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Recently I've been trying to learn more about probability theory, stochastic process, and stochastic analysis and came upon the following set of lecture notes:

Lunardi–Miranda–Pallara, Infinite Dimensional Analysis, 2015–2016. [PDF].

The notes treat a number of topics which seem really fantastic, including Gaussian measures in infinite dimensional spaces, the Cameron–Martin space, abstract Wiener spaces, and Ornstein–Uhlenbeck operators.

However, I haven't been able to figure out exactly how these topics fit in the larger landscape of mathematics/physics, and what are the connections between them and other topics. To this end, I wanted to ask:

Question. What are some applications of these topics to other areas of mathematics (not necessarily only analysis) and physics?

For instance, I've read in the Wikipedia page for abstract Wiener space that part of the motivation behind it is making sense of (special cases of) path integrals in quantum mechanics and quantum field theory. Is there some place where I could read more about this?

Emily
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  • (Let me also comment that I'd be fine with it even if it turned out to be zero applications of these topics; I do think they are certainly very cool/interesting enough on their own) – Emily May 11 '23 at 18:09
  • In this answer I elaborate why neither the concrete nor the abstract Wiener space is helpful to rigorously define the path integral that is relevant in quantum physics. To this date there is no satisfactory definition of the Feynman path integral. Nonetheless: I agree with you that Wiener spaces are a cool subject. As a start to read about serious applications let me draw your attention to Malliavin calculus. – Kurt G. May 11 '23 at 18:20
  • maybe Black-Scholes equation, or stochastic diffussion processes could be some examples – Joako May 11 '23 at 20:22
  • @KurtG. Oh, this is lovely, I was actually planning to learn Malliavin calculus too, so it's really nice to know there will be applications of these topics there. Thanks! – Emily May 12 '23 at 01:51
  • @Joako Do you have some references on this? I couldn't find much stuff connecting the two areas sadly =/ – Emily May 12 '23 at 01:52

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