type estimates for discrete solutions |
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82 | (7) |
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4.1.1 Discrete variational inequalities |
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82 | (2) |
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4.1.2 Piecewise affine interpolation and comparison results |
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84 | (5) |
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4.2 Convergence of discrete solutions |
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89 | (4) |
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4.2.1 Convergence when the initial datum u0 E D(φ) |
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89 | (3) |
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4.2.2 Convergence when the initial datum u0 E D(φ) |
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92 | (1) |
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4.3 Regularizing effect, uniqueness and the semigroup property |
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93 | (4) |
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4.4 Optimal error estimates |
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97 | (8) |
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97 | (2) |
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99 | (4) |
| II Gradient Flow in the Space of Probability Measures |
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103 | (218) |
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5 Preliminary Results on Measure Theory |
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105 | (28) |
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5.1 Narrow convergence, tightness, and uniform integrability |
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106 | (12) |
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5.1.1 Unbounded and 1.s.c. integrands |
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109 | (4) |
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5.1.2 Hilbert spaces and weak topologies |
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113 | (5) |
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5.2 Transport of measures |
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118 | (3) |
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5.3 Measure-valued maps and disintegration theorem |
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121 | (3) |
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5.4 Convergence of plans and convergence of maps |
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124 | (4) |
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5.5 Approximate differentiability and area formula in Euclidean spaces |
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128 | (5) |
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6 The Optimal Transportation Problem |
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133 | (18) |
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6.1 Optimality conditions |
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135 | (4) |
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6.2 Optimal transport maps and their regularity |
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139 | (12) |
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6.2.1 Approximate differentiability of the optimal transport map |
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142 | (4) |
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6.2.2 The infinite dimensional case |
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146 | (2) |
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6.2.3 The quadratic case p = 2 |
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148 | (3) |
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7 The Wasserstein Distance and its Behaviour along Geodesics |
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151 | (16) |
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7.1 The Wasserstein distance |
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151 | (7) |
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7.2 Interpolation and geodesics |
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158 | (2) |
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7.3 The curvature properties of P2(X) |
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160 | (7) |
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8 A.C. Curves in Pp(X) and the Continuity Equation |
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167 | (34) |
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8.1 The continuity equation in Rd |
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169 | (9) |
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8.2 A probabilistic representation of solutions of the continuity equation |
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178 | (4) |
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8.3 Absolutely continuous curves in Pp(X) |
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182 | (7) |
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8.4 The tangent bundle to Pp(X) |
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189 | (5) |
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8.5 Tangent space and optimal maps |
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194 | (7) |
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9 Convex Functionals in Pp(X) |
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201 | (26) |
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9.1 λ-geodesically convex functionals in Pp(X) |
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202 | (3) |
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9.2 Convexity along generalized geodesics |
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205 | (4) |
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9.3 Examples of convex functionals in Pp(X) |
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209 | (6) |
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9.4 Relative entropy and convex functionals of measures |
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215 | (12) |
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9.4.1 Log-concavity and displacement convexity |
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220 | (7) |
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10 Metric Slope and Subdifferential Calculus in Pp(X) |
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227 | (52) |
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10.1 Subdifferential calculus in Pr2(X): the regular case |
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229 | (5) |
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10.1.1 The case of λ-convex functionals along geodesics |
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231 | (1) |
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10.1.2 Regular functionals |
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232 | (2) |
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10.2 Differentiability properties of the p-Wasserstein distance |
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234 | (6) |
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10.3 Subdifferential calculus in &Pp(X): the general case |
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240 | (14) |
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10.3.1 The case of λ-convex functionals along geodesics |
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244 | (2) |
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10.3.2 Regular functionals |
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246 | (8) |
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10.4 Example of subdifferentials |
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254 | (25) |
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10.4.1 Variational integrals: the smooth case |
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254 | (1) |
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10.4.2 The potential energy |
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255 | (2) |
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10.4.3 The internal energy |
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257 | (8) |
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10.4.4 The relative internal energy |
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265 | (2) |
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10.4.5 The interaction energy |
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267 | (2) |
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10.4.6 The opposite Wasserstein distance |
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269 | (3) |
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10.4.7 The sum of internal, potential and interaction energy |
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272 | (4) |
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10.4.8 Relative entropy and Fisher information in infinite dimensions |
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276 | (3) |
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11 Gradient Flows and Curves of Maximal Slope in Pp(X) |
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279 | (28) |
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11.1 The gradient flow equation and its metric formulations |
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280 | (15) |
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11.1.1 Gradient flows and curves of maximal slope |
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283 | (1) |
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11.1.2 Gradient flows for λ-convex functionals |
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284 | (2) |
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11.1.3 The convergence of the "Minimizing Movement" scheme |
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286 | (9) |
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11.2 Gradient flows in Pp2(X) for λ-convex functionals along generalized geodesics |
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295 | (9) |
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11.2.1 Applications to Evolution PDE's |
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298 | (6) |
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11.3 Gradient flows in Pp(X) for regular functionals |
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304 | (3) |
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307 | (14) |
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12.1 Caratheodory and normal integrands |
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307 | (1) |
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12.2 Weak convergence of plans and disintegrations |
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308 | (2) |
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12.3 PC metric spaces and their geometric tangent cone |
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310 | (4) |
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12.4 The geometric tangent spaces in P2(X) |
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314 | (7) |
| Bibliography |
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321 | (10) |
| Index |
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331 | |
Lisainfo e-raamatute kohta