KPZ-type equation from growth driven by a non-Markovian diffusion
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912483444260864 |
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| author | Dembo, Amir Yang, Kevin |
| author_facet | Dembo, Amir Yang, Kevin |
| contents | We study a stochastic PDE model for an evolving set $\mathbb{M}(t)\subseteq\mathbb{R}^{\mathrm{d}+1}$ that resembles a continuum version of origin-excited or reinforced random walk. We show that long-time fluctuations of an associated height function are given by a regularized Kardar-Parisi-Zhang (KPZ)-type PDE on a hypersurface in $\mathbb{R}^{\mathrm{d}+1}$, modulated by a Dirichlet-to-Neumann operator. We also show that for $\mathrm{d}+1=2$, the regularization in this KPZ-type equation can be removed after renormalization. To our knowledge, this gives the first instance of KPZ-type behavior in Laplacian growth, which was asked about (for somewhat different models) in Parisi-Zhang '84 and Ramirez-Sidoravicius '04. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_16095 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | KPZ-type equation from growth driven by a non-Markovian diffusion Dembo, Amir Yang, Kevin Probability 82C24, 60H15, 58J65, 35R60 We study a stochastic PDE model for an evolving set $\mathbb{M}(t)\subseteq\mathbb{R}^{\mathrm{d}+1}$ that resembles a continuum version of origin-excited or reinforced random walk. We show that long-time fluctuations of an associated height function are given by a regularized Kardar-Parisi-Zhang (KPZ)-type PDE on a hypersurface in $\mathbb{R}^{\mathrm{d}+1}$, modulated by a Dirichlet-to-Neumann operator. We also show that for $\mathrm{d}+1=2$, the regularization in this KPZ-type equation can be removed after renormalization. To our knowledge, this gives the first instance of KPZ-type behavior in Laplacian growth, which was asked about (for somewhat different models) in Parisi-Zhang '84 and Ramirez-Sidoravicius '04. |
| title | KPZ-type equation from growth driven by a non-Markovian diffusion |
| topic | Probability 82C24, 60H15, 58J65, 35R60 |
| url | https://arxiv.org/abs/2311.16095 |