Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces

TL;DR

This paper develops a comprehensive framework for optimal transport and nonsmooth analysis in infinite-dimensional extended metric spaces, linking Cheeger energies, Dirichlet forms, and curvature conditions.

Contribution

It introduces extended metric-topological measure spaces and explores their properties, connecting dynamic transport distances with Dirichlet forms and curvature conditions in a novel way.

Findings

Establishes properties of dynamic transport distances in extended spaces

Links Cheeger energies with Dirichlet forms and curvature conditions

Analyzes contractivity and EVI formulation of Heat semigroup

Abstract

We introduce the setting of extended metric-topological measure spaces as a general "Wiener-like" framework for optimal transport problems and nonsmooth metric analysis in infinite dimension. After a brief review of optimal transport tools for general Radon measures, we discuss the notions of the Cheeger energy, of the Radon measures concentrated on absolutely continuous curves, and of the induced "dynamic transport distances." We study their main properties and their links with the theory of Dirichlet forms and the Bakry-\'Emery curvature condition, in particular concerning the contractivity properties and the EVI formulation of the induced Heat semigroup.