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Branching process with a random medium

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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 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.

How to Cite This Entry:
Branching process with a random medium. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process_with_a_random_medium&oldid=46153
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article