A PDE hierarchy for directed polymers in random environments

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2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)7335-7370
Number of pages36
Issue number10
StatePublished - Oct 2021


  • directed polymer
  • reaction-diffusion equation
  • stochastic heat equation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


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