Kantorovich duality for general transport costs and applications

By Nathael Gozlan, Cyril Roberto, Paul-Marie Samson, Prasad Tetali. 2014

191 citations recorded by the source corpus.

We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/2b480419cf4755dfebd506412cc6ca25150cafff

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In Applied Mathematics

Citations in the corpus

On a mixture of Brenier and Strassen Theorems (2018, methodology): “Let θ : R → R be some convex function and μ, ν ∈ P1(R); using the terminology of [33], the barycentric optimal transport cost between μ and ν is defined by”
Stability of martingale optimal transport and weak optimal transport (2019, unknown): “[23] proposed the following nonlinear generalization of (OT): Given a cost function C : X × P(Y) → R the optimal weak transport problem is”
Applications of weak transport theory (2020, unknown): “[34, 33] are mainly motivated by applications to geometric inequalities.”
The Wasserstein space of stochastic processes (2021, background): “the survey of Leonard [55]), martingale optimal transport, the weak transport problem (see [36, 6] among many others), the theory of linear transfers ([25]), and mathematical finance / martingale optimal transport, we refer to [10, 15, 2, 12].”
Sampling of probability measures in the convex order and approximation of Martingale Optimal Transport problems (2017, unknown): “x m Q(x)j%(dx) under the constraint that Qis a Markov kernel such that Q= : Note that this problem is a particular case of the general transport costs considered by Gozlan et al. in Denition 2.3 [9]. When the X iare distinct, (3.1) is recovered by setting Q(x;dy) = (P J j=1 q ij Y j (dy) if x= X if”
A new class of costs for optimal transport planning (2018, background): “Before concluding this introduction, we point out that, while completing the present study, we became aware of the recently published paper [11] (and of its arXiv 2014 version) where the authors consider problem (1.”
Martingale Benamou–Brenier: A probabilistic perspective (2017, background): “Similarly, the martingale version (MBB) can be reformulated as discrete time problem, more precisely, a transport problem for general transport costs in the sense of [28].”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, methodology): “[25].”
Adapted Wasserstein distances and stability in mathematical finance (2019, background): “ss the choice of cost function and give some examples. Moreover we derive a contraction principle (Theorem 3.10) which relates adapted Wasserstein distance with a ‘weak’ (in the sense of Gozlan et al [29]) transport distance. This result forms the basis for the proofs of the results mentioned in the”
Concentration inequalities for random tensors (2019, unknown): “Significant efforts were made to extend Gaussian concentration to more general, not necessarily bounded, distributions, see e.g. [2, 6, 15] and the references therein.”