论文标题
带有常规初始数据的KPZ方程的时间增量
Temporal increments of the KPZ equation with general initial data
论文作者
论文摘要
我们考虑（1+1）-Dimensional kpz方程$ \ MATHCAL {H}^f（t，X）$的Cole-Hopf解决方案，始于初始数据$ f $。在本文中，我们研究了kpz时间过程的示例路径属性$ \ mathcal {h} _t^f：= \ Mathcal {h}^f（t，0）$。我们表明，对于非常大的初始数据，包括任何确定的连续初始数据，其增长速度比抛物线较慢（$ f（x）\ ll 1 + x^2 $），以及狭窄的楔形和平稳的初始数据，$ \ nathcal {h} _t^f $的时间增量的时间为$ brownian $ brownian $ hurs $ hurs $ hurs $ hurs $ hurs $ hurs $ hurs的时间均匀近似。结果，我们获得了KPZ时间属性的几个样本路径特性，包括时间过程的变化，迭代对数定律，连续性模量和一组点的Hausdorff尺寸，并具有异常大的增量。
We consider the Cole-Hopf solution of the (1+1)-dimensional KPZ equation $\mathcal{H}^f(t,x)$ started with initial data $f$. In this article, we study the sample path properties of the KPZ temporal process $\mathcal{H}_t^f := \mathcal{H}^f(t,0)$. We show that for a very large class of initial data which includes any deterministic continuous initial data that grows slower than parabola ($f(x) \ll 1 + x^2$) as well as narrow wedge and stationary initial data, temporal increments of $\mathcal{H}_t^f$ are well approximated by increments of a fractional Brownian motion of Hurst parameter $\frac14$. As a consequence, we obtain several sample path properties of KPZ temporal properties including variation of the temporal process, law of iterated logarithms, modulus of continuity and Hausdorff dimensions of the set of points with exceptionally large increments.
