Fine properties of geodesics and geodesic $\lambda$-convexity for the Hellinger-Kantorovich distance

TL;DR

This paper investigates the regularity and geometric properties of geodesics in the Hellinger-Kantorovich space, providing conditions for convexity and new insights into optimal transport dynamics between positive measures.

Contribution

It introduces new regularity results for dual potentials and Hamilton-Jacobi solutions, enabling the construction of transport flows and establishing geodesic convexity conditions.

Findings

Regularity conditions for dual potentials and Hamilton-Jacobi solutions

Construction of characteristic transport-dilation flows for geodesics

Conditions ensuring geodesic $mbda$-convexity of measure functionals

Abstract

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton--Jacobi equation arising in the dual dynamic formulation of HK, which are sufficiently strong to construct a characteristic transport-dilation flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of HK geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic $\lambda$-convexity with respect to the Hellinger--Kantorovich distance. Examples of geodesically convex functionals are provided.