Evolutionary Variational Inequalities on the Hellinger-Kantorovich and Spherical Hellinger-Kantorovich spaces

TL;DR

This paper investigates the convergence of a minimizing movement scheme in Hellinger-Kantorovich spaces, demonstrating that interpolated curves satisfy Evolutionary Variational Inequalities as the time step approaches zero.

Contribution

It establishes the convergence of geodesic interpolations to EVI solutions in Hellinger-Kantorovich spaces, leveraging their geometric properties.

Findings

Convergence of interpolated curves to EVI solutions.

Validation of the minimizing movement scheme in these spaces.

Enhanced understanding of the geometric structure of Hellinger-Kantorovich spaces.

Abstract

We study the minimizing movement scheme for families of geodesically semi-convex functionals defined on either the Hellinger-Kantorovich or the Spherical Hellinger-Kantorovich space. By exploiting some of the finer geometric properties of those spaces, we prove that the sequence of curves, which are produced by geodesically interpolating the points generated by the minimizing movement scheme, converges to curves that satisfy the Evolutionary Variational Inequality (EVI), when the time step goes to 0.