论文标题
半空间固定的kardar-parisi-zhang方程之外
Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
论文作者
论文摘要
我们在半线$ x \ geqslant 0 $上研究Kardar-Parisi-Zhang方程，并使用Neumann型边界条件。在最近的工作中表征了KPZ动力学的固定度量：它们取决于两个参数，即动力学的边界参数$ u $，以及无穷大初始条件的漂移$ -V $。当这些固定过程之一给出初始条件时，我们考虑高度场的波动。在很大程度上$ t $，自然地将参数重新分组为$（u，v）= t^{ - 1/3}（a，b）$以研究关键区域。在特殊情况下，$ a+b = 0 $，在以前的作品中处理过，固定过程只是布朗尼人。但是，这些布朗式固定措施在边界阶段特别相关（$ a <0 $），但在未绑定阶段却不相关。例如，从平坦或液滴初始数据开始，边界附近的高度字段以$ a> 0 $和$ b = 0 $收敛到固定过程，这不是棕色。对于$ a+b \ geqslant 0 $，我们准确地确定了高度函数$ h（0，t）$的大时间分布$ f_ {a，b}^{\ rm stat} $。作为一个应用程序，我们以半行的确切协方差为半行，两次$ 1 \ ll t_1 \ ll t_2 $从固定初始数据开始，以及从液滴初始数据开始，以限制$ t_1/t_2 \至1 $。
We study the Kardar-Parisi-Zhang equation on the half-line $x \geqslant 0$ with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter $u$ of the dynamics, and the drift $-v$ of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time $t$, it is natural to rescale parameters as $(u,v)=t^{-1/3}(a,b)$ to study the critical region. In the special case $a+b=0$, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ($a<0$) but not in the unbound phase. For instance, starting from the flat or droplet initial data, the height field near the boundary converges to the stationary process with $a>0$ and $b=0$, which is not Brownian. For $a+b\geqslant 0$, we determine exactly the large time distribution $F_{a,b}^{\rm stat}$ of the height function $h(0,t)$. As an application, we obtain the exact covariance of the height field in a half-line at two times $1\ll t_1\ll t_2$ starting from stationary initial data, as well as estimates, when starting from droplet initial data, in the limit $t_1/t_2\to 1$.
