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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.06260 |
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| _version_ | 1866908524516212736 |
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| author | Castillo, Bryan Dunlap, Alexander |
| author_facet | Castillo, Bryan Dunlap, Alexander |
| contents | We study a large class of scaling-critical reaction-diffusion equations in two spatial dimensions, where the initial data is white noise mollified at scale $\varepsilon^2$ and the reaction term is attenuated by a factor of $(\log\varepsilon^{-1})^{-1}$. We show that as $\varepsilon\to 0$, the solution converges to the solution of a McKean-Vlasov equation, which is Gaussian with standard deviation given by the solution to an ODE. Our result covers the case of the reaction term $f(u)=u^3$, and thus gives a new proof of the limiting behavior for the Allen-Cahn equation discovered in the recent work of Gabriel, Rosati, and Zygouras (Probab. Theory Related Fields 192: 1373-1446, 2025). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data Castillo, Bryan Dunlap, Alexander Probability Analysis of PDEs We study a large class of scaling-critical reaction-diffusion equations in two spatial dimensions, where the initial data is white noise mollified at scale $\varepsilon^2$ and the reaction term is attenuated by a factor of $(\log\varepsilon^{-1})^{-1}$. We show that as $\varepsilon\to 0$, the solution converges to the solution of a McKean-Vlasov equation, which is Gaussian with standard deviation given by the solution to an ODE. Our result covers the case of the reaction term $f(u)=u^3$, and thus gives a new proof of the limiting behavior for the Allen-Cahn equation discovered in the recent work of Gabriel, Rosati, and Zygouras (Probab. Theory Related Fields 192: 1373-1446, 2025). |
| title | McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2509.06260 |