Hydrodynamics of a $d$-dimensional long jumps symmetric exclusion with a slow barrier
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913382865567744 |
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| author | Cardoso, Pedro Gonçalves, Patrícia Jiménez-Oviedo, Byron |
| author_facet | Cardoso, Pedro Gonçalves, Patrícia Jiménez-Oviedo, Byron |
| contents | We obtain the hydrodynamic limit of symmetric long-jumps exclusion in $\mathbb{Z}^d$ (for $d \geq 1$), where the jump rate is inversely proportional to a power of the jump's length with exponent $γ+1$, where $γ\geq 2$. Moreover, movements between $\mathbb{Z}^{d-1} \times \mathbb{Z}_{-}^{*}$ and $\mathbb{Z}^{d-1} \times \mathbb N$ are slowed down by a factor $αn^{-β}$ (with $α>0$ and $β\geq 0$). In the hydrodynamic limit we obtain the heat equation in $\mathbb{R}^d$ without boundary conditions or with Neumann boundary conditions, depending on the values of $β$ and $γ$. The (rather restrictive) condition in \cite{casodif} (for $d=1$) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01152 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hydrodynamics of a $d$-dimensional long jumps symmetric exclusion with a slow barrier Cardoso, Pedro Gonçalves, Patrícia Jiménez-Oviedo, Byron Probability Analysis of PDEs We obtain the hydrodynamic limit of symmetric long-jumps exclusion in $\mathbb{Z}^d$ (for $d \geq 1$), where the jump rate is inversely proportional to a power of the jump's length with exponent $γ+1$, where $γ\geq 2$. Moreover, movements between $\mathbb{Z}^{d-1} \times \mathbb{Z}_{-}^{*}$ and $\mathbb{Z}^{d-1} \times \mathbb N$ are slowed down by a factor $αn^{-β}$ (with $α>0$ and $β\geq 0$). In the hydrodynamic limit we obtain the heat equation in $\mathbb{R}^d$ without boundary conditions or with Neumann boundary conditions, depending on the values of $β$ and $γ$. The (rather restrictive) condition in \cite{casodif} (for $d=1$) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped. |
| title | Hydrodynamics of a $d$-dimensional long jumps symmetric exclusion with a slow barrier |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2304.01152 |