  The application of the Feynman-Kac formula to Polaron models of quantum
theory leads to the path measure of Brownian motion perturbed by a pair
potential that is translation invariant both in space and time. An important
problem in this context is the validity of a central limit theorem in infinite
volume. We show both the existence of the relevant infinite volume limits and a
functional central limit theorem in a generality that includes the Fr\"ohlich
polaron for all coupling constants. The proofs are based on an extension of a
novel method by Mukherjee and Varadhan.
