A new optimal transport distance on the space of finite Radon measures

TL;DR

This paper introduces a novel optimal transport distance for finite Radon measures with varying masses, based on non-conservative continuity equations, and explores its mathematical properties and applications to population dynamics.

Contribution

It presents a new optimal transport metric for measures with different masses, along with its theoretical properties and applications to modeling population distribution as a gradient flow.

Findings

Established topological and geometrical properties of the new metric space

Derived a formal Riemannian structure and differential calculus

Applied the framework to prove long-time convergence in population models

Abstract

We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula. We establish various topological and geometrical properties of the resulting metric space, derive some formal Riemannian structure, and develop differential calculus following F. Otto's approach. Finally, we apply these ideas to identify an ideal free distribution model of population dynamics as a gradient flow and obtain new long-time convergence results.