We consider a particle moving in one dimension, its velocity being a reversible diffusion process, with constant diffusion coefficient, of which the invariant measure behaves like $$(1+|v|)^{-\beta }$$ ( 1 + | v | ) - β for some $$\beta >0$$ β > 0 . We prove that, under a suitable rescaling, the position process resembles a Brownian motion if $$\beta \ge 5$$ β ≥ 5 , a stable process if $$\beta \in [1,5)$$ β ∈ [ 1 , 5 ) and an integrated symmetric Bessel process if $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) . The critical cases $$\beta =1$$ β = 1 and $$\beta =5$$ β = 5 require special rescalings. We recover some results of Nasreddine and Puel (ESAIM Math Model Numer Anal 49:1–17, 2015), Cattiaux et al. (Kinet Relat Models, to appear), Lebeau and Puel (Commun Math Phys, to appear. arXiv:1711.03060 ) and Barkai et al. (Phys Rev X 4:021036, 2014) on the kinetic Fokker–Planck equation, with an alternative approach.

中文翻译：

---

一维临界动力学 Fokker-Planck 方程、贝塞尔和稳定过程

我们考虑一个在一维运动的粒子，它的速度是一个可逆的扩散过程，具有恒定的扩散系数，其不变测度表现为 $$(1+|v|)^{-\beta }$$ ( 1 + | v | ) - β 对于某些 $$\beta >0$$ β > 0 。我们证明，在适当的重新标度下，如果 $$\beta \ge 5$$ β ≥ 5 ，则位置过程类似于布朗运动，如果 $$\beta \in [1,5)$$ β ∈ [ 1 , 5) 和一个积分对称 Bessel 过程，如果 $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) 。临界情况 $$\beta =1$$ β = 1 和 $$\beta =5$$ β = 5 需要特殊的重新缩放。我们恢复了 Nasreddine 和 Puel (ESAIM Math Model Numer Anal 49:1–17, 2015)、Cattiaux 等人的一些结果。（Kinet Relat 模型，出现）、Lebeau 和 Puel（Commun Math Phys，出现。arXiv：1711.03060）和 Barkai 等人。(Phys Rev X 4:021036,