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.
Categories: math.PR math.AP