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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.

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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

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

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  • $\begingroup$ Will look them up for sure. Are you aware of anyone who worked out their proofs (and wrote it up somewhere); some of which are quite unintuitive, to say the least. $\endgroup$
    – horaceT
    Apr 8, 2017 at 17:42
  • $\begingroup$ whose proof? Ito's proof is sometimes unintuitive because he think the symbolic calculus will be a justification. I think Ito has another long book explaining how he thinks about his symbolic calculus(now known as Ito calculus), which I did not have a chance to read. Ito, Kiyosi. Foundations of stochastic differential equations in infinite dimensional spaces. Society for Industrial and Applied Mathematics, 1984. $\endgroup$
    – Henry.L
    Apr 8, 2017 at 17:45
  • $\begingroup$ @horaceT updated another source and hope helps! $\endgroup$
    – Henry.L
    Apr 8, 2017 at 17:48
  • $\begingroup$ AFAIK, japanese mathematics is not exactly the most readable in the world.....let me start another thread about one of their proofs. $\endgroup$
    – horaceT
    Apr 8, 2017 at 17:58
  • $\begingroup$ @horaceT sure, happy to see! $\endgroup$
    – Henry.L
    Apr 8, 2017 at 17:58

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