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-growth 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 λ-convexity with respect to the Hellinger-Kantorovich distance. Examples of geodesically convex functionals are provided.
Fine Properties of Geodesics and Geodesic λ-Convexity for the Hellinger-Kantorovich Distance
Savarè, Giuseppe
2023
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-growth 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 λ-convexity with respect to the Hellinger-Kantorovich distance. Examples of geodesically convex functionals are provided.File | Dimensione | Formato | |
---|---|---|---|
s00205-023-01941-1.pdf
accesso aperto
Descrizione: Paper
Tipologia:
Pdf editoriale (Publisher's layout)
Licenza:
Creative commons
Dimensione
1.19 MB
Formato
Adobe PDF
|
1.19 MB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.