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2008 | Buch

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Gradient Flows

in Metric Spaces and in the Space of Probability Measures

verfasst von: Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré

Verlag: Birkhäuser Basel

Buchreihe : Lectures in Mathematics ETH Zürich

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

This book is devoted to a theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure. It is made of two parts, the first one concerning gradient flows in metric spaces and the second one devoted to gradient flows in the L²-Wasserstein space of probability measures on a separable Hilbert space X endowed with the Wasserstein L² metric (we consider the L^p-Wasserstein distance, p ∈ (1, ∞), as well).

Notation

Gradient Flow in Metric Spaces

Frontmatter

Chapter 1. Curves and Gradients in Metric Spaces

Abstract

As we briefly discussed in the introduction, the notion of gradient flows in a metric space

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relies on two elementary but basic concepts: the metric derivative of an absolutely continuous curve with values in

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and the upper gradients of a functional defined in

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. The related definitions are presented in the next two sections (a more detailed treatment of this topic can be found for instance in [20]); the last one deals with curves of maximal slope.

Chapter 2. Existence of Curves of Maximal Slope and their Variational Approximation

Abstract

The main object of our investigation is the solution of the following Cauchy problem in the complete metric space (

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, d): Problem 2.0.1. Given a functional φ:

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→ (−∞,+∞] and an initial datum u₀ ∈ D(φ), find a (p-)curve u of maximal slope in (0,+∞) for φ such that u(0+) = u₀.

Chapter 3. Proofs of the Convergence Theorems

Abstract

We divide the proof of the main convergence theorems in four steps: first of all, we study a single minimization problem of the scheme (2.0.4); stability estimates are then derived for discrete solutions which yield Proposition 2.2.3 by a compactness argument. Finally, convergence is obtained by combining the a priori energy estimates with the gradient properties of the relaxed slope. We will conclude this section with the proof of Theorem 2.4.15.

Chapter 4. Uniqueness, Generation of Contraction Semigroups, Error Estimates

Abstract

In all this section we consider the “quadratic” approximation scheme (2.0.3b), (2.0.4) for 2-curves of maximal slope and we identify the “weak” topology σ with the “strong” one induced by the distance d as in Remark 2.1.1: thus we are assuming that

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(4.0.1)

but we are not imposing any compactness assumptions on the sublevels of φ. Existence, uniqueness and semigroup properties for minimizing movement u ∈ MM(Φ; u₀) (and not simply the generalized ones, recall Definition 2.0.6) are well known in the case of lower semicontinuous convex functionals in Hilbert spaces [38]. In this framework the resolvent operator in J_τ [·] (3.1.2) is single valued and non expansive, i.e.

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(4.0.2)

this property is a key ingredient, as in the celebrated Crandall-Ligget generation Theorem [58], to prove the uniform convergence of the exponential formula (cf. (2.0.9))

$$ u\left( t \right) = \mathop {lim}\limits_{n \to \infty } \left( {J_{{t \mathord{\left/ {\vphantom {t n}} \right. \kern-\nulldelimiterspace} n}} } \right)^n \left[ {u_0 } \right],d\left( {u\left( t \right),\left( {J_{{t \mathord{\left/ {\vphantom {t n}} \right. \kern-\nulldelimiterspace} n}} } \right)^n \left[ {u_0 } \right]} \right) \leqslant \frac{{2\left| {\partial \varphi } \right|\left( {u_0 } \right)t}} {{\sqrt n }}, $$

(4.0.3)

and therefore to define a contraction semigroup on $ \overline {D\left( \varphi \right)} $. Being generated by a convex functional, this semigroup exhibits a nice regularizing effect [37], since u(t) ∈ D(|∂φ|) whenever t > 0 even if the starting vale u₀ simply belongs to $ \overline {D\left( \varphi \right)} $.

Gradient Flow in the Space of Probability Measures

Frontmatter

Chapter 5. Preliminary Results on Measure Theory

Abstract

In this chapter we introduce, mostly without proofs, some basic measure-theoretic tools needed in the next chapters. We decided to present the most significant result in the quite general framework of separable metric spaces in view of possible applications to infinite dimensional Hilbert (or Banach) spaces, thus avoiding any local compactness assumption (we refer to the treatises [126, 71, 72, 136, 67] for comprehensive presentations of this subject).

