Fine Properties of Geodesics and Geodesic $$\lambda $$-Convexity for the Hellinger–Kantorovich Distance
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Published:2023-11-29
Issue:6
Volume:247
Page:
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ISSN:0003-9527
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Container-title:Archive for Rational Mechanics and Analysis
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language:en
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Short-container-title:Arch Rational Mech Anal
Author:
Liero MatthiasORCID, Mielke AlexanderORCID, Savaré GiuseppeORCID
Abstract
AbstractWe study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem ($$\textsf{H}\!\!\textsf{K}$$
H
K
), 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 $$\textsf{H}\!\!\textsf{K}$$
H
K
, which are sufficiently strong to construct a characteristic transport-growth flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of $$\textsf{H}\!\!\textsf{K}$$
H
K
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 $$\lambda $$
λ
-convexity with respect to the Hellinger–Kantorovich distance. Examples of geodesically convex functionals are provided.
Funder
Ministero dell’Istruzione, dell’Universitá e della Ricerca European Research Council German Research Foundation
Publisher
Springer Science and Business Media LLC
Subject
Mechanical Engineering,Mathematics (miscellaneous),Analysis
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