The stationary AKPZ equation : logarithmic superdiffusivity

We study the two‐dimensional Anisotropic KPZ equation (AKPZ) formally given by ∂ t H = 1 2 Δ H + λ ( ( ∂ 1 H ) 2 − ( ∂ 2 H ) 2 ) + ξ , $$\begin{equation*} \hspace*{3.4pc}\partial _t H=\frac{1}{2}\Delta H+\lambda ((\partial _1 H)^2-(\partial _2 H)^2)+\xi , \end{equation*}$$ 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 | ∇ H | 2 = ( ∂ 1 H ) 2 + ( ∂ 2 H ) 2 $|\nabla H|^2=(\partial _1 H)^2+(\partial _2 H)^2$ , 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 (GFF)) is superdiffusive: its diffusion coefficient diverges for large times as log t $\sqrt {\mathop {\mathrm{log}}\nolimits t}$ up to log log t $\mathop {\mathrm{log}}\nolimits \mathop {\mathrm{log}}\nolimits t$ corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like t 1 / 2 × ( log t ) 1 / 4 $t^{1/2}\times (\mathop {\mathrm{log}}\nolimits t)^{1/4}$ . Moreover, we show that if the process is rescaled diffusively ( t → t / ε 2 , x → x / ε , ε → 0 $t\rightarrow t/\varepsilon ^2, x\rightarrow x/\varepsilon , \varepsilon \rightarrow 0$ ), then it evolves non‐trivially already on time‐scales of order approximately 1 / | log ε | ≪ 1 $1/\sqrt {|\mathop {\mathrm{log}}\nolimits \varepsilon |}\ll 1$ . 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 to the two‐dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e., the case λ = 0 $\lambda =0$ )