In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d⩾1, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form (1+|v|2)−β2, in the range β∈]d,d+4[. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). 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:=∫Rdfdv, with a fractional Laplacian κ(−Δ)β−d+26 and a positive diffusion coefficient κ.

中文翻译：

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具有重尾平衡的 Fokker–Planck 方程的分数扩散：任意维度的 à la Koch 谱方法

在本文中，我们将开发的谱方法（Dechicha 和 Puel (2023)）扩展到任何维度 d⩾1，以便为具有重尾平衡的 Fokker–Planck 算子构建特征解，其形式为 (1+ |v|2)−β2，在 βε]d,d+4[ 范围内。在 1 维中开发的方法受到 H. Koch 关于非线性 KdV 方程的工作的启发 (Nonlinearity 28 (2015) 545)。本文的策略与维度 1 相同，但工具不同，因为维度 1 基于 ODE 方法。作为我们构建的直接结果，我们获得了动力学 Fokker-Planck 方程的分数扩散极限，对于正确的密度 ρ:=∫Rdfdv，具有分数拉普拉斯 κ(−Δ)β−d+26 和正扩散系数κ。