The stationary AKPZ equation: logarithmic superdiffusivity

Abstract

We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given by \begin{equation*} \partial_t H=\frac12\Delta H+\lambda((\partial_1 H)^2-(\partial_2 H)^2)+\xi\,, \end{equation*} where $\xi$ is a space-time white noise and $\lambda$ 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 $\lambda$ 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 $\lambda=0$).