Weak coupling limit of the Anisotropic KPZ equation

Abstract

In the present work, we study the two-dimensional anisotropic KPZ equation (AKPZ), which is formally given by \begin{equation*} \partial_t h=\tfrac12 Δh + λ((\partial_1 h)^2)-(\partial_2 h)^2) +ξ\,, \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{ν_{\rm eff}}{2} Δh + \sqrt{ν_{\rm eff}}ξ\,, \end{equation*} where ${\nu }_{\mathrm{eff}}>1$ 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.

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Cited by in corpus (2)

• The stationary AKPZ equation: logarithmic superdiffusivity
• Scaling Limits of Stochastic Transport Equations on Manifolds