A PDE hierarchy for directed polymers in random environments

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