广义分数Poisson过程及其相关随机动力学
Generalized fractional Poisson process and related stochastic dynamics
摘要：
本文综述了广义分数泊松过程（GFPP）。GFPP是一个更新过程，推广了Laskin的分数泊松计数过程，最早由Cahoy和Polito提出。GFPP包含具有容许范围$0<β\leq 1$、$α>0$的两个索引参数和表征时间尺度的参数。GFPP涉及Prabhakar广义Mittag-Leffler函数，并包含Laskin分数泊松过程，Erlang过程和标准泊松过程的参数的特殊选择。我们证明了这一点，通过明确的公式。我们开发的Montroll-Weiss连续时间随机游走（CTRW）的GFPP无向网络上的Prabhakar分布的等待时间之间的跳跃的步行者。对于这次行走，我们推导出一个广义分数Kolmogorov-Feller方程，其中涉及控制网络随机运动的Prabhakar广义分数算子。我们分析在$d$维度的“尺度良好”的扩散限制，并得到一个分数扩散方程，这是相同类型的步行与Mittag-Leffler分布的等待时间。GFPP具有捕获某些复杂系统动力学的各个方面的潜力。
英文摘要：
We survey the 'generalized fractional Poisson process' (GFPP). The GFPP is a renewal process generalizing Laskin's fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges $0<β\leq 1$, $α>0$ and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in $d$ dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.
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