The stationary AKPZ equation: Logarithmic superdiffusivity

Abstract We study the two‐dimensional Anisotropic KPZ equation (AKPZ) formally given by where ξ is a space‐time white noise and λ strictly positive constant. While classical equation, whose nonlinearity , can be linearised via Cole‐Hopf transformation, this not case for AKPZ. prove that stationary solution to AKPZ (whose invariant measure Gaussian Free Field (GFF)) superdiffusive: its diffusion coefficient diverges large times as up corrections, in Tauberian sense. Morally, says correlation length grows with time like . Moreover, we show if process rescaled diffusively (), then it evolves non‐trivially already on time‐scales of order approximately Both claims hold soon non‐zero. These results are contrast belief, common mathematics community, diffusive at scales and, under simple scaling, converges Stochastic Heat Equation (2dSHE) additive (i.e., ).