A critical stochastic heat equation with long-range noise
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916974707081216 |
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| author | Dunlap, Alexander Hairer, Martin Li, Xue-Mei |
| author_facet | Dunlap, Alexander Hairer, Martin Li, Xue-Mei |
| contents | We consider a semilinear stochastic heat equation in spatial dimension at least $3$, forced by a noise that is white in time with a covariance kernel that decays like $\lvert x\rvert^{-2}$ as $\lvert x\rvert\to\infty$. We show that in an appropriate diffusive scaling limit with a logarithmic attenuation of the noise, the pointwise statistics of the solution can be approximated by the solution to a forward-backward stochastic differential equation (FBSDE). The scaling and structure of the problem is similar to that of the two-dimensional stochastic heat equation forced by an approximation of space-time white noise considered by the first author and Gu (Ann. Probab., 2022). However the resulting FBSDE is different due to the long-range correlations of the noise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_23790 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A critical stochastic heat equation with long-range noise Dunlap, Alexander Hairer, Martin Li, Xue-Mei Probability Analysis of PDEs We consider a semilinear stochastic heat equation in spatial dimension at least $3$, forced by a noise that is white in time with a covariance kernel that decays like $\lvert x\rvert^{-2}$ as $\lvert x\rvert\to\infty$. We show that in an appropriate diffusive scaling limit with a logarithmic attenuation of the noise, the pointwise statistics of the solution can be approximated by the solution to a forward-backward stochastic differential equation (FBSDE). The scaling and structure of the problem is similar to that of the two-dimensional stochastic heat equation forced by an approximation of space-time white noise considered by the first author and Gu (Ann. Probab., 2022). However the resulting FBSDE is different due to the long-range correlations of the noise. |
| title | A critical stochastic heat equation with long-range noise |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2509.23790 |