Fine properties of geodesics and geodesic lambda-convexity for the Hellinger--Kantorovich distance
- Liero, Matthias
- Mielke, Alexander
- Savaré, Giuseppe
2020 Mathematics Subject Classification
- 28A33 54E35 49Q2 49J35 49J40 49K35 46G99
- Hellinger--Kantorovich distance, regularity geodesic curves, optimality conditions for dual potentials, geodesic semiconvexity
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.