So, the statement you make is that the $C_0$-semigroup would actually be a group? As in the Schrödinger equation, which is a Weyl rotation of the heat equation? I understood the 'cannot be made smoother' in this context as the fact that you can 'go back in time' and would make things less smooth. I guess this is what you are saying but more mathematically. I still have to wrap my head around why a rotation in the complex plane would give such profound consequences in the behavior of the heat equation. Perhaps, that is for another question when I thought more about it. (...)

@George: Nice answer. Are you a stochastic analyst?

@Nate: Gradient Flows in Metric Spaces and in the Space of Probability Measures by Ambrosio et al uses the definition I give (pg 113). Do you have a reference where I can find your version?

@Gerald: Okay, but that happens for instance if our space is separable and complete (that all finite Borel measures are tight) and in this question we don't assume separability. Am I missing something?

@Nate: Okay, done!

@Ricky Demer: Good point, I have changed it to your suggestion.

Looks good! Thanks for the links. Interestingly P. Clément is an emeritus professor at my department.

Maybe someone has written lecture notes about this subject, it doesn't have to be a book.