Tamed spaces -- Dirichlet spaces with distribution-valued Ricci bounds

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-\'Emery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with either interior singularities or singular boundary behavior.