论文标题
在随机环境中定向聚合物的PDE层次结构
A PDE hierarchy for directed polymers in random environments
论文作者
论文摘要
对于在高斯随机环境中定向的聚合物，$ q（t，\ cdot）$表示淬灭的端点密度，\ [q_n（t，x_1，\ ldots，x_n）= \ m athbf {e} [q（t，x_1）\ ldots q（t，x_1）\ ldots q（t，x__n） $ \ {q_n \} _ {n \ geq1} $。我们介绍了系统的两个应用：（i）我们计算$ \ {μ_t（dx）= q（t，x）dx \} _ {t \ geq0} $的生成器，用于某些特殊功能，其中$ \ {μ_t（dx）（dx）\} _ { （ii）在$ d \ geq 3 $的高温状态下，我们证明了对扩散重新缩放聚合物路径的退火端点分布的定量中心限制定理。我们还研究了由发电机动机的非本地扩散反应方程，并建立了超扩散$ O（t^{2/3}）$缩放。
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 $\{Q_n\}_{n\geq1}$. We present two applications of the system: (i) we compute the generator of $\{μ_t(dx)=q(t,x)dx\}_{t\geq0}$ for some special functionals, where $\{μ_t(dx)\}_{t\geq0}$ is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with $d\geq 3$, 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.
