Again I had to point out my favorite book on diffusion process below. The authors belong to Ito school, so their understanding is quite insightful and consistent with Ito's.

> Ikeda, Nobuyuki, and Shinzo Watanabe. Stochastic differential
> equations and diffusion processes. Vol. 24. Elsevier, 2014.

If you are more interested in the geometric aspect of these notions you mentioned, probably Ambrosio's works is of some interest.

> Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. Gradient flows: in
> metric spaces and in the space of probability measures. Springer
> Science & Business Media, 2008.