A PDE hierarchy for directed polymers in random environments

Abstract

For a Brownian directed polymer in a Gaussian random environment, with $q(t,\cdot )$ denoting the quenched endpoint density and \[ Q_n(t,x_1,\ldots,x_n)=\mathbf{E}[q(t,x_1)\ldots q(t,x_n)], \] we derive a hierarchical PDE system satisfied by $\{Q_n{\}}_{n\ge 1}$. We present two applications of the system: (i) we compute the generator of $\{{\mu }_t(dx)=q(t,x)dx{\}}_{t\ge 0}$ for some special functionals, where $\{{\mu }_t(dx){\}}_{t\ge 0}$ is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with $d\ge 3$, 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.

arXiv comment28 pages; v3, final version

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Cited by in corpus (2)

• High temperature behaviors of the directed polymer on a cylinder
• Long-time behavior for a nonlocal model from directed polymers