Chapter 6. The Optimal Transportation Problem

Abstract

Let X, Y be separable metric spaces such that any Borel probability measure in X, Y is tight (5.1.9), i.e. Radon spaces, according to Definition 5.1.4, and let c : X × Y → [0,+∞] be a Borel cost function. Given μ ∈

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(X), ν ∈

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(Y) the optimal transport problem, in Monge’s formulation, is given by

$$ \inf \left\{ {\smallint _X c\left( {x,t\left( x \right)} \right)d\mu \left( x \right):t_\# \mu = \nu } \right\}. $$

(6.0.1)

This problem can be ill posed because sometimes there is no transport map t such that t_#μ = ν (this happens for instance when μ is a Dirac mass and ν is not a Dirac mass). Kantorovich’s formulation

$$ \min \left\{ {\smallint _{X \times Y} c\left( {x,y} \right)d\gamma \left( {x,y} \right):\gamma \in \Gamma \left( {\mu ,\nu } \right)} \right\} $$

(6.0.2)

circumvents this problem (as μ× ν ∈ Г(μ, ν)). The existence of an optimal transpoplan, when c is l.s.c., is provided by (5.1.15) and by the tightness of Г(μ, ν) (this property is equivalent to the tightness of μ, ν, a property always guaranteed in Radon spaces).

Chapter 7. The Wasserstein Distance and its Behaviour along Geodesics

Abstract

In this chapter we will introduce the p-th Wasserstein distance W_p(μ, ν) between two measures μ, ν ∈

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(X). The first section is devoted to its preliminary properties, in connection with the optimal transportation problems studied in the previous chapter and with narrow convergence: the main topological results are valid in general metric spaces.

Chapter 8. Absolutely Continuous Curves in p (X) and the Continuity Equation

Abstract

In this chapter we endow

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_p(X), when X is a separable Hilbert space, with a kind of differential structure, consistent with the metric structure introduced in the previous chapter. Our starting point is the analysis of absolutely continuous curves μ_t : (a, b) →

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_p(X) and of their metric derivative |μ′|(t): recall that these concepts depend only on the metric structure of

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(X), by Definition 1.1.1 and (1.1.3). We show in Theorem 8.3.1 that for p > 1 this class of curves coincides with (distributional, in the duality with smooth cylindrical test functions) solutions of the continuity equation

$$ \frac{\partial } {{\partial t}}\mu _t + \nabla .\left( {\upsilon _t \mu _t } \right) = 0inX \times \left( {a,b} \right). $$

More precisely, given an absolutely continuous curve μ_t, one can find a Borel time-dependent velocity field v_t : X → X such that $ \left\| {\upsilon _t } \right\|_{L^p \left( {\mu _t } \right)} \leqslant \left| {\mu '} \right|\left( t \right) $ for ℒ¹-a.e. t ∈ (a, b) and the continuity equation holds. Conversely, if μt solve the continuity equation for some Borel velocity field v_t with $ \smallint _a^b \left\| {\upsilon _t } \right\|_{L^p \left( {\mu _t } \right)} dt < + \infty $, then μ_t is an absolutely continuous curve and $ \left\| {\upsilon _t } \right\|_{L^p \left( {\mu _t } \right)} \geqslant \left| {\mu '} \right|\left( t \right) $ for ℒ¹-a.e. t ∈ (a, b).

Chapter 9. Convex Functionals in p (X)

Abstract

The importance of geodesically convex functionals in Wasserstein spaces was firstly pointed out by McCann [111], who introduced the three basic examples we will discuss in detail in 9.3.1, 9.3.4, 9.3.6. His original motivation was to prove the uniqueness of the minimizer of an energy functional which results from the sum of the above three contributions.

Chapter 10. Metric Slope and Subdifferential Calculus in (X)

Abstract

As we have seen in Section 1.4, in the classical theory of subdifferential calculus for proper, lower semicontinuous functionals φ : X → (−∞,+∞] defined in a Hilbert space X, the Fréchet Subdifferential ∂φ : X → 2^X of φ is a multivalued operator defined as

$$ \xi \in \partial \varphi \left( \upsilon \right) \Leftrightarrow \upsilon \in D\left( \varphi \right),\mathop {\lim \inf }\limits_{w \to \upsilon } \frac{{\varphi \left( w \right) - \varphi \left( \upsilon \right) - \left\langle {\xi ,w - \upsilon } \right\rangle }} {{\left| {w - \upsilon } \right|}} \geqslant 0, $$

(10.0.1)

which we will also write in the equivalent form for v ∈ D(φ)

$$ \xi \in \partial \varphi \left( \upsilon \right) \Leftrightarrow \varphi \left( w \right) \geqslant \varphi \left( \upsilon \right) + \left\langle {\xi ,w - \upsilon } \right\rangle + o\left( {\left| {w - \upsilon } \right|} \right)asw \to \upsilon . $$

(10.0.2)

As usual in multivalued analysis, the proper domain D(∂φ) ⊂ D(φ) is defined as the set of all v ∈ X such that ∂φ(v) ≠ φ; we will use this convention for all the multivalued operators we will introduce.

