What surprises me is there is little discussion about characteristic functionals, except for some 40 year old papers.

It seems surprising to me that physicists, being used to analogies from simple systems to complex ones, are not used to emphasize the existence of characteristic functions in probability, and how its obvious generalization is the partition function. Maybe this is explained somewhere, but it is not "typical".

I read that the Nelson approach, trying to define QM as "stochastic paths", à la Markov processes, was a dead end.

A bit similar with Jaynes' Max Ent principle: I see there is a guy who recently has written about recovering the Schroedinger equation from Max Ent. By looking at Jaynes arguments, it seems the Gibbs measure could be justified quite rationally under this framework. But in general (and apart from the recent trend on entropy with black holes) this idea is not used much, and the Gibbs measure is used without much justification.

Is there any referece where the concept of characteristic functionals, and measure theory for infinite dimensional systems in QFT, is discussed? Or in general, other probabilistic issues?