A new class of transport distances between measures

By Jean Dolbeault, Bruno Nazaret, Giuseppe Savar'e. 2008

213 citations recorded by the source corpus.

We introduce a new class of distances between nonnegative Radon measures in $${\mathbb{R}^d}$$ . They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou and Brenier (Numer Math 84:375–393, 2000) and provide a wide family interpolating between the Wasserstein and the homogeneous $${W^{-1,p}_\gamma}$$ -Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/5a5436587bfd41c3f29b3ed73a490b9ab065bc75

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In Applied Mathematics

Citations in the corpus

Gradient flows of the entropy for jump processes (2012, methodology): “The proof follows from the very same arguments as in [14], Theorem 5.”
A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions (2010, unknown): “Let us observe that the choice of the distance on the space of probability measures plays a key role, and by changing it one can construct solutions to more general classes of evolution equations, see for instance [1, 5, 7].”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, background): “Similar ideas have already been developed in a continuous setting in [17], where a general class of transportation metrics is constructed based on the usual continuity equation in Rn.”
Convexity properties of Dirichlet integrals and Picone-type inequalities (2014, background): “For generalizations of this transport problem involving functions of the form H(z) s−β the reader can consult [10] and [14].”
A new optimal transport distance on the space of finite Radon measures (2015, methodology): “In [20] the authors constructed a new class of pseudo-distances on the space of non-negative Radon measures via modified Benamou-Brenier formulas, and as a result some techniques are similar to the ones here.”
Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics (2012, background): “It has been proven only recently by Dolbeault, Nazaret and the third author [12] that even for certain non-linear mobilities m, the formal metric structure indicated in (8) still leads to a genuine metric Wm on the space of positive measures.”
Gradient flows of the entropy for finite Markov chains (2011, background): “In a continuous setting, metrics with these properties have been studied in the recent papers [6, 9].”
Computational Optimal Transport (2018, background): “A possible choice for this mobility functional is proposed in Dolbeault et al. [2009] ∀ (a, b) ∈ R+ × R, θ(a, b) = as−p ‖b‖ (7.”
Geodesics for a class of distances in the space of probability measures (2012, unknown): “In the recent paper [8], Dolbeault, Nazaret and Savaré, introduced a new class of distances between probability measures through the introduction of some concave increasing nonnegative nonlinear mobility function m.”