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Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, GTM 25) would certainly be a good starting point. Yet, they stay within the limits of ordinary calculus. My question is about a calculus that has physical measurements in the center of its considerations. Velocity will no longer be the quotient of two real numbers. Instead, it would be the result of the quotient of two probability distributions: distance versus time, both as random variables.

Are there or have there been attempts to develop analysis along those lines, or something that goes in that direction? Something like "Wick's ideas thought to the end". I'm aware of the fact that this description is lousy. I cannot specify it further since I haven't seen something I'm looking for, yet.

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    $\begingroup$ I don't understand which equations that are centered. Also, can you please explain your idea of the analysis of velocity? $\endgroup$ Commented Oct 13, 2023 at 1:39
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    $\begingroup$ Can you perhaps describe some concrete/specific math toy problem to understand what you mean? $\endgroup$ Commented Oct 13, 2023 at 3:21
  • $\begingroup$ So the closer I can think of is Gubinelli-derivative mathoverflow.net/questions/361920/… Here we basically "Taylor"-expand the process Y in terms of some other process X. $\endgroup$ Commented Oct 13, 2023 at 3:38
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    $\begingroup$ To me this concept seems to be hopeless. Consider for example the pair distance/time. The problem is that with 2 probability measures it seems to be impossible to describe such concepts as correlation, covariance ... Maybe if all random variables are independent that this concept is o.k. $\endgroup$ Commented Oct 13, 2023 at 8:27

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Perhaps you need the algebra of random variables. By using this algebra and the standard techniques of calculus you can, at least in principle, determine the PDF of functions $f(X_1, \dots, X_n)$ of $n$ random variables from the PDF of their arguments: this of course includes the standard difference and sum of random variables but also their product and quotient. A now classical text is [1] which, moreover, is available at the Internet Archive (with some restrictions).

Reference

[1] Melvin Dale Springer, The algebra of random variables (English) Wiley Series in Probability and mathematical Statistics. New York etc.: John Wiley & Sons, pp. XIX+470 (1979), ISBN:0-471-01406-0, MR519342, Zbl 0399.60002.

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