Chapter 11. Gradient Flows and Curves of Maximal Slope in p (X)

Abstract

In this chapter we state some of the main results of the paper, concerning existence, uniqueness, approximation, and qualitative properties of gradient flows μ_t generated by a proper, l.s.c. functional φ in

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_p, X being a separable Hilbert space. Taking into account the first part of this book and the (sub)differential theory developed in the previous chapter, there are at least four possible approaches to gradient flows which can be adapted to the framework of Wasserstein spaces:

The “Minimizing Movement” approximation. We can simply consider any limit curve of the variational approximation scheme we introduced at the beginning of Chapter 2 (see Definition 2.0.6), i.e. a “Generalized minimizing movement” GMM(Φ; μ₀) in the terminology suggested by E. De Giorgi. In the context of

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₂(ℝ^d) this procedure has been first used in [94, 121, 122, 120, 123] and subsequently it has been applied in many different contexts, e.g. by [93, 115, 124, 84, 85, 89, 78, 45, 46, 2, 86, 76, 15, 19].

Curves of Maximal Slope. We can look for absolutely continuous curves $ \mu _t \in AC_{loc}^p $((0,+∞);

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_p(X)) which satisfy the differential form of the Energy inequality

$$ \frac{d} {{dt}}\varphi \left( {\mu _t } \right) \leqslant - \frac{1} {p}\left| {\mu '} \right|^p \left( t \right) - \frac{1} {q}\left| {\partial \varphi } \right|^q \left( {\mu _t } \right) \leqslant - \left| {\partial \varphi } \right|\left( {\mu _t } \right) \cdot \left| {\mu '} \right|\left( t \right) $$

(11.0.1)

for ℒ¹-a.e. t ∈ (a, b). Notice that in the present case of

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_p(X), we established in Chapter 8 a precise description of absolutely continuous curve (in terms of the continuity equation) and of the metric velocity (in terms of the L^p(μ_t;X)-norm of the related velocity vector field); moreover, in Chapter 10 we have shown an equivalent differential characterization of the slope |∂φ| in terms of the L^q(μ_t;X)-norm of the Fréchet subdifferential of φ

The pointwise differential formulation. Since we have at our disposal a notion of tangent space and the related concepts of velocity vector field v_t and (sub)differential ∂φ(μ_t), we can reproduce the simple definition of gradient flow modeled on smooth Riemannian manifold, i.e.

$$ \upsilon _t \in - \partial \varphi \left( {\mu _t } \right), $$

(11.0.2)

trying to adapt it to the case p ≠ 2 and to extended plan subdifferentials.

Systems of Evolution Variational Inequalities (E.V.I.). When p = 2, in the case of λ-convex functionals along geodesics in

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₂(X), we can try to find solutions of the family of “metric” variational inequalities

$$ \frac{1} {2}\frac{d} {{dt}}W_2^2 \left( {\mu _t ,\nu } \right) - \varphi \left( \nu \right) - \varphi \left( {\mu _t } \right) - \frac{\lambda } {2}W_2^2 \left( {\mu _t ,\nu } \right)\forall \nu \in D\left( \varphi \right). $$

(11.0.3)

This formulation provides the best kind of solutions, for which in particular one can prove not only uniqueness, but also error estimates. On the other hand it imposes severe restrictions on the space (p = 2) and on the functional (λ-convexity along generalized geodesics).

Chapter 12. Appendix

Abstract

In this section we recall some standard facts about integrands depending on two variables, measurable w.r.t. the first one, and more regular w.r.t. the second one.

Backmatter

Titel: Gradient Flows
verfasst von: Luigi Ambrosio
Nicola Gigli
Giuseppe Savaré
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8722-8
Print ISBN: 978-3-7643-8721-1
DOI: https://doi.org/10.1007/978-3-7643-8722-8