Generalized Wasserstein distance and its application to transport equations with source

Abstract

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized Wasserstein distance to study a transport equation with source, in which both the vector field and the source depend on the measure itself. We prove existence and uniqueness of the solution to the Cauchy problem when the vector field and the source are Lipschitzian with respect to the generalized Wasserstein distance.

References in corpus (3)

• Multiscale modeling of granular flows with application to crowd dynamics
• Pedestrian flows in bounded domains with obstacles
• Existence and approximation of probability measure solutions to models of collective behaviors

Cited by in corpus (52)

• Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures
• Scaling Algorithms for Unbalanced Transport Problems
• Unnormalized Optimal Transport
• A new optimal transport distance on the space of finite Radon measures
• Topological Speed Limit
• Sinkhorn Divergences for Unbalanced Optimal Transport
• Linearized Optimal Transport for Collider Events
• Mean-field and graph limits for collective dynamics models with time-varying weights
• An Interpolating Distance between Optimal Transport and Fisher-Rao
• Differentiated cell behavior: a multiscale approach using measure theory
• Which Metric on the Space of Collider Events?
• A varifold approach to surface approximation
• A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows
• SHAPER: Can You Hear the Shape of a Jet?
• Transport of measures on networks
• Generalized Unnormalized Optimal Transport and its fast algorithms
• Semi-invariant Riemannian metrics in hydrodynamics
• Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces
• An optimal transport approach for solving dynamic inverse problems in spaces of measures
• Outlier-Robust Optimal Transport
• A new perspective on Wasserstein distances for kinetic problems
• On the ubiquity of the Cauchy distribution in spectral problems
• Mean-field limit of a hybrid system for multi-lane multi-class traffic
• Understanding Mass Transfer Directions via Data-Driven Models with Application to Mobile Phone Data
• Randomized low-rank approximation for symmetric indefinite matrices
• Entropy-Transport distances between unbalanced metric measure spaces
• On properties of the Generalized Wasserstein distance
• Duality and quotient spaces of generalized Wasserstein spaces
• Moving in a crowd: human perception as a multiscale process
• A rare mutation model in a spatial heterogeneous environment: the Hawk and Dove game
• The Wasserstein-Fisher-Rao metric for waveform based earthquake location
• Hybrid control for optimal visiting problems for a single player and for a crowd
• Mean-Field Pontryagin Maximum Principle
• A Geometric Approach for Real-time Monitoring of Dynamic Large Scale Graphs: AS-level graphs illustrated
• Stable Super-Resolution of Images: A Theoretical Study
• The Linearized Hellinger--Kantorovich Distance
• Control to flocking of the kinetic Cucker-Smale model
• Graph-Based Equilibrium Metrics for Dynamic Supply-Demand Systems with Applications to Ride-sourcing Platforms
• Entropy Partial Transport with Tree Metrics: Theory and Practice
• Enhancing Rotated Object Detection via Anisotropic Gaussian Bounding Box and Bhattacharyya Distance
• Small perturbations of polytopes
• Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis
• Application of an unbalanced optimal transport distance and a mixed L1/Wasserstein distance to full waveform inversion
• Asymptotic behaviour of a structured population model on a space of measures
• Localization of directed polymers in continuous space
• Locally Private k-Means in One Round
• Leader formation with mean-field birth and death models
• Chaos propagation in genetic algorithms: An optimal transport approach
• Existence and differentiability in parameter of the measure solution to a perturbed non-linear transport equation
• Weak optimal total variation transport problems and generalized Wasserstein barycenters
• Fast Unbalanced Optimal Transport on a Tree
• Well-posedness and propagation of chaos for multi-agent models with strategies and diffusive effects