A stochastic Burgers equation from a class of microscopic interactions
A stochastic Burgers equation from a class of microscopic interactions
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DOI：
10.1214/13-aop878
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发表时间：
2012-09
期刊：
影响因子：
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通讯作者：
P. Gonccalves;M. Jara;S. Sethuraman
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文献类型：
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作者：
P. Gonccalves;M. Jara;S. Sethuraman
我们考虑一类最近邻弱非对称质量守恒粒子系统在$\mathbb{Z}$上的演化，它包括零程和排斥过程的类型，从定态的扰动开始.当弱不对称性为O（n^{-\gamma}）$（1/2<\gamma\leq1 $）时，我们证明了涨落场的标度极限是一个广义的Ornstein-Uhlenbeck过程.然而，在临界弱不对称时，$\gamma=1/2$，我们表明，所有的极限点满足一个鞅公式，可以解释在一个随机Burgers方程来自采取KPZ方程的梯度。证明使用了一个尖锐的“玻尔兹曼-吉布斯”估计，提高了早期的界限。
We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on $\mathbb{Z}$, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order $O(n^{-\gamma})$ for $1/2<\gamma\leq1$, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein-Uhlenbeck process. However, at the critical weak asymmetry when $\gamma=1/2$, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp "Boltzmann-Gibbs" estimate which improves on earlier bounds.