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My second thought is that someone else has thought of this before and worked it all out circa 1950. Norbert Wiener, perhaps. There are hints of this in the preface to the text, that when taking a more sophisticated look at the wave equation it is necessary to use measures and Lebesque integrals. The author is not going to do this, and writes that most physicists don't worry about it and consult an expert if necessary. So has anyone out there heard of a treatment of quantum mechanics in this way, based on a random walk that diverges inversely to dt, and could give me a few keywords to search on?

In the meanwhile I'll continue with the elementary text, which is working very well for me.