Quenched central limit theorem for the stochastic heat equation in weak disorder

We continue with the study of the mollified stochastic heat equation in $d\ge 3$ given by $d u_{\epsilon,t}=\frac 12\Delta u_{\epsilon,t}+ \beta \epsilon^{(d-2)/2} ,u_{\epsilon,t} ,d B_{\epsilon,t}$ with spatially smoothened cylindrical Wiener process $B$, whose (renormalized) Feynman-Kac solution describes the partition function of the continuous directed polymer. In an earlier work (\cite{MSZ16}), a phase transition was obtained, depending on the value of $\beta >0$ in the limiting object of the smoothened solution $u_{ϵ}$ as the smoothing parameter $ϵ\to 0$ This partition function naturally defines a quenched polymer path measure and we prove that as long as $\beta >0$ stays small enough while $u_{ϵ}$ converges to a strictly positive non-degenerate random variable, the distribution of the diffusively rescaled Brownian path converges under the aforementioned polymer path measure to standard Gaussian distribution.