Spherical Hellinger--Kantorovich Gradient Flows

We study nonlinear degenerate parabolic equations of Fokker-Planck type that 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.