A new class of transport distances
arXiv:0803.1235 · doi:10.1007/s00526-008-0182-5
Abstract
We introduce a new class of distances between nonnegative Radon measures in Euclidean spaces. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou-Brenier and provide a wide family interpolating between the Wasserstein and the homogeneous (dual) 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. 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.
References in corpus (1)
Cited by in corpus (42)
- Optimal Transport with Proximal Splitting
- Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures
- Ricci curvature of finite Markov chains via convexity of the entropy
- Geometrical Bounds of the Irreversibility in Markovian Systems
- Cahn-Hilliard and Thin Film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics
- Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape
- Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
- One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: Remarks on duality and flows
- Housekeeping and excess entropy production for general nonlinear dynamics
- Jump processes as Generalized Gradient Flows
- Gradient flow structure for McKean-Vlasov equations on discrete spaces
- Instability, rupture and fluctuations in thin liquid films: Theory and computations
- Transport distances and geodesic convexity for systems of degenerate diffusion equations
- Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit
- Entropic and gradient flow formulations for nonlinear diffusion
- Dimensional contraction via Markov transportation distance
- Computational Mean-field information dynamics associated with Reaction diffusion equations
- Generalized Dynamic Programming Principle and Sparse Mean-Field Control Problems
- On gradient flow and entropy solutions for nonlocal transport equations with nonlinear mobility
- Incompressible immiscible multiphase flows in porous media: a variational approach
- The Landau equation as a Gradient Flow
- Exponential decay of Rényi divergence under Fokker-Planck equations
- Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances
- Cosh gradient systems and tilting
- Hamilton--Jacobi equations for controlled gradient flows: the comparison principle
- Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility
- A new perspective on Wasserstein distances for kinetic problems
- Controlling conservation laws II: compressible Navier-Stokes equations
- Geometric thermodynamics of reaction-diffusion systems: Thermodynamic trade-off relations and optimal transport for pattern formation
- Cellular gradient flow structure connects single-cell-level rules and population-level dynamics
- Optimal transport on gas networks
- Covariance-modulated optimal transport and gradient flows
- On a novel gradient flow structure for the aggregation equation
- Carathéodory Theory and A Priori Estimates for Continuity Inclusions in the Space of Probability Measures
- Wasserstein geometry and Ricci curvature bounds for Poisson spaces
- Evolution models for mass transportation problems
- Path constrained unbalanced optimal transport
- Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces
- Drift-diffusion equations with saturation
- A gradient flow perspective on McKean-Vlasov equations in econophysics
- Aggregation-Confinement-Diffusion Evolutions with Saturation: Regularity and Long-Time Asymptotics
- Continuous symmetrizations and uniqueness of solutions to nonlocal equations