“…In this section, we describe some recent results contained in the second author's PhD thesis [Tre14], where existence and uniqueness problems are investigated for diffusion operators, in the same abstract framework as above; two papers [Tre15a] and [Tre15b], the first one dealing with refined results in Euclidean spaces and the second one dealing with the abstract setting are in preparation. Motivations for such an extension come at least from two sides: first, the theory of infinitesimally Hilbertian metric measure spaces, mentioned in Example 4.1 and currently under intensive development, requires new calculus tools, and diffusion processes, strongly connected with parabolic partial differential equations, provide a natural extension of flows and transport equations techniques.…”