The stationary AKPZ equation: logarithmic superdiffusivity

Abstract

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 $\xi$ 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 ($t\to t/{\epsilon }^2,x\to x/\epsilon ,\epsilon \to 0$), then it evolves non-trivially already on time-scales of order approximately $1/\sqrt{|\log \epsilon |}\ll 1$. 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 $\lambda =0$).

arXiv commentv3: Main result strengthened to $\sqrt{\log t}$ super-diffusivity

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Cited by in corpus (1)

• Weak coupling limit of the Anisotropic KPZ equation