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.