Fractional diffusion for Fokker-Planck equation with heavy tail equilibrium: an à la Koch spectral method in any dimension
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929609017131008 |
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| author | Dechicha, Dahmane Puel, Marjolaine |
| author_facet | Dechicha, Dahmane Puel, Marjolaine |
| contents | In this paper, we extend the spectral method developed [Dechicha and Puel, 2023] to any dimension $d\geqslant 1$, in order to construct an eigen-solution for the Fokker-Planck operator with heavy tail equilibria, of the form $(1+|v|^2)^{-\fracβ{2}}$, in the range $β\in ]d,d+4[$. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation [Koch, Nonlinearity, 2015]. The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker-Planck equation, for the correct density $ρ:= \int_{\mathbb{R}^d} f \mathrm{d}v$, with a fractional Laplacian $κ(-Δ_x)^{\frac{β-d+2}{6}}$ and a positive diffusion coefficient $κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_07162 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fractional diffusion for Fokker-Planck equation with heavy tail equilibrium: an à la Koch spectral method in any dimension Dechicha, Dahmane Puel, Marjolaine Analysis of PDEs 35Q84 (35B27 35R11) In this paper, we extend the spectral method developed [Dechicha and Puel, 2023] to any dimension $d\geqslant 1$, in order to construct an eigen-solution for the Fokker-Planck operator with heavy tail equilibria, of the form $(1+|v|^2)^{-\fracβ{2}}$, in the range $β\in ]d,d+4[$. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation [Koch, Nonlinearity, 2015]. The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker-Planck equation, for the correct density $ρ:= \int_{\mathbb{R}^d} f \mathrm{d}v$, with a fractional Laplacian $κ(-Δ_x)^{\frac{β-d+2}{6}}$ and a positive diffusion coefficient $κ$. |
| title | Fractional diffusion for Fokker-Planck equation with heavy tail equilibrium: an à la Koch spectral method in any dimension |
| topic | Analysis of PDEs 35Q84 (35B27 35R11) |
| url | https://arxiv.org/abs/2303.07162 |