KPZ equation, its renormalization and invariant measures
KPZ equation, its renormalization and invariant measures
复制标题
DOI：
10.1007/s40072-015-0046-x
复制
发表时间：
2015-04
期刊：
影响因子：
--
通讯作者：
T. Funaki;J. Quastel
中科院分区：
文献类型：
--
作者：
T. Funaki;J. Quastel
Kardar-Parisi-Zhang(KPZ)方程是一个随机偏微分方程，由于强迫噪声的非线性性和粗糙性之间的不一致性，该方程是不适定的。然而，它的Cole-Hopf解，定义为带有乘性噪声的线性随机热方程(SHE)的解的对数，是一个数学上定义良好的对象。事实上，海尔(Ann Math 178：559-694,2013)最近证明了，在周期边界条件下，通过适当的重整化，KPZ方程的解实际上可以通过Cole-Hopf变换得到。不幸的是，这种变换并不能很好地适应于研究这些马尔可夫过程的不变度量。本文给出了周期边界条件下全直线上KPZ方程的一种不同类型的正则化，这种正则化从研究不变测度的角度来看是适当的。应用于该方程的Cole-Hopf变换导致具有拖尾噪声的SHE具有额外复杂的非线性项。在时间平均和定常情况下，证明了这一项可以用一个简单的线性项来代替，从而使极限方程是一个带系数的额外线性项的线性方程。这些方法本质上是随机分析的：使用维纳-伊特展开和建立玻尔兹曼-吉布斯原理的类似方法。结果表明，由勒贝格测度给出的具有高度位移的双边几何布朗运动的分布在由她决定的演化下是不变的。
The Kardar–Parisi–Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because of the inconsistency between the nonlinearity and the roughness of the forcing noise. However, its Cole–Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer (Ann Math 178:559–694, 2013) has recently proved that the solution of SHE can actually be derived through the Cole–Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic boundary conditions. This transformation is unfortunately not well adapted to studying the invariant measures of these Markov processes. The present paper introduces a different type of regularization for the KPZ equation on the whole lineor under periodic boundary conditions, which is appropriate from the viewpoint of studying the invariant measures. The Cole–Hopf transform applied to this equation leads to an SHE with a smeared noise having an extra complicated nonlinear term. Under time average and in the stationary regime, it is shown that this term can be replaced by a simple linear term, so that the limit equation is the linear SHE with an extra linear term with coefficient. The methods are essentially stochastic analytic: The Wiener–Itô expansion and a similar method for establishing the Boltzmann–Gibbs principle are used. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE on.