The book Quantum dissipative systems by Weiss dedicates a subsection to the Feynman Vernon method, see also the original reference. See also this article and chapter 18.8 of the book by Kleinert.

It's applied to the Caldeira-Leggett model, which is a toy model for a particle in contact with a heat bath. There are a number of mesoscopic systems out there in which a Feynman-Vernon functional of similar type pops up. I don't have any references, but tunneling junctions in fractional quantum Hall edges, impurities in Luttinger liquids and SQUID devices form three examples. I'm sure the book by Weiss has some references as well.

The Keldysh-Schwinger or real-time formalism is required to treat systems out of equilibrium. For a list of references see this thread here. But this formalism by itself is not enough. You need to make some assumptions regarding the degrees of freedom of the heat bath, the coupling between the subsystem and the external heat bath and also the initial (untangled or not) state of the system as a whole.

The idea as follows: you model the system under consideration in contact with a heat bath. In the Caldeira-Leggett model the heat bath is a macroscopic number of harmonic oscillators, each of which is in contact with the degrees of freedom of the system under consideration. The Feynman-Vernon functional is obtained by integrating out the degrees of freedom associated with the heat bath, all by using a path integral formalism. We can think of this functional as describing the time evolution of the reduced density matrix.

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