Composite fractional time evolutions of cumulants in fractional generalized birth processes

Abstract

A fractional generalized pure birth process is studied based on the master equation approach. The exact analytic solution of the generating function of the probability density for the process is given in terms of the Gauss hypergeometric function. The expressions of moments are also obtained in an explicit form. The effects of fixed and distributed initial conditions are also elucidated. The effect of memory is demonstrated quantitatively in terms of the mean, variance, and the Fano factor. It is also shown how to discriminate a fractional generalized birth process from the fractional Yule-Furry process. Further, the appearance of composite fractional time evolutions of cumulants is elucidated in conjunction with the fractional Poisson and the fractional Yule-furry processes.

Fractional Linear Birth-Death Stochastic Process—An Application of Heun's Differential Equation

“…Although we have pointed out that the power law nature appears when considering the asymptotic property of the Mittag-Leffler function in Eqs. (29) and (32), the power law t α nature can appear when the critical conditions are realized, as shown below.…”

“…Namely, they are nonlinear functions of t α , they cannot be described by a finite number of scaling functions ("multi-fractal" nature) (cf. References [29,30,32]).…”

“…The mean and variance in Eqs. (29) and (32) are expected under the situation that a long time memory is relevant. Figure 1 shows the time evolutions of (i) the mean and (ii) the variance as a function of the fractional index of α.…”

“…In previous papers, the fractional Poisson process (FPP), [29,30] the fractional generalized birth process (FGBP) [31,32] and a linear birth-death process [33] were studied. It was shown that the generating function u (GF) for the FGBP with λ n = λn+ν and µ n = 0 in the Laplace domain is described by the Gauss hypergeometric equation…”

“…(B3) in Appendix B of Ref. [32]. Also it was shown that the Laplace transformed generating function (GF) for a simple fractional birth-death process with λ n = ν and µ n = nµ in Eq.…”