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

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.