A multispecies birth death-immigration process and its diffusion approximation

作者：Di Crescenzo Antonio^*; Martinucci Barbara; Rhandi Abdelaziz

来源：Journal of Mathematical Analysis and Applications, 2016, 442(1): 291-316.

DOI：10.1016/j.jmaa.2016.04.059

• Birth-death process
• Diffusion process
• Permutations with k components
• Polylogarithm function

We consider an extended birth death immigration process defined on a lattice formed by the integers of d semiaxes joined at the origin. When the process reaches the origin, then it may jump toward any semiaxis with the same rate. The dynamics on each ray evolves according to a one-dimensional linear birth death process with immigration. We investigate the transient and asymptotic behavior of the process via its probability generating function. The stationary distribution, when existing, is a zero-modified negative binomial distribution. We also study a diffusive approximation of the process, which involves a diffusion process with linear drift and infinitesimal variance on each ray. It possesses a gamma-type transient density admitting a stationary limit. As a byproduct of our study, we obtain a closed form of the number of permutations with a fixed number of components, and a new series form of the polylogarithm function expressed in terms of the Gauss hypergeometric function.

• 出版日期2016-10-1