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The Fractional Poisson Process and the Inverse Stable Subordinator


 
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1. Title Title of document The Fractional Poisson Process and the Inverse Stable Subordinator
 
2. Creator Author's name, affiliation, country Mark M Meerschaert; Michigan State University; United States
 
2. Creator Author's name, affiliation, country Erkan Nane; Auburn University; United States
 
2. Creator Author's name, affiliation, country P. Vellaisamy; Indian Institute of Technology Bombay; India
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Fractional Poisson process; Inverse stable subordinator; Renewal process; Mittag-Leffler waiting time; Fractional difference-differential equations; Caputo fractional derivative; Generalized Mittag-leffler function; Continuous time random walk limit; Di
 
3. Subject Subject classification 60K05; 33E12; 26A33
 
4. Description Abstract The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) The research of MMM was partially supported by NSF grants DMS-1025486, DMS-0803360, EAR-0823965, and NIH grant R01-EB012079-01.
 
7. Date (YYYY-MM-DD) 2011-08-28
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/920
 
10. Identifier Digital Object Identifier 10.1214/EJP.v16-920
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 16
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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