KPZ line ensemble
KPZ line ensemble
复制标题
DOI：
10.1007/s00440-015-0651-7
复制
发表时间：
2013-12
影响因子：
2
通讯作者：
Ivan Corwin;A. Hammond
中科院分区：
文献类型：
--
作者：
Ivan Corwin;A. Hammond
对于每个，我们构造具有三个属性的随机连续曲线的指数系综：（1）最低指数曲线分布为具有窄楔形初始数据的Kardar-Parisi-Zhang（KPZ）随机偏微分方程的时间tHopf-Cole解；（2）整个系综满足我们称为布朗吉布斯属性的重采样不变性[与]；（3）最低指数曲线的增量，当以 为中心并垂直缩小和水平缩小时，随着时间趋向无穷大，相对于布朗桥保持均匀绝对连续（即具有紧密的 Radon-Nikodym 导数）。这种结构使用 O’Connell 发现的扩散作为输入（Ann Probab 40:437–458, 2012），与 O’Connell-Yor 半离散布朗聚合物有关，这是 Moreno Flores 等人的收敛结果。 （准备中）该扩散的最低指数曲线到具有窄楔形初始数据的 KPZ 方程的解，以及 Amir 等人证明的单点分布公式。 (Commun Pure Appl Math 64:466–537, 2011) 使用窄楔初始数据求解 KPZ 方程。我们提供了这种结构的四个主要应用：（1）具有窄楔形初始数据的 KPZ 方程的时间解的均匀（趋向无穷大）布朗绝对连续性，即使在垂直和水平缩放时也是如此；（2）具有一般初始数据的 KPZ 方程解的单点（垂直）涨落尺度的通用性；（3）连续介质定向随机聚合物端点的尺度浓度；（4）指数上限和指数上限具有一般初始数据的 KPZ 方程在固定时间的解的下尾界。
For eachwe construct an-indexed ensemble of random continuous curves with three properties:(1)the lowest indexed curve is distributed as the timetHopf–Cole solution to the Kardar–Parisi–Zhang (KPZ) stochastic partial differential equation with narrow wedge initial data;(2)the entire ensemble satisfies a resampling invariance which we call the-Brownian Gibbs property[with];(3)increments of the lowest indexed curve, when centered byand scaled down vertically byand horizontally by, remain uniformly absolutely continuous (i.e. have tight Radon–Nikodym derivatives) with respect to Brownian bridges as timetgoes to infinity.This construction uses as inputs the diffusion that O’Connell discovered (Ann Probab 40:437–458, 2012) in relation to the O’Connell–Yor semi-discrete Brownian polymer, the convergence result of Moreno Flores et al. (in preparation) of the lowest indexed curve of that diffusion to the solution of the KPZ equation with narrow wedge initial data, and the one-point distribution formula proved by Amir et al. (Commun Pure Appl Math 64:466–537, 2011) for the solution of the KPZ equation with narrow wedge initial data. We provide four main applications of this construction:(1)uniform (astgoes to infinity) Brownian absolute continuity of the timetsolution to the KPZ equation with narrow wedge initial data, even when scaled vertically byand horizontally by;(2)universality of theone-point (vertical) fluctuation scale for the solution of the KPZ equation with general initial data;(3)concentration in thescale for the endpoint of the continuum directed random polymer;(4)exponential upper and lower tail bounds for the solution at fixed time of the KPZ equation with general initial data.