The Allen-Cahn equation with weakly critical random initial datum
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915396754341888 |
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| author | Gabriel, Simon Rosati, Tommaso Zygouras, Nikos |
| author_facet | Gabriel, Simon Rosati, Tommaso Zygouras, Nikos |
| contents | This work considers the two-dimensional Allen-Cahn equation
$$ \partial_t u = \frac{1}{2}Δu + \mathfrak{m}\, u -u^3\;, \quad u(0,x)= η(x)\;, \qquad \forall (t,x) \in [0, \infty) \times \mathbb{R}^{2} \;, $$ where the initial condition $ η$ is a two-dimensional white noise, which lies in the scaling critical space of initial data to the equation. In a weak coupling scaling, we establish a Gaussian limit with nontrivial size of fluctuations, thus casting the nonlinearity as marginally relevant. The result builds on a precise analysis of the Wild expansion of the solution and an understanding of the underlying stochastic and combinatorial structure. This gives rise to a representation for the limiting variance in terms of Butcher series associated to the solution of an ordinary differential equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_04628 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Allen-Cahn equation with weakly critical random initial datum Gabriel, Simon Rosati, Tommaso Zygouras, Nikos Probability Analysis of PDEs This work considers the two-dimensional Allen-Cahn equation $$ \partial_t u = \frac{1}{2}Δu + \mathfrak{m}\, u -u^3\;, \quad u(0,x)= η(x)\;, \qquad \forall (t,x) \in [0, \infty) \times \mathbb{R}^{2} \;, $$ where the initial condition $ η$ is a two-dimensional white noise, which lies in the scaling critical space of initial data to the equation. In a weak coupling scaling, we establish a Gaussian limit with nontrivial size of fluctuations, thus casting the nonlinearity as marginally relevant. The result builds on a precise analysis of the Wild expansion of the solution and an understanding of the underlying stochastic and combinatorial structure. This gives rise to a representation for the limiting variance in terms of Butcher series associated to the solution of an ordinary differential equation. |
| title | The Allen-Cahn equation with weakly critical random initial datum |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2311.04628 |