Central limit theorem for a class of one-dimensional kinetic equations
Central limit theorem for a class of one-dimensional kinetic equations
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DOI：
10.1007/s00440-010-0269-8
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发表时间：
2011-06-01
影响因子：
2
通讯作者：
Matthes, Daniel
中科院分区：
文献类型：
--
作者：
Bassetti, Federico;Ladelli, Lucia;Matthes, Daniel
我们引入了一类关于真实的直线的动力学型方程，它是经典Kac漫画的推广。碰撞增益算子由具有一般性质的光滑变换定义。通过建立与中心极限问题的联系，我们能够证明方程的解朝着极限分布的长时间收敛。例如，我们证明了如果初始条件属于某个稳定律nu（alpha）的正常吸引域，则极限是nu（alpha）的尺度混合。在一些额外的假设下，计算了Wasserstein度量收敛到平衡点的显式指数速度，并证明了概率密度的强收敛性。
We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law nu (alpha), then the limit is a scale mixture of nu (alpha). Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.