Singularity and regularity of the critical 2D Stochastic Heat Flow
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916844359647232 |
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| author | Caravenna, Francesco Sun, Rongfeng Zygouras, Nikos |
| author_facet | Caravenna, Francesco Sun, Rongfeng Zygouras, Nikos |
| contents | The Critical 2D Stochastic Heat Flow (SHF) provides a natural candidate solution to the ill-posed 2D Stochastic Heat Equation with multiplicative space-time white noise. In this paper, we initiate the investigation of the spatial properties of the SHF. We prove that, as a random measure on $\mathbb{R}^2$, it is a.s. singular w.r.t. the Lebesgue measure. This is obtained by probing a "quasi-critical" regime and showing the asymptotic log-normality of the mass assigned to vanishing balls, as the disorder strength is sent to zero at a suitable rate, accompanied by similar results for critical 2D directed polymers. We also describe the regularity of the SHF, showing that it is a.s. Hölder $C^{-ε}$ for any $ε>0$, implying the absence of atoms, and we establish local convergence to zero in the long time limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06128 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularity and regularity of the critical 2D Stochastic Heat Flow Caravenna, Francesco Sun, Rongfeng Zygouras, Nikos Probability Mathematical Physics Primary: 82B44, Secondary: 35R60, 60H15, 82D60 The Critical 2D Stochastic Heat Flow (SHF) provides a natural candidate solution to the ill-posed 2D Stochastic Heat Equation with multiplicative space-time white noise. In this paper, we initiate the investigation of the spatial properties of the SHF. We prove that, as a random measure on $\mathbb{R}^2$, it is a.s. singular w.r.t. the Lebesgue measure. This is obtained by probing a "quasi-critical" regime and showing the asymptotic log-normality of the mass assigned to vanishing balls, as the disorder strength is sent to zero at a suitable rate, accompanied by similar results for critical 2D directed polymers. We also describe the regularity of the SHF, showing that it is a.s. Hölder $C^{-ε}$ for any $ε>0$, implying the absence of atoms, and we establish local convergence to zero in the long time limit. |
| title | Singularity and regularity of the critical 2D Stochastic Heat Flow |
| topic | Probability Mathematical Physics Primary: 82B44, Secondary: 35R60, 60H15, 82D60 |
| url | https://arxiv.org/abs/2504.06128 |