Fine Properties of Geodesics and Geodesic λ-Convexity for the Hellinger–Kantorovich Distance

We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem ( 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–Jacob...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2023-12, Vol.247 (6), p.112-112, Article 112
Main Authors: Liero, Matthias, Mielke, Alexander, Savaré, Giuseppe
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Convexity
Digital Object Identifier
Entropy
Fluid- and Aerodynamics
Geodesy
Hamilton-Jacobi equation
Interpolation
Mathematical and Computational Physics
Physics
Physics and Astronomy
Regularity
Theoretical
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Summary:We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem ( 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 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 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 λ -convexity with respect to the Hellinger–Kantorovich distance. Examples of geodesically convex functionals are provided.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-023-01941-1