A PDE hierarchy for directed polymers in random environments
Abstract
For a Brownian directed polymer in a Gaussian random environment, with q(t, ṡ) denoting the quenched endpoint density and ${Q}_{n}(t,{x}_{1},\dots ,{x}_{n})=\mathbf{E}[q(t,{x}_{1})\dots q(t,{x}_{n})],$ we derive a hierarchical PDE system satisfied by ${\left\{{Q}_{n}\right\}}_{n{\geqslant}1}$. We present two applications of the system: (i) we compute the generator of ${\left\{{\mu }_{t}(\mathrm{d}x)=q(t,x)\mathrm{d}x\right\}}_{t{\geqslant}0}$ for some special functionals, where ${\left\{{\mu }_{t}(\mathrm{d}x)\right\}}_{t{\geqslant}0}$ 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(t2/3) scaling.
- Publication:
-
Nonlinearity
- Pub Date:
- October 2021
- DOI:
- 10.1088/1361-6544/ac23b7
- arXiv:
- arXiv:2002.02799
- Bibcode:
- 2021Nonli..34.7335G
- Keywords:
-
- directed polymer;
- stochastic heat equation;
- reaction-diffusion equation;
- 60H15;
- 35K57;
- 82D60;
- Mathematics - Probability;
- Mathematical Physics;
- Mathematics - Analysis of PDEs
- E-Print:
- 28 pages