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 -deformed logarithms in their charts, and are isomorphic to the (whole) model spaces, . These are weighted Sobolev spaces with Lebesgue exponents as small as , 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/-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.

中文翻译：

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Fisher-Rao 和 Kantorovich-Wasserstein-Otto 度量的全局可微结构

我们开发了一类非参数的 Banach-Sobolev 概率测度流形，尽管其拓扑相对较弱，但支持 Fisher-Rao 和 Kantorovich-Wasserstein-Otto (KWO) 黎曼度量。流形在其图表中采用 Kaniadakis 变形对数，并且与（整个）模型空间 是同构的。这些是加权索博列夫空间，勒贝格指数小至 ，使其适合近似。 KWO 度量是通过一组具有不规则一阶项的生成器的马尔可夫半群来分析的。获得相关抛物线方程的强先验估计。与弱庞加莱/收敛定理一起，它们被用来建立切向量的“速度场”表示的霍尔德连续性——这是 KWO 度量继承的属性。流形为研究涉及这两个度量的问题提供了一个简单的框架。