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Koenigs functions in the subcritical and critical Markov branching processes with Poisson probability reproduction of particles

Penka Mayster, Assen Tchorbadjieff

TL;DR

This work develops a real-time Poisson-reproducing Markov branching process by applying Abel and Schröder frameworks to Koenigs functions, focusing on subcritical and critical regimes. It provides three equivalent analytical routes to explicit Koenigs-function representations, leveraging exponential Bell polynomials and the exponential-integral functions $\mathrm{Ei}$ and $\mathrm{Ein}$ to characterize limit laws and invariant measures. The approach yields high-precision, computable series expansions near crucial points ($s=0$ and $s=1$) and yields insights into the limit conditional law and stationary measures. The findings advance the analytic toolbox for branching processes and offer techniques transferable to related Abel/Schröder-type problems in stochastic dynamics.

Abstract

Special functions have always played a central role in physics and in mathematics, arising as solutions of nonlinear differential equations, as well as in the theory of branching processes, which extensively uses probability generating functions. The theory of iteration of real functions leads to limit theorems for the discrete-time and real-time Markov branching processes. The Poisson reproduction of particles in real time is analysed through the integration of the Kolmogorov equation. These results are further extended by employing graphical representations of Koenigs functions under subcritical and critical branching mechanisms. The limit conditional law in the subcritical case and the invariant measure for the critical case are discussed, as well. The obtained explicit solutions contain the exponential Bell polynomials and the modified exponential-integral function $\rm{Ein} (z)$.

Koenigs functions in the subcritical and critical Markov branching processes with Poisson probability reproduction of particles

TL;DR

This work develops a real-time Poisson-reproducing Markov branching process by applying Abel and Schröder frameworks to Koenigs functions, focusing on subcritical and critical regimes. It provides three equivalent analytical routes to explicit Koenigs-function representations, leveraging exponential Bell polynomials and the exponential-integral functions and to characterize limit laws and invariant measures. The approach yields high-precision, computable series expansions near crucial points ( and ) and yields insights into the limit conditional law and stationary measures. The findings advance the analytic toolbox for branching processes and offer techniques transferable to related Abel/Schröder-type problems in stochastic dynamics.

Abstract

Special functions have always played a central role in physics and in mathematics, arising as solutions of nonlinear differential equations, as well as in the theory of branching processes, which extensively uses probability generating functions. The theory of iteration of real functions leads to limit theorems for the discrete-time and real-time Markov branching processes. The Poisson reproduction of particles in real time is analysed through the integration of the Kolmogorov equation. These results are further extended by employing graphical representations of Koenigs functions under subcritical and critical branching mechanisms. The limit conditional law in the subcritical case and the invariant measure for the critical case are discussed, as well. The obtained explicit solutions contain the exponential Bell polynomials and the modified exponential-integral function .

Paper Structure

This paper contains 10 sections, 4 theorems, 144 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Poisson probability reproduction in real time
  3. Recurrence relations for the Koenigs functions derivatives when $0<\lambda<1$ and $\lambda=1,$
  4. Subcritical MBP $X(t)$, $0<\lambda<1$,
  5. Integration of the reciprocal power series in the neighbourhood of $s=0$
  6. Subcritical MBP in the neighborhood of $s=1$
  7. Backward Kolmogrov equation for the Poisson reproduction in critical case
  8. In the neighborhood of the point $s=0$
  9. In the neighborhood of the point $s=1$
  10. Conclusion

Key Result

Theorem 3.1

Let the function The explicit form of the function $A(s)$ for $0<\lambda<1$ is represented by the series expantion The increasing and decreasing factorials, are defined by

Figures (7)

  • Figure 1: Graphics of h(s) for sub-, critical and supper- critical branching processes. The subcritical BP of $\lambda = 1/3, 1/2, 2/3$ is plotted on the left side. The critical ($\lambda=1$) and supercritical ($\lambda=2, 3$) are on the right side.
  • Figure 2: Graphics of the function $B(s)$ using eq. \ref{['eq:BS']} for $\lambda =1/6$ (black), $\lambda =5/6$ (red) and $\lambda =17/18$ (green).
  • Figure 3: Histogram of $f_n=\frac{-B^{(n)}(0)}{n!}$ for different $\lambda$.
  • Figure 4: Graphics of $A(s)$ (left) and $B(s)=e^{A(s)}$ (right).
  • Figure 5: Graphics of $B(s)=(1-s)e^{G(s)}$ for $\lambda=1/3$(blue), $\lambda=1/2$(red) and $\lambda=2/3(green)$.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 4.1