## Abstract

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 language | English (US) |
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Pages (from-to) | 7335-7370 |

Number of pages | 36 |

Journal | Nonlinearity |

Volume | 34 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2021 |

## Keywords

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

## ASJC Scopus subject areas

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