k 及以后秩序的分波波波斯松进程 (Fractional Poisson Processes of Order k and Beyond)

In this article, we introduce fractional Poisson felds of order k in n-dimensional Euclidean space $R_n^+$. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k in one dimensional Euclidean space $R_1^+$. These processes are defined in terms of fractional compound Poisson processes. Time-fractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poisson process of order k, allows on average infinite number of arrivals in any interval. We derive the marginal probabilities, governing difference-differential equations of the introduced processes. We also provide Watanabe martingale characterization for some time-changed Poisson processes.

翻译：在本篇文章中,我们引入了微小的 Poisson felds k 在正维的 Euclidean 空间 $R_n ⁇ $。我们还致力于时间偏差的 Poisson 进程 k, 空间偏差的 Poisson 进程 k 以及时间偏差的 Poisson 进程 k 在一维的 Euclidean 空间的时间偏差的 Poisson 进程 $R_1 ⁇ $。这些进程是以分数的化合物 Poisson 进程来定义的。时间偏差的 Poisson 进程 k 自然地将 Poisson 进程和 Poisson 进程宽到一个繁重的尾注时间计进程。空间偏差的 Poisson 进程 k 允许在任何间隔内平均无限的到达者。我们从边际的概率中得出, 管理引入的进程的差别的方程式。我们还为一些时间变化的 Poisson 进程提供 Watanabe martingale 的马丁格尔特征定性。