Title: Weak coupling limit of the Anisotropic KPZ equation Show Chinese title

Title: 各向异性KPZ方程的弱耦合极限

Abstract: In the present work, we study the two-dimensional anisotropic KPZ equation (AKPZ), which is formally given by \begin{equation*} \partial_t h=\tfrac12 \Delta h + \lambda ((\partial_1 h)^2)-(\partial_2 h)^2) +\xi\,, \end{equation*} where $\xi$ denotes a space-time white noise and $\lambda>0$ is the so-called coupling constant. The AKPZ equation is a {\it critical} SPDE, meaning that not only it is analytically ill-posed but also the breakthrough path-wise techniques for singular SPDEs [M. Hairer, Ann. Math. 2014] and [M. Gubinelli, P. Imkeller and N. Perkowski, Forum of Math., Pi, 2015] are not applicable. As shown in [G. Cannizzaro, D. Erhard, F. Toninelli, arXiv, 2020], the equation regularised at scale $N$ has a diffusion coefficient that diverges logarithmically as the regularisation is removed in the limit $N\to\infty$. Here, we study the \emph{weak coupling limit} where $\lambda=\lambda_N=\hat\lambda/\sqrt{\log N}$: this is the correct scaling that guarantees that the nonlinearity has a still non-trivial but non-divergent effect. In fact, as $N\to\infty$ the sequence of equations converges to the linear stochastic heat equation \begin{equation*} \partial_t h =\tfrac{\nu_{\rm eff}}{2} \Delta h + \sqrt{\nu_{\rm eff}}\xi\,, \end{equation*} where $\nu_{\rm eff} >1$ is explicit and depends non-trivially on $\hat\lambda$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation. Show Chinese abstract

Abstract: 在本工作中，我们研究了二维各向异性KPZ方程（AKPZ），其形式上由\begin{equation*} \partial_t h=\tfrac12 \Delta h + \lambda ((\partial_1 h)^2)-(\partial_2 h)^2) +\xi\,, \end{equation*}给出，其中$\xi$表示时空白噪声，$\lambda>0$是所谓的耦合常数。AKPZ方程是一个{\it 临界}SPDE，这意味着它不仅在分析上是不适定的，而且对于奇异SPDE的突破性路径技巧 [M. Hairer, Ann. Math. 2014] 和 [M. Gubinelli, P. Imkeller and N. Perkowski, Forum of Math., Pi, 2015] 也不适用。如 [G. Cannizzaro, D. Erhard, F. Toninelli, arXiv, 2020] 所示，以尺度$N$正则化的方程具有一个在正则化被移除时对数发散的扩散系数，极限为$N\to\infty$。 在这里，我们研究 \emph{弱耦合极限} 其中 $\lambda=\lambda_N=\hat\lambda/\sqrt{\log N}$：这是正确的缩放，保证非线性仍然有非平凡但不发散的效果。 事实上，当 $N\to\infty$ 时，方程序列收敛到线性随机热方程 \begin{equation*} \partial_t h =\tfrac{\nu_{\rm eff}}{2} \Delta h + \sqrt{\nu_{\rm eff}}\xi\,, \end{equation*} 其中 $\nu_{\rm eff} >1$ 是显式的，并且非平凡地依赖于 $\hat\lambda$。 这是对于一个临界、奇异的SPDE的第一个完整的正规化类型结果，该SPDE不能通过Cole-Hopf或其他变换进行线性化。