A PDE hierarchy for directed polymers in random environments

For a Brownian directed polymer in a Gaussian random environment, with $q(t,\cdot )$ denoting the quenched endpoint density and $$Q_n(t,x_1,\dots ,x_n)=\mathbf{E}[q(t,x_1)\dots q(t,x_n)],$$ we derive a hierarchical PDE system satisfied by $\{Q_n{\}}_{n\ge 1}$. We present two applications of the system: (i) we compute the generator of $\{{\mu }_t(dx)=q(t,x)dx{\}}_{t\ge 0}$ for some special functionals, where $\{{\mu }_t(dx){\}}_{t\ge 0}$ 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 $O(t^{2\mathrm{/}3})$ scaling.