I have a masters degree in probability and statistics but am new to quantum physics. I have been reading an elementary text about the Schroedinger equation and I keep thinking that the Heisenberg Uncertainty Principle could emerge from a random walk that has the characteristic that as the interval of time gets smaller the momentum becomes increasingly uncertain. The random walk would be like a fractal, except while a fractal has the same level of detail at all levels our random walk would have more detail and be more jagged the more closely one looks at it. It does this without bound, so it is divergent. So integrating in the dt way in which variation over tiny units of time is assumed to be smooth is going to work reasonably well but somewhat oddly, thus giving rise to the uncertainty principle and all this other phenomena. 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.