论文标题
固定的AKPZ方程：对数超富度
The stationary AKPZ equation: logarithmic superdiffusivity
论文作者
论文摘要
我们研究由\ begin {方程*}正式给出的二维各向异性KPZ方程（AKPZ） \ partial_t h = \frac12Δh+λ（（（\ partial_1 h）^2 - （\ partial_2 h）^2）^2）+ξ\ ,, \ end {equation*}其中$ξ$是时空白噪声，而$λ$是严格的正常数。尽管经典的二维KPZ方程（其非线性为$ | \ nabla h |^2 =（\ partial_1 h）^2+（\ partial_2 h）^2 $可以通过Cole-hopf变换来线性化，但AKPZ并非如此。我们证明，对AKPZ的固定解决方案（其不变度的度量是高斯自由场）是超延伸的：其扩散系数在很大的时期散开，$ \ sqrt {\ log t} $最高$ \ log \ log \ log \ log t $ t $校正，从tauberian的意义上讲。在道德上，这说明相关长度随时间增长，例如$ t^{1/2} \ times（\ log t）^{1/4} $。此外，我们表明，如果该过程被扩散地重新定制（$ t \ to t/\ varepsilon^2，x \ to x/\ x/\ varepsilon，\ varepsilon \ to0 $），那么它已经在订单上已经在订单上的时间表上已经大约$ 1/\ sqrt form pogles |一旦非线性的系数$λ$均为非零，这两项索赔立即持有。这些结果与数学社区中常见的信念相反，即Akpz方程在较大的尺度上是扩散的，在简单的扩散缩放下，将带有附加噪声的二维随机热方程（2DSHE）收敛（即情况$λ= 0 $）。
We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given by \begin{equation*} \partial_t H=\frac12ΔH+λ((\partial_1 H)^2-(\partial_2 H)^2)+ξ\,, \end{equation*} where $ξ$ is a space-time white noise and $λ$ is a strictly positive constant. While the classical two-dimensional KPZ equation, whose nonlinearity is $|\nabla H|^2=(\partial_1 H)^2+(\partial_2 H)^2$, can be linearised via the Cole-Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field) is superdiffusive: its diffusion coefficient diverges for large times as $\sqrt{\log t}$ up to $\log\log t$ corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like $t^{1/2}\times (\log t)^{1/4}$. Moreover, we show that if the process is rescaled diffusively ($t\to t/\varepsilon^2, x\to x/\varepsilon, \varepsilon\to0$), then it evolves non-trivially already on time-scales of order approximately $1/\sqrt{|\log\varepsilon|}\ll1$. Both claims hold as soon as the coefficient $λ$ of the nonlinearity is non-zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges the two-dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e. the case $λ=0$).
