A user’s guide to optimal transport Luigi Ambrosio ∗ Nicola Gigli y Contents 1 The optimal transport problem 4 1.1 Monge and Kantorovich formulations of the optimal transport problem . .4 1.2 Necessary and sufficient optimality conditions . .7 1.3 The dual problem . 13 1.4 Existence of optimal maps . 15 1.5 Bibliographical notes . 25 2 The Wasserstein distance W2 27 2.1 X Polish space . 27 2.2 X geodesic space . 34 2.3 X Riemannian manifold . 44 2.3.1 Regularity of interpolated potentials and consequences . 44 2.3.2 The weak Riemannian structure of (P2(M);W2) ................. 48 2.4 Bibliographical notes . 53 3 Gradient flows 54 3.1 Hilbertian theory of gradient flows . 54 3.2 The theory of Gradient Flows in a metric setting . 56 3.2.1 The framework . 56 3.2.2 General l.s.c. functionals and EDI . 61 3.2.3 The geodesically convex case: EDE and regularizing effects . 65 3.2.4 The compatibility of Energy and distance: EVI and error estimates . 70 3.3 Applications to the Wasserstein case . 74 d 3.3.1 Elements of subdifferential calculus in (P2(R );W2) .............. 76 3.3.2 Three classical functionals . 77 3.4 Bibliographical notes . 84 ∗[email protected] [email protected] 1 4 Geometric and functional inequalities 85 4.1 Brunn-Minkowski inequality . 85 4.2 Isoperimetric inequality . 85 4.3 Sobolev Inequality . 86 4.4 Bibliographical notes . 87 5 Variants of the Wasserstein distance 88 5.1 Branched optimal transportation . 88 5.2 Different action functional . 89 5.3 An extension to measures with unequal mass . 90 5.4 Bibliographical notes . 92 6 More on the structure of (P2(M);W2) 93 6.1 “Duality” between the Wasserstein and the Arnold Manifolds . 93 6.2 On the notion of tangent space . 95 6.3 Second order calculus . 97 6.4 Bibliographical notes . 117 7 Ricci curvature bounds 118 7.1 Convergence of metric measure spaces . 120 7.2 Weak Ricci curvature bounds: definition and properties . 123 7.3 Bibliographical notes . 135 Introduction The opportunity to write down these notes on Optimal Transport has been the CIME course in Cetraro given by the first author in 2009. Later on the second author joined to the project, and the initial set of notes has been enriched and made more detailed, in particular in connection with the differentiable structure of the Wasserstein space, the synthetic curvature bounds and their analytic implications. Some of the results presented here have not yet appeared in a book form, with the exception of [44]. It is clear that this subject is expanding so quickly that it is impossible to give an account of all developments of the theory in a few hours, or a few pages. A more modest approach is to give a quick mention of the many aspects of the theory, stimulating the reader’s curiosity and leaving to more detailed treatises as [6] (mostly focused on the theory of gradient flows) and the monumental book [80] (for a -much - broader overview on optimal transport). In Chapter 1 we introduce the optimal transport problem and its formulations in terms of transport maps and transport plans. Then we introduce basic tools of the theory, namely the duality formula, the c-monotonicity and discuss the problem of existence of optimal maps in the model case cost=distance2. In Chapter 2 we introduce the Wasserstein distance W2 on the set P2(X) of probability measures with finite quadratic moments and X is a generic Polish space. This distance naturally arises when considering the optimal transport problem with quadratic cost. The connections between geodesics in 2 P2(X) and geodesics in X and between the time evolution of Kantorovich potentials and the Hopf- Lax semigroup are discussed in detail. Also, when looking at geodesics in this space, and in particular when the underlying metric space X is a Riemannian manifold M, one is naturally lead to the so-called time-dependent optimal transport problem, where geodesics are singled out by an action minimization principle. This is the so-called Benamou-Brenier formula, which is the first step in the interpretation of P2(M) as an infinite-dimensional Riemannian manifold, with W2 as Riemannian distance. We then further exploit this viewpoint following Otto’s seminal work [67]. In Chapter 3 we make a quite detailed introduction to the theory of gradient flows, borrowing almost all material from [6]. First we present the classical theory, for λ-convex functionals in Hilbert spaces. Then we present some equivalent formulations that involve only the distance, and therefore are applicable (at least in principle) to general metric space. They involve the derivative of the distance from a point (the (EVI) formulation) or the rate of dissipation of the energy (the (EDE) and (EDI) formulations). For all these formulations there is a corresponding discrete version