  For a Brownian directed polymer in a Gaussian random environment, with
$q(t,\cdot)$ denoting the quenched endpoint density and \[
Q_n(t,x_1,\ldots,x_n)=\mathbf{E}[q(t,x_1)\ldots q(t,x_n)], \] we derive a
hierarchical PDE system satisfied by $\{Q_n\}_{n\geq1}$. We present two
applications of the system: (i) we compute the generator of
$\{\mu_t(dx)=q(t,x)dx\}_{t\geq0}$ for some special functionals, where
$\{\mu_t(dx)\}_{t\geq0}$ is viewed as a Markov process taking values in the
space of probability measures; (ii) in the high temperature regime with $d\geq
3$, we prove a quantitative central limit theorem for the annealed endpoint
distribution of the diffusively rescaled polymer path. We also study a nonlocal
diffusion-reaction equation motivated by the generator and establish a
super-diffusive $O(t^{2/3})$ scaling.
