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''branching process in a random medium''
 
''branching process in a random medium''
  
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[[Category:Branching processes]]
 
[[Category:Branching processes]]
  
A time-inhomogeneous branching process in which the inhomogeneity is random. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175601.png" /> be a stationary sequence of random variables (the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175602.png" /> is interpreted as the state of the "medium" at the moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175603.png" />), and let to each possible state of the medium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175604.png" /> correspond a probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175605.png" /> of the number of descendants of a single particle:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175606.png" /></td> </tr></table>
+
$$
 +
p _ {k} ( \overline \xi \; )  \geq  0,\ \
 +
\sum _ {k = 0 } ^  \infty 
 +
p _ {k} ( \overline \xi \; = 1,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175607.png" /></td> </tr></table>
+
$$
 +
F _ {\overline \xi \; }  (s)  = \sum _ {k = 0 } ^  \infty  p _ {k} ( \overline \xi \; ) s  ^ {k} .
 +
$$
  
In order to construct a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175608.png" /> of a branching process in a random medium the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b0175609.png" /> and the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756010.png" /> of the states of the medium are fixed, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756011.png" /> is determined for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756012.png" /> as a sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756013.png" /> independent random variables with distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756014.png" />. Such a complication of the branching [[Galton–Watson process|Galton–Watson process]] is natural if, for example, the branching process in a random medium is regarded as a model of a biological population.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756015.png" />, under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756016.png" />, has the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
{\mathsf E} \{ s ^ {\mu (t) }
 +
\mid  \mu (0) = 1 \}
 +
= {\mathsf E} _ {\overline \xi \; }  F _ {\xi _ {0}  }
 +
(F _ {\xi _ {1}  } ( \dots (F _ {\xi _ {t-1 }  } (s) ) \dots ))
 +
$$
  
(for a branching Galton–Watson process, i.e. for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756018.png" />, the right-hand side of (*) equals the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756019.png" />-fold iteration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756020.png" />). Branching processes in a random medium may be subcritical, critical and supercritical: the "criticality parameter" here (see {{Cite|AN}}) is the variable
+
(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 {{Cite|AN}}) is the variable
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756021.png" /></td> </tr></table>
+
$$
 +
\rho  = {\mathsf E} _ {\xi _ {0}  }  \mathop{\rm ln} \
 +
\sum _ {k = 0 } ^  \infty 
 +
kp _ {k} ( \xi _ {0} )  = \
 +
{\mathsf E} _ {\xi _ {0}  }  \mathop{\rm ln} \
 +
F _ {\xi _ {0}  } ^ { \prime } (1)
 +
$$
  
(for ordinary branching processes, the "criticality parameter" is the mathematical expectation of the number of "descendants" of a single particle). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756022.png" />, the branching process in a random medium is said to be subcritical and, for the random variable
+
(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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756023.png" /></td> </tr></table>
+
$$
 +
q ( \overline \xi \; = \
 +
\lim\limits _ {t \rightarrow \infty }  {\mathsf P}
 +
\{ \mu (t) = 0
 +
\mid  \mu (0) = 1, \overline \xi \; \}
 +
$$
  
which is the probability of extinction of the branching process in a random medium for a given trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756024.png" />, the relation
+
which is the probability of extinction of the branching process in a random medium for a given trajectory $  \overline \xi \; $,  
 +
the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756025.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ q ( \overline \xi \; ) = 1 \}  = 1
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756026.png" /> the limits
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756027.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow \infty }  {\mathsf P}
 +
\{ \mu (t) = k
 +
\mid  \mu (0) = 1,\
 +
\mu (t) > 0, \overline \xi \; \}
 +
= p _ {k}  ^ {*} ( \overline \xi \; )
 +
$$
  
 
exist and satisfy
 
exist and satisfy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756028.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 1 } ^  \infty 
 +
p _ {k}  ^ {*} ( \overline \xi \; = 1.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756029.png" />, the branching process in a random medium is said to be critical, and
+
If $  \rho = 0 $,  
 +
the branching process in a random medium is said to be critical, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756030.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ q ( \overline \xi \; ) = 1 \}  = 1
 +
$$
  
and, for almost all realizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756031.png" />,
+
and, for almost all realizations of $  \overline \xi \; $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756032.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow \infty }  {\mathsf P}
 +
\{ \mu (t) = k
 +
\mid  \mu (0) = 1,\
 +
\mu (t) > 0, \overline \xi \; \}  = 0.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756033.png" />, the branching process in a random medium is said to be supercritical; in such a case
+
If $  \rho > 0 $,  
 +
the branching process in a random medium is said to be supercritical; in such a case
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756034.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ q ( \overline \xi \; ) < 1 \}  = 1
 +
$$
  
and, if certain additional conditions are met, there exists for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756035.png" /> a non-negative random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756036.png" />,
+
and, if certain additional conditions are met, there exists for almost all $  \overline \xi \; $
 +
a non-negative random variable $  W $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017560/b01756037.png" /></td> </tr></table>
+
$$
 +
= \lim\limits _ {t \rightarrow \infty }
 +
 +
\frac{\mu (t) }{F _ {\xi _ {0}  } ^ { \prime } (1) \dots
 +
F _ {\xi _ {t-1 }  } ^ { \prime } (1) }
 +
,\ \
 +
{\mathsf E} W  = 1.
 +
$$
  
 
====References====
 
====References====

Latest revision as of 06:29, 30 May 2020


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.

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