Fractional Poisson Process
暫譯: 分數型 Poisson 過程

Laskin Nick

Fractional Poisson Process
  • 出版商: World Scientific Pub
  • 出版日期: 2026-10-04
  • 售價: $6,420
  • 貴賓價: 9.5 折 $6,099
  • 語言: 英文
  • 頁數: 482
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 9819833094
  • ISBN-13: 9789819833092
  • 相關分類: 機率統計學 Probability-and-statistics

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商品描述

This book provides a systematic, comprehensive, and self-contained treatment of the fractional Poisson process and the fractional Poisson distribution, covering both their theory and wide-ranging applications. It explores four main approaches to studying the fractional Poisson process: the fractional generalization of the Kolmogorov-Feller equation; renewal theory, replacing the exponential waiting time distribution of a Poisson process with one modeled by the Mittag-Leffler function; the inverse stable subordinator method, which constructs a fractional Poisson process from a known Poisson process by subordinating the time variable to an independent inverse stable subordinator; and the direct design of the probability mass function using the complete monotonicity of the Mittag-Leffler function and its generalizations, without relying on fractional differential or integral operators.

The book demonstrates the use of various Mittag-Leffler type functions, Fox-Wright and Fox's H-functions, the Lévy α-stable distribution, and techniques such as the Laplace and Mellin transforms and the Mellin-Barnes representation. These tools are shown to be especially effective for analyzing fractional stochastic processes and their probability mass and density functions.

This is the first book devoted to the fractional Poisson process, written by its inventor, who introduced the concept in 2003.

商品描述(中文翻譯)

本書系統性、全面且自成一體地介紹分數階 Poisson 過程（fractional Poisson process）與分數階 Poisson 分布（fractional Poisson distribution），涵蓋其理論與廣泛的應用。書中探討研究分數階 Poisson 過程的四種主要方法：Kolmogorov–Feller 方程的分數階推廣；更新理論（renewal theory），以 Mittag-Leffler 函數所描述的等待時間分布，取代 Poisson 過程中的指數等待時間分布；反穩定次級過程（inverse stable subordinator）方法，透過將時間變數從屬於一個獨立的反穩定次級過程，從已知的 Poisson 過程建構分數階 Poisson 過程；以及利用 Mittag-Leffler 函數及其推廣函數的完全單調性（complete monotonicity），直接設計機率質量函數，而不依賴分數階微分或積分算子。

本書示範各種 Mittag-Leffler 型函數、Fox–Wright 函數與 Fox 的 H 函數、Lévy α-穩定分布（Lévy α-stable distribution），以及 Laplace 變換、Mellin 變換和 Mellin–Barnes 表示法等技術的運用。書中說明，這些工具對於分析分數階隨機過程，以及其機率質量函數與密度函數，特別具有成效。

這是第一本專門探討分數階 Poisson 過程的著作，由該概念的發明者親自撰寫；他於 2003 年提出了這個概念。

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