A PDE hierarchy for directed polymers in random environments

For a Brownian directed polymer in a Gaussian random environment, with q(t, ·) denoting the quenched endpoint density and Qn(t,x1,?,xn)=E[q(t,x1) q(txn)], we derive a hierarchical PDE system satisfied by Qn}n=1. We present two applications of the system: (i) we compute the generator of µt(dx)=q(t,xdx}t=0 for some special functionals, where µt(dx)t=0 is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with d 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.