A PDE hierarchy for directed polymers in random environments. (16th September 2021) - Gu, Yu; Henderson, Christopher
Abstract: For a Brownian directed polymer in a Gaussian random environment, with q ( t, ⋅) denoting the quenched endpoint density and Q n ( t, x 1, …, x n ) = E [ q ( t, x 1 ) … q ( t, x n ) ], we derive a hierarchical PDE system satisfied by Q n n ⩾ 1 . We present two applications of the system: (i) we compute the generator of μ t ( d x ) = q ( t, x ) d x t ⩾ 0 for some special functionals, where μ t ( d x ) 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.