Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation

We consider a class of Gibbs measures defined with respect to increments $\{\omega(t)-\omega(s)\}_{s<t}$ of $d$-dimensional Wiener measure, with the underlying Hamiltonian carrying interactions of the form $H(t-s,\omega(t)-\omega(s))$ that are invariant under uniform translations of paths. In such interactions we allow {\it{long-range}} dependence in the time variable (including power law decay up to $t\mapsto (1+t)^{-(2+\varepsilon)}$ for $\varepsilon>0$) and unbounded (singular) interactions (including singularities of the form $x\mapsto 1/|x|^p$ in $d\geq 3$ or $x\mapsto \delta_0(x)$ in $d=1$) attached to the space variables. These assumptions on the interaction seem to be sharp and cover quantum mechanical models like the Nelson model and the polaron problem with ultraviolet cut off (both carrying bounded spatial interactions with power law decay in time) as well as the Fr\"ohlich polaron with a short range interaction in time but carrying Coulomb singularity in space. In this set up, we develop a unified approach for proving a central limit theorem for the rescaled process of increments for any coupling parameter and obtain an explicit expression for the limiting variance which is strictly positive. As a further application, we study the solution of the multiplicative-noise stochastic heat equation in spatial dimensions $d\geq 3$. When the noise is mollified both in time and space, we show that the averages of the diffusively rescaled solutions converge pointwise to the solution of a diffusion equation whose coefficients are homogenized in this limit.