Branching process with a random medium

branching process in a random medium

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

A time-inhomogeneous branching process in which the inhomogeneity is random. Let $ \overline \xi \; = \{ \xi _ {0} , \xi _ {1} ,\dots \} $ be a stationary sequence of random variables (the value of $ \xi _ {t} $ is interpreted as the state of the "medium" at the moment of time $ t $), and let to each possible state of the medium $ \overline \xi \; $ correspond a probability distribution $ \{ p _ {k} ( \overline \xi \; ) \} $ of the number of descendants of a single particle:

$$ p _ {k} ( \overline \xi \; ) \geq 0,\ \ \sum _ {k = 0 } ^ \infty p _ {k} ( \overline \xi \; ) = 1, $$

$$ F _ {\overline \xi \; } (s) = \sum _ {k = 0 } ^ \infty p _ {k} ( \overline \xi \; ) s ^ {k} . $$

In order to construct a trajectory $ \{ \mu (0), \mu (1) ,\dots \} $ of a branching process in a random medium the value of $ \mu (0)= m $ and the trajectory $ \overline \xi \; $ of the states of the medium are fixed, and $ \mu (t+ 1) $ is determined for each $ t = 0, 1 \dots $ as a sum of $ \mu (t) $ independent random variables with distribution $ \{ p _ {k} ( \xi _ {t} ) \} $. Such a complication of the branching Galton–Watson process is natural if, for example, the branching process in a random medium is regarded as a model of a biological population.

The properties of branching processes in a random medium are analogous to those of ordinary branching processes. For instance, the generating distribution function of $ \mu (t) $, under the condition $ \mu (0) = 1 $, has the form

$$ \tag{* } {\mathsf E} \{ s ^ {\mu (t) } \mid \mu (0) = 1 \} = {\mathsf E} _ {\overline \xi \; } F _ {\xi _ {0} } (F _ {\xi _ {1} } ( \dots (F _ {\xi _ {t-1 } } (s) ) \dots )) $$

(for a branching Galton–Watson process, i.e. for $ {\mathsf P} \{ \xi _ {t} \equiv 0 \} = 1 $, the right-hand side of (*) equals the $ t $- fold iteration of $ F _ {0} (s) $). Branching processes in a random medium may be subcritical, critical and supercritical: the "criticality parameter" here (see [AN]) is the variable

$$ \rho = {\mathsf E} _ {\xi _ {0} } \mathop{\rm ln} \ \sum _ {k = 0 } ^ \infty kp _ {k} ( \xi _ {0} ) = \ {\mathsf E} _ {\xi _ {0} } \mathop{\rm ln} \ F _ {\xi _ {0} } ^ { \prime } (1) $$

(for ordinary branching processes, the "criticality parameter" is the mathematical expectation of the number of "descendants" of a single particle). If $ \rho < 0 $, the branching process in a random medium is said to be subcritical and, for the random variable

$$ q ( \overline \xi \; ) = \ \lim\limits _ {t \rightarrow \infty } {\mathsf P} \{ \mu (t) = 0 \mid \mu (0) = 1, \overline \xi \; \} $$

which is the probability of extinction of the branching process in a random medium for a given trajectory $ \overline \xi \; $, the relation

$$ {\mathsf P} \{ q ( \overline \xi \; ) = 1 \} = 1 $$

is valid. There is also the analogue of the limit theorem of the subcritical Galton–Watson branching process: For almost all realizations of the sequence $ \overline \xi \; $ the limits

$$ \lim\limits _ {t \rightarrow \infty } {\mathsf P} \{ \mu (t) = k \mid \mu (0) = 1,\ \mu (t) > 0, \overline \xi \; \} = p _ {k} ^ {*} ( \overline \xi \; ) $$

exist and satisfy

$$ \sum _ {k = 1 } ^ \infty p _ {k} ^ {*} ( \overline \xi \; ) = 1. $$

If $ \rho = 0 $, the branching process in a random medium is said to be critical, and

$$ {\mathsf P} \{ q ( \overline \xi \; ) = 1 \} = 1 $$

and, for almost all realizations of $ \overline \xi \; $,

$$ \lim\limits _ {t \rightarrow \infty } {\mathsf P} \{ \mu (t) = k \mid \mu (0) = 1,\ \mu (t) > 0, \overline \xi \; \} = 0. $$

If $ \rho > 0 $, the branching process in a random medium is said to be supercritical; in such a case

$$ {\mathsf P} \{ q ( \overline \xi \; ) < 1 \} = 1 $$

and, if certain additional conditions are met, there exists for almost all $ \overline \xi \; $ a non-negative random variable $ W $,

$$ W = \lim\limits _ {t \rightarrow \infty } \frac{\mu (t) }{F _ {\xi _ {0} } ^ { \prime } (1) \dots F _ {\xi _ {t-1 } } ^ { \prime } (1) } ,\ \ {\mathsf E} W = 1. $$

References

[AN]	K.B. Athreya, P.E. Ney, "Branching processes" , Springer (1972) MR0373040 Zbl 0259.60002

Comments

Additional references can be found in the article Branching process.