Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866918324822081536 |
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| author | Dunlap, Alexander Graham, Cole |
| author_facet | Dunlap, Alexander Graham, Cole |
| contents | We consider a two-dimensional stochastic heat equation with noise correlated at scale $ρ\ll 1$ and of strength $|\logρ|^{-1/2}σ(v)$ depending nonlinearly on the solution $v$. Under certain conditions, the first author and Gu have shown that the one-point statistics of $v$ converge in law as $ρ\to 0$ to the terminal value of an associated forward-backward SDE. Here, we show that the 2D stochastic heat equation is stable under renormalization with a new effective nonlinearity tied to the decoupling function of the forward-backward SDE. This allows us to extend the pointwise results to a much broader class of nonlinearities. We also show that these limiting pointwise statistics are insensitive to the fine details of the noise, and thus universal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11850 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality Dunlap, Alexander Graham, Cole Probability Analysis of PDEs 60H15 (Primary), 35R60, 60H10 (Secondary) We consider a two-dimensional stochastic heat equation with noise correlated at scale $ρ\ll 1$ and of strength $|\logρ|^{-1/2}σ(v)$ depending nonlinearly on the solution $v$. Under certain conditions, the first author and Gu have shown that the one-point statistics of $v$ converge in law as $ρ\to 0$ to the terminal value of an associated forward-backward SDE. Here, we show that the 2D stochastic heat equation is stable under renormalization with a new effective nonlinearity tied to the decoupling function of the forward-backward SDE. This allows us to extend the pointwise results to a much broader class of nonlinearities. We also show that these limiting pointwise statistics are insensitive to the fine details of the noise, and thus universal. |
| title | Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality |
| topic | Probability Analysis of PDEs 60H15 (Primary), 35R60, 60H10 (Secondary) |
| url | https://arxiv.org/abs/2308.11850 |