Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon
Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon
复制标题
无限视界平面洛伦兹过程的极限定律和递推
DOI：
10.1007/s10955-007-9367-0
复制
发表时间：
2007
影响因子：
1.6
通讯作者：
Tamás Varjú
中科院分区：
文献类型：
--
作者：
D. Szász;Tamás Varjú
摘要
正如 Bleher (J. Stat. Phys. 66(1):315–373, 1992) 观察到的，平面、无限地平线、周期性洛伦兹过程的自由飞行矢量 {Sn∣n=0,1,2,…} 属于高斯定律的非标准吸引力域，实际上与
$\sqrt{n\log n}$
缩放。我们的首要目标是证实他的猜想，事实上，
$\frac{S_{n}}{\sqrt{n\log n}}$
分布收敛于高斯定律（全局极限定理）。 Bálint 和 Gouëzel 最近的方法（Commun. Math. Phys. 263:461–512, 2006）帮助我们从本质上简化了早期粗略证明的思想（Szász, D., Varjú, T. in Modern Dynamical Systems and applications, pp. 433–445, 2004）。此外，我们还可以推导出 (a) 全局极限定理的局部版本，(b) 平面、无限视界、周期性洛伦兹过程的递推，最后 (c) 其无限不变测度的遍历性。
Abstract
As Bleher (J. Stat. Phys. 66(1):315–373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {Sn∣n=0,1,2,…} belongs to the non-standard domain of attraction of the Gaussian law—actually with the
$\sqrt{n\log n}$
scaling. Our first aim is to establish his conjecture that, indeed,
$\frac{S_{n}}{\sqrt{n\log n}}$
converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of Bálint and Gouëzel (Commun. Math. Phys. 263:461–512, 2006), helped us to essentially simplify the ideas of our earlier sketchy proof (Szász, D., Varjú, T. in Modern dynamical systems and applications, pp. 433–445, 2004). Moreover, we can also derive (a) the local version of the Global Limit Theorem, (b) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally (c) the ergodicity of its infinite invariant measure.