Geometric branching reproduction Markov processes

Abstract

We present a model of a continuous-time Markov branching process with the infinitesimal generating function defined by the geometric probability distribution. It is proved that the solution of the backward Kolmogorov equation is expressed by the composition of special functions-Wright function in the subcritical case and Lambert-W function in the critical case. We found the explicit form of conditional limit distribution in the subcritical branching reproduction. In the critical case, the extinction probability and probability mass function are expressed as a series containing Bell polynomial, Stirling numbers, and Lah numbers.

Factorial moments of the critical Markov branching process with geometric reproduction of particles

“…A branching process model driven by geometric reproduction of particles was introduced in [12]. By design, it is a time-homogeneous Markov branching process X(t), t > 0, with probability generating function F (t, s), |s| < 1.…”

“…The p.g.f. F (t, s), |s| < 1, is determined in [12] as the unique solution to the Kolmogorov equations [2,7,10] after the Lagrange inversion method following the classical theory [5,8,9]. The factorial moments in the subcritical case are computed in [12].…”

“…F (t, s), |s| < 1, is determined in [12] as the unique solution to the Kolmogorov equations [2,7,10] after the Lagrange inversion method following the classical theory [5,8,9]. The factorial moments in the subcritical case are computed in [12]. They are used to obtain the probability mass function, central moments and variance-to-mean ratio (VMR) for the limiting random variable.…”

“…The agreement between Lagrange inversion method and series expansion of the Lambert-W function is confirmed by the approximation of extinction probability. The detailed explanation can be found in [12].…”

“…However, the use of the Lambert-W function on its principal branch W 0 (x) of x ≥ −1/e is limited to obtaining values of F (t, s) only for s < 1 [12]. Thus, 5), ( 6)) for different values of Kt (computation details can be found in [12]).…”