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Convex Analysisの二,三の進展について

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.

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The geometry of dissipative evolution equations: the porous medium equation

Summary. The 
$L^2$
 Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 1. Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . 120 2. Background on optimal measures . . . . . . . . . . . . . . . . . . . 126 Part I. Strictly convex costs . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3. Existence and uniqueness of optimal maps . . . . . . . . . . . . . 133 4. Characterization of the optimal map . . . . . . . . . . . . . . . . . 137 Part II. Costs which are strictly concave as a function of d i s t a n c e . . . 141 5. The role of optimal maps . . . . . . . . . . . . . . . . . . . . . . . . 141 6. Uniqueness of optimal solutions . . . . . . . . . . . . . . . . . . . . 144 Part III. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A. Legendre transforms and conjugate costs . . . . . . . . . . . . . . 148 B. Examples of c-concave potentials . . . . . . . . . . . . . . . . . . . 152 C. Regularity of c-concave potentials . . . . . . . . . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

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The geometry of optimal transportation

Introduction Uniform estimates on the $p$-Laplacian, limits as $p\to\infty$ The transport set and transport rays Differentiability and smoothness properties of the potential Generic properties of transport rays Behavior of the transport density along rays Vanishing of the transport density at the ends of rays Approximate mass transfer plans Passage to limits a.e. Optimality Appendix: Approximating semiconcave and semiconvex functions by $C^2$ functions Bibliography.

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Differential equations methods for the Monge-Kantorovich mass transfer problem

1. Degree theory for continuous functions 2. Degree theory in finite dimensional spaces 3. Some applications of the degree theory to Topology 4. Measure theory and Sobolev spaces 5. Properties of the degree for Sobolev functions 6. Local invertibility of Sobolev functions. Applications 7. Degree in infinite dimensional spaces References Index

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Degree Theory in Analysis and Applications

Let μ1,…, μN be Borel probability measures on ℝd. Denote by Γ(μ1,…, μN) the set of all N-tuples T=(T1,…, TN) such that Ti:ℝd ℝd (i = 1,…, N) are Borel-measurable and satisfy μ1 = μi[V] for all Borel V ⊂ ℝd. The multidimensional Monge-Kantorovich problem investigated in this paper consists of finding S=(S1,…, SN) ∈ Γ(μ1,…, μN) minimizing over the set Γ(μ1, ···, μN). We study the case where the μi's have finite second moments and vanish on (d - 1)-rectifiable sets. We prove existence and uniqueness of optimal maps S when we impose that S1(x) ≡ x and give an explicit form of the maps Si. The result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality. As a consequence, we obtain uniqueness and characterization of an optimal measure for the multidimensional Kantorovich problem. © 1998 John Wiley & Sons, Inc.

Optimal maps for the multidimensional Monge-Kantorovich problem

In this paper we consider a Hamiltonian H on P 2 (R 2d ), the set of probability measures with finite quadratic moments on the phase space R 2d = R d × R d , which is a metric space when endowed with the Wasserstein distance W 2 . We study the initial value problem dμ t /dt+∇·(J d V t μ t ) = 0, where J d is the canonical symplectic matrix, μ 0 is prescribed, and V t is a tangent vector to P 2 (R 2d ) at μ t , belonging to ∂H(μ t ), the subdifferential of H at μ t . Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ 0 is absolutely continuous. It ensures that fit remains absolutely continuous and V t = ∇H(μ t ) is the element of minimal norm in ∂H(μt). The second method handles any initial measure μ 0 . If we further assume that H is λ-convex, proper, and lower-semicontinuous on P 2 (R 2d ), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H(μ t ) = H(μ 0 ).

https://people.math.gatech.edu/%7Egangbo/publications/H15.pdf

Wilfrid Gangbo

Papers

The geometry of optimal transportation

Differential equations methods for the Monge-Kantorovich mass transfer problem

Degree Theory in Analysis and Applications

Optimal maps for the multidimensional Monge-Kantorovich problem

Hamiltonian ODEs in the Wasserstein space of probability measures