A Critical Drift-Diffusion Equation: Connections to the Diffusion on $\textbf{SL}(2)$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909916123365376 |
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| author | Morfe, Peter Otto, Felix Wagner, Christian |
| author_facet | Morfe, Peter Otto, Felix Wagner, Christian |
| contents | In this note, we connect two seemingly unrelated objects: On the one hand is a two-dimensional drift-diffusion process $X$ with divergence-free and time-independent drift $b$. The drift is given by a stationary Gaussian ensemble, and we focus on the critical case where a small-scale cut-off is necessary for well-posedness and the large-scale cancellations lead to a borderline super-diffusive behavior. On the other hand is the natural diffusion $F$ on the Lie group $\textbf{SL}(2)$ of matrices of determinant one. As a consequence of this connection, the strongly non-Gaussian character of $F$ transmits to how $X$ depends on its starting point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15983 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Critical Drift-Diffusion Equation: Connections to the Diffusion on $\textbf{SL}(2)$ Morfe, Peter Otto, Felix Wagner, Christian Probability Analysis of PDEs In this note, we connect two seemingly unrelated objects: On the one hand is a two-dimensional drift-diffusion process $X$ with divergence-free and time-independent drift $b$. The drift is given by a stationary Gaussian ensemble, and we focus on the critical case where a small-scale cut-off is necessary for well-posedness and the large-scale cancellations lead to a borderline super-diffusive behavior. On the other hand is the natural diffusion $F$ on the Lie group $\textbf{SL}(2)$ of matrices of determinant one. As a consequence of this connection, the strongly non-Gaussian character of $F$ transmits to how $X$ depends on its starting point. |
| title | A Critical Drift-Diffusion Equation: Connections to the Diffusion on $\textbf{SL}(2)$ |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2410.15983 |