Spherical Hellinger-Kantorovich gradient flows

TL;DR

This paper investigates nonlinear degenerate parabolic equations as gradient flows under the spherical Hellinger-Kantorovich distance, proving solvability, convergence, and related transportation inequalities without assuming geodesic convexity of the entropy.

Contribution

It introduces new analysis of gradient flows with respect to the spherical Hellinger-Kantorovich distance, including solvability and convergence results without geodesic convexity assumptions.

Findings

Proved solvability of the equations.

Established exponential convergence to equilibrium.

Derived transportation inequalities for the distances.

Abstract

We study nonlinear degenerate parabolic equations of Fokker-Planck type which can be viewed as gradient flows with respect to the recently introduced spherical Hellinger-Kantorovich distance. The driving entropy is not assumed to be geodesically convex. We prove solvability of the problem and the entropy-entropy production inequality, which implies exponential convergence to the equilibrium. As a corollary, we obtain some related results for the Wasserstein gradient flows. We also deduce transportation inequalities in the spirit of Talagrand, Otto and Villani for the spherical and conic Hellinger-Kantorovich distances.