Branching process with a random medium

2020 Mathematics Subject Classification: Primary: 60J80 [MSN][ZBL]

A model of a branching process in which the lifetime of a particle is an arbitrary non-negative random variable, while the number of daughter particles depends on its age at the moment of transformation. In the single-type particle model each particle has a random duration of life $ \tau $ with distribution function

$$ {\mathsf P} \{ \tau \leq t \} = G (t). $$

At the end of its life the particle is transformed into $ k $ daughter particles of age zero with a probability $ p _ {k} (u) $ if the transformation took place when the age attained by the original particle was $ u $. Let $ \mu (t) $ be the number of particles at the moment of time $ t $. The generating function $ F(t; x) $ of the probability distribution of $ \mu (t) $ for a process beginning with one particle of age zero satisfies the equation

$$ \tag{* } F (t; x) = \ \int\limits _ { 0 } ^ { t } h (u, F (t - u ; x)) dG (u) + x (1 - G (t)), $$

where

$$ h (u, x) = \ \sum _ {k = 0 } ^ \infty p _ {k} (u) x ^ {k} . $$

Put

$$ \left . a (u) = \ \frac{\partial h }{\partial x } \ \right | _ {x = 1 } ,\ \ \left . b (u) = \ \frac{\partial ^ {2} h }{\partial x ^ {2} } \ \right | _ {x = 1 } , $$

$$ A = \int\limits _ { 0 } ^ \infty a (u) dG (u),\ B = \int\limits _ { 0 } ^ \infty b (u) dG (u). $$

An age-dependent branching process is said to be subcritical, critical or supercritical if $ A > 1 $, $ A = 1 $ and $ B > 0 $, or $ A > 1 $, respectively. The behaviour of the process as $ t \rightarrow \infty $ substantially depends on its criticality. Subcritical and critical processes die out with probability one, i.e.

$$ \lim\limits _ {t \rightarrow \infty } \ {\mathsf P} \{ \mu (t) = 0 \} = 1. $$

The following results have been obtained for these processes [S]: asymptotic formulas for the moments $ \mu (t) $, necessary and sufficient conditions of extinction, conditions of existence and uniqueness of a solution of equation (*) and asymptotic formulas as $ t \rightarrow \infty $ for

$$ Q (t) = \ {\mathsf P} \{ \mu (t) > 0 \} . $$

The limit distributions have also been determined. In the critical case, as $ t \rightarrow \infty $:

$$ Q(t) \approx \ 2 \frac{\int\limits _ { 0 } ^ \infty u a (u) dG (u) }{Bt } , $$

$$ {\mathsf P} \left \{ \frac{\mu (t) }{ {\mathsf E} ( \mu (t) \mid \mu (t) > 0) } < x \mid \mu (t) > 0 \right \} \rightarrow 1 - e ^ {-x} ,\ x > 0. $$

If $ h(u, x) $ is independent of $ u $, the age-dependent branching process is a Bellman–Harris process. The model just described has been generalized to include processes with several types of particles, and also to processes for which a particle may generate new particles several times during its lifetime [S2], [M].

References

[S]	B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) MR0408018 Zbl 0291.60039
[S2]	B.A. Sewastjanow, "Age-dependent branching processes" Theory Probab. Appl. , 9 : 4 (1964) pp. 521–537 Teor. Veroyatnost. i Primenen. , 9 : 4 (1964) pp. 577–594 MR0170396 Zbl 0248.60059
[M]	C.J. Mode, "Multitype branching processes" , Elsevier (1971) MR0279901 Zbl 0219.60061

Comments

Additional references can be found in the article Branching process.