Weak coupling limit of the Anisotropic KPZ equation
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| Formato: | Preprint |
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2021
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| author | Cannizzaro, Giuseppe Erhard, Dirk Toninelli, Fabio |
| author_facet | Cannizzaro, Giuseppe Erhard, Dirk Toninelli, Fabio |
| contents | 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 $ξ$ denotes a space-time white noise and $λ>0$ 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 $λ=λ_N=\hatλ/\sqrt{\log N}$: 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 $ν_{\rm eff} >1$ is explicit and depends non-trivially on $\hatλ$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_09046 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Weak coupling limit of the Anisotropic KPZ equation Cannizzaro, Giuseppe Erhard, Dirk Toninelli, Fabio Probability Analysis of PDEs 60H17, 82C27 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 $ξ$ denotes a space-time white noise and $λ>0$ 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 $λ=λ_N=\hatλ/\sqrt{\log N}$: 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 $ν_{\rm eff} >1$ is explicit and depends non-trivially on $\hatλ$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation. |
| title | Weak coupling limit of the Anisotropic KPZ equation |
| topic | Probability Analysis of PDEs 60H17, 82C27 |
| url | https://arxiv.org/abs/2108.09046 |