of the gradient flow formulation given by the implicit Euler scheme. We will then show that there is convergence of the scheme to the continuous solution as the time discretization parameter tends to 0. The (EVI) formulation is the stronger one, in terms of uniqueness, contraction and regularizing effects. On the other hand this formulation depends on a compatibility condition between energy and distance; this condition is fulfilled in Non Positively Curved spaces in the sense of Alexandrov if the energy is convex along geodesics. Luckily enough, n the compatibility condition holds even for some important model functionals in P2(R ) (sum of the so-called internal, potential and interaction energies), even though the space is Positively Curved in the sense of Alexandrov. In Chapter 4 we illustrate the power of optimal transportation techniques in the proof of some classi- cal functional/geometric inequalities: the Brunn-Minkowski inequality, the isoperimetric inequality and the Sobolev inequality. Recent works in this area have also shown the possibility to prove by optimal transportation methods optimal effective versions of these inequalities: for instance we can quantify the closedness of E to a ball with the same volume in terms of the vicinity of the isoperimetric ratio of E to the optimal one. Chapter 5 is devoted to the presentation of three recent variants of the optimal transport problem, which lead to different notions of Wasserstein distance: the first one deals with variational problems giving rise to branched transportation structures, with a ‘Y shaped path’ opposed to the ‘V shaped one’ typical of the mass splitting occurring in standard optimal transport problems. The second one involves modification in the action functional on curves arising in the Benamou-Brenier formula: this leads to many different optimal transportation distances, maybe more difficult to describe from the Lagrangian viepoint, but still with quite useful implications in evolution PDE’s and functional inequalities. The last one deals with transportation distance between measures with unequal mass, a variant useful in the modeling problems with Dirichlet boundary conditions. d Chapter 6 deals with a more detailed analysis of the differentiable structure of P2(R ): besides the analytic tangent space arising from the Benamou-Brenier formula, also the “geometric” tangent space, based on constant speed geodesics emanating from a given base point, is introduced. We also present Otto’s viewpoint on the duality between Wasserstein space and Arnold’s manifolds of measure- preserving diffeomorphisms. A large part of the chapter is also devoted to the second order differentiable properties, involving curvature. The notions of parallel transport along (sufficently regular) geodesics and Levi-Civita connection in the Wasserstein space are discussed in detail. 3 Finally, Chapter 7 is devoted to an introduction to the synthetic notions of Ricci lower bounds for metric measure spaces introduced by Lott & Villani and Sturm in recent papers. This notion is based on suitable convexity properties of a dimension-dependent internal energy along Wasserstein geodesics. Synthetic Ricci bounds are completely consistent with the smooth Riemannian case and stable under measured-Gromov-Hausdorff limits. For this reason these bounds, and their analytic implications, are a useful tool in the description of measured-GH-limits of Riemannian manifolds. 1 The optimal transport problem 1.1 Monge and Kantorovich formulations of the optimal transport problem Given a Polish space (X; d) (i.e. a complete and separable metric space), we will denote by P(X) the set of Borel probability measures on X. By support supp(µ) of a measure µ 2 P(X) we intend the smallest closed set on which µ is concentrated. If X; Y are two Polish spaces, T : X ! Y is a Borel map, and µ 2 P(X) a measure, the measure T#µ 2 P(Y ), called the push forward of µ through T is defined by −1 T#µ(E) = µ(T (E)); 8E ⊂ Y; Borel. The push forward is characterized by the fact that Z Z fdT#µ = f ◦ T dµ, for every Borel function f : Y ! R [ {±∞}, where the above identity has to be understood in the following sense: one of the integrals exists (possibly attaining the value ±∞) if and only if the other one exists, and in this case the values are equal. Now fix a Borel cost function c : X ×Y ! R[f+1g. The Monge version of the transport problem is the following: Problem 1.1 (Monge’s optimal transport problem) Let µ 2 P(X), ν 2 P(Y ). Minimize Z T 7! cx; T (x) dµ(x) X among all transport maps T from µ to ν, i.e. all maps T such that T#µ = ν. Regardless of the choice of the cost function c, Monge’s problem can be ill-posed because: • no admissible T exists (for instance if µ is a Dirac delta and ν is not).