Abstract 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 ≤ 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.

中文翻译：

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广义分数泊松过程和相关的随机动力学

摘要 我们调查了“广义分数泊松过程”（GFPP）。GFPP 是一个更新过程，它概括了 Laskin 的分数泊松计数过程，并首先由 Cahoy 和 Polito 引入。GFPP 包含两个指标参数，其允许范围为 0 < β ≤ 1，α > 0 和一个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。我们通过显式公式来证明这一点。我们为无向网络上的 GFPP 开发了 Montroll-Weiss 连续时间随机游走 (CTRW)，该网络在步行者的跳跃之间具有 Prabhakar 分布的等待时间。对于这次步行，我们推导出一个广义分数阶 Kolmogorov-Feller 方程，它涉及控制网络上随机运动的 Prabhakar 广义分数阶算子。我们在 d 维中分析了“良好缩放”的扩散极限，并获得了一个分数扩散方程，该方程与具有 Mittag-Leffler 分布式等待时间的步行具有相同的类型。GFPP 有可能捕捉某些复杂系统动态的各个方面。