Space-fractional versions of the negative binomial and Polya-type processes

TLDR: In this article, a space fractional negative binomial process (SFNB) was introduced by time-changing the Space fractional Poisson process by a gamma subordinator and its one-dimensional distributions were derived in terms of generalized Wright functions and their governing equations were obtained.

Abstract: In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Levy process and the corresponding Levy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process.