In the book "Conditional Measures and Applications", it was pointed out that "Reynolds operators have not yet been studied from the potential theoretic point of view ." Have there been any research done in that area?
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$\begingroup$ Are these Reynolds operators in the sense of averaging, and if so, what do they have to do witH Fluid Dynamics? $\endgroup$– Yemon ChoiCommented Jun 5, 2014 at 2:56
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$\begingroup$ Yes, they are the averaging operators. They are used in models of turbulence. $\endgroup$– guest11Commented Jun 5, 2014 at 3:43
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for what it's worth, here is one rather recent & pessimistic assessment:
Gian-Carlo Rota wrote that "The Reynolds operators are the potentials, in the language of probabilistic potential theory, of semigroups of measure-preserving transformations". I do not believe that this powerful but formal analogy has attracted any researchers. I even have the impression that Rota knew or rather sensed that it might not lead anywhere.
Jean Dhombres, in Gian-Carlo Rota on Analysis and Probability (2003)
answered Jun 5, 2014 at 8:21