Weak coupling limit of the anisotropic KPZ equation

DOI10.1215/00127094-2022-0094arXiv2108.09046MaRDI QIDQ6183961

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Publication date: 5 January 2024

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Abstract: In the present work, we study the two-dimensional anisotropic KPZ equation (AKPZ), which is formally given by �egin{equation*} partial_t h= frac12 Delta h + lambda ((partial_1 h)^2)-(partial_2 h)^2) +xi,, 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 $Noinfty$. Here, we study the emph{weak coupling limit} where $lambda=lambd{a}_N=hatlambda/sqrtlogN$: this is the correct scaling that guarantees that the nonlinearity has a still non-trivial but non-divergent effect. In fact, as $Noinfty$ the sequence of equations converges to the linear stochastic heat equation �egin{equation*} partial_t h = frac{

u_{ m eff}}{2} Delta h + sqrt{

u_{ m eff}}xi,, end{equation*} where $u_{meff}>1$ is explicit and depends non-trivially on $hatlambda$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation.

Full work available at URL: https://arxiv.org/abs/2108.09046

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