Global differentiable structures for the Fisher-Rao and Kantorovich-Wasserstein-Otto metrics

Abstract

We develop a class of non-parametric, Banach-Sobolev manifolds of probability measures that, despite having comparatively weak topologies, support the Fisher-Rao and Kantorovich-Wasserstein-Otto (KWO) Riemannian metrics. The manifolds employ the Kaniadakis κ-deformed logarithms in their charts, and are isomorphic to the (whole) model spaces, W1,λ(μ). These are weighted Sobolev spaces with Lebesgue exponents as small as λ=4, making them suitable for approximations. The KWO metric is analysed through a bundle of Markov semigroups having generators with irregular first-order terms. Strong a-priori estimates are obtained for the associated parabolic equations. Together with a weak Poincaré/Lp-convergence theorem, these are used to establish the Hölder continuity of the “velocity field” representation of tangent vectors—a property inherited by the KWO metric. The manifolds provide a simple framework for the study of problems involving both metrics.