This thesis studies heat flows acting on different objects on possibly singular spaces that admit synthetic lower Ricci curvature bounds by constants, functions, or signed measures. Geometric properties of such spaces and probabilistic features of diffusion processes on these are related to functional inequalities for the involved semigroups. Moreover, heat flow methods are used to set up a second order calculus in the general presence of such measure-valued lower Ricci bounds.
First, for a given RCD space, we prove the equivalence of the following synthetic characterizations (with respect to a given lower semicontinuous function k) of the "Ricci curvature at every point being bounded from below by k": geodesic semiconvexity of the relative entropy, the evolution variational inequality, Bochner’s inequality, gradient bounds for the functional heat flow, transport estimates, and the pathwise coupling property.
Second, on arbitrary weighted Riemannian manifolds, we prove the equivalence of the previous pathwise coupling property with respect to k and pointwise lower boundedness of the Bakry–Émery Ricci tensor by k, only assuming continuity of k. Under an additional exponential integrability condition on k, which holds if k is in the functional Kato class of the weighted manifold, we prove conservativeness and Bismut–Elworthy–Li’s derivative formula.
Third, we extend the second order calculus for RCD spaces from Gigli to Dirichlet spaces which are tamed by a signed extended Kato class measure in the sense of Erbar, Rigoni, Sturm and Tamanini. Inter alia, nonsmooth analogues of Hessians, covariant and exterior derivatives, and the Ricci curvature are defined. Employing these objects, in turn, we define heat flows on 1-forms and vector fields and, along with their basic properties, prove domination of the latter by certain semigroups acting on functions.
Fourth, again in the setting of RCD spaces, we obtain improved functional inequalities and regularization properties of the heat flow on 1-forms. The spectrum of its generator, the Hodge Laplacian, is studied as well. Finally, we construct a heat kernel for this heat flow and prove Gaussian upper bounds on its pointwise operator norm
