Branching process with a random medium

branching process in a random medium

2010 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