A PDE hierarchy for directed polymers in random environments. (16th September 2021)

• Record Type: Journal Article
• Authors: Gu, Yu; Henderson, Christopher
• Is Part Of: Nonlinearity. Volume 34:Number 10(2021)
• Page Start: 7335
• Page End: 7370
• Publication Date: 2021-09-16
• Subjects: directed polymer -- stochastic heat equation -- reaction-diffusion equation; 60H15 -- 35K57 -- 82D60; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac23b7
• Languages: English
• ISSNs: 0951-7715

Abstract

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.