The $\infty$-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps

Abstract

We consider the nonlinear optimal transportation problem of minimizing the cost functional C ∞ (λ) = λ-ess sup (x,y)∈Ω 2 |y − x| in the set of probability measures on Ω 2 having prescribed marginals. This corresponds to the question of characterizing the measures that realize the innite Wasserstein distance. We establish the existence of local solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.

The Monge problem for strictly convex norms in $\mathbb{R}^d$

“…The following lemma, although quite simple, is an important step in the proof of Proposition 5.2 and Theorem 6.1 below. Its proof is a straightforward adaptation of that of Lemma 5.2 from [13] and we detail it for the convenience of the reader. Lemma 4.3.…”

“…This section is independent of the transport problem (1.3), and the definitions and techniques detailed below are refinements of similar ones which were first applied in [13] in the framework of nonclassical transportation problems involving cost functionals not in integral form.…”

“…The originality of our method for the proof of Theorem 1.1 above is that it does not require disintegration of measures and relies on a simple but powerful regularity result (see Lemma 4.3 below) which has been used in some transport problem with cost functional in non-integral form [13]. In §2 we recall some well known results on duality and optimality conditions for problem (1.3).…”

Уравнение Агрегации С Анизотропной Диффузией

“…Indeed, since the level sets of f 2 are traveling at speed 1 and the level sets off 2 are traveling with speed v(h), for each τ we can find a transport plan between them with maximal displacement L ∞ distance at most 2τ in its support. Since these densities are both in L ∞ , there is an optimal transport map for the ∞-Wasserstein such that |T (τ, x) − x| ≤ 2τ , see [29].…”

Approximation by finitely supported measures

“…The case p = ∞ is obtained as a limit of the finite case, see [3]. Let μ 0 and μ 1 be compactly supported measures of the same mass and let Π be a transport plan between μ 0 and μ 1 .…”