This is a guest post by Peter Guttorp. Peter and I are working on updating Peter's book Stochastic Modeling of Scientific Data. To provide the reader with productive distractions, we are sprinkling the book with "historical boxes." Also, let's be honest, it keeps us entertained as well, because we are both math history junkies. Well, Peter does real research in the history of mathematics (e.g., here) , so he is a real historian, but I am definitely a junkie. While working on one of the background chapters, we put a historical box for Chapman and Kolmogorov next to the first appearance of the Chapman-Kolmogorov equation, but then wondered why we did this, because Peter knew that neither Chapman nor Kolmogorov were responsible for its discovery. This is, of course, not surprising in light of the Stigler's law, but then who gave this equation its modern name? Peter started digging and this is what he has found so far:

The origin of the Chapman-Kolmogorov equation dates at least back to Bachelier’s 1900 dissertation on the theory of speculation (but it may be earlier than that). Bachelier did not give it a name.

In physics it is sometimes called Smoluchovsky’s equation, after Smoluchovsky’s 1906 rediscovery of Brownian motion (preceded by Bachelier and by Einstein (1905)). Smoluchovksy’s work was extended by Chapman (1928) to the case of nonstationary transition kernels. It was independently developed by Kolmogorov in his fundamental 1933 Mathematische Annalen paper.

But where did the name come from? The first uses of the Chapman-Kolmogorov name are in Feller’s 1940 Transactions of the AMS paper, where he says that it is “known as the equation of Chapman and Kolmogoroff” . The 1940 thesis of Lundberg also uses this term, referring to Frechet (1938) and Kolmorogov (1931). Going to Frechet’s book, he does not give the equation a name, but designates it “as a mnemotechnical tool” with the letter I.

In 1936, Feller writes in his Mathematische Annalen paper “Finally, from the rule of combination of probabilities, what Chapman and von Smoluchowski called the fundamental equality follows”.

So what happened between 1936 and 1940? Did the name get invented in the corridors of the University of Stockholm (where Feller was a lecturer, Lundberg a PhD student, and Cramer the professor)?If you know anything about the origin of the name Chapman-Kolmogorov equation, please get in touch with Peter and/or me. We are very curious!

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