  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.
