标题： 连续时间随机游走、Mittag-Leffler等待时间与分数扩散：数学方面 显示英文标题

标题： Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects

摘要： 我们展示了通用幂律等待时间分布与时间分数阶CTRW特有的Mittag-Leffler等待时间分布之间的渐近长时间等价性。这种渐近等价性是通过结合“时间重标定”和“相关更新过程的加速”，然后进行极限过程来实现的，为此我们需要重标定和加速参数之间合适的对应关系。 将注意力转向具有通用幂律跳跃分布的一维空间CTRWs，“空间重标定”可以被解释为另一种“加速”，然后在相关参数之间适当的关系下，最终在极限情况下导致时空分数阶扩散方程。 最后，我们将“时间分数阶漂移”过程视为Mittag-Leffler更新过程计数的适当缩放极限。 显示英文摘要

摘要： We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.