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?
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)