This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their proofs? Specifically, things like local times, killing, and shunts. 


[1] K Ito, H McKean, Jr, Diffusion Processes and their Sample Paths, Springer, 1974.



Any modern/recent version of Ito & McKean?
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:[Geometric Characterization of Martingales](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.
> 
> 
> 


