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. The understanding of his statement really depends on how you understand random measures.

> 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. Also look at another answer here:https://mathoverflow.net/questions/227720/geometric-characterization-of-martingales/265068#265068

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