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I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend?

I'm looking for a book that has a unified approach and stress the interplay between both subjects. My interest is broad and ranges from the study of Markov semigroups to the study of Gaussian Sobolev spaces for example.

The book closest to this is : Functional analysis for probability and stochastic process by Adam Brobowski. The book seems great but I would like to know the state of the art on this literature by people with more experience than me in these fields.

Thank you a lot !

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    $\begingroup$ Bobrowski's book contains quite a number of annoying inaccuracies. For example, it is claimed that the dual unit ball of a Banach space is not only weak$^*$ compact (by Alaoglu's theorem) but also weak$^*$sequentially compact which, in general, is wrong. $\endgroup$
    – Jochen Wengenroth
    Commented Jun 18 at 8:42
  • $\begingroup$ @JochenWengenroth Thank you for your comment ! I remember his statement of Alaoglu’s theorem but I missed the claim involving the weak star sequentially compactness, I will look at this. $\endgroup$
    – FASAP
    Commented Jun 18 at 9:09
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    $\begingroup$ "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis Solid foundation in functional analysis with applications to PDEs, which extend to stochastic PDEs and probabilistic interpretations. It covers Sobolev spaces in detail, essential for understanding Gaussian Sobolev spaces. $\endgroup$
    – DrJay
    Commented Jun 18 at 9:59
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    $\begingroup$ cambridge.org/core/books/… $\endgroup$
    – Michael Greinecker
    Commented Jun 18 at 11:24
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    $\begingroup$ Ledoux-Talagrand is of course a great reference link.springer.com/book/10.1007/978-3-642-20212-4 $\endgroup$
    – Fabrice Baudoin
    Commented Jun 18 at 13:07

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