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A property of branching processes ensuring that the number of particles at any moment of time is finite. The problem of the regularity of a branching process is usually reduced to the problem of uniqueness of the solution of some differential or integral equation. For instance, in a continuous-time branching process the differential equation
 
A property of branching processes ensuring that the number of particles at any moment of time is finite. The problem of the regularity of a branching process is usually reduced to the problem of uniqueness of the solution of some differential or integral equation. For instance, in a continuous-time branching process the differential equation
  
<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/b017590/b0175901.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  F (t; s) }{\partial  t }
 +
  = \
 +
f (F (t; s))
 +
$$
 +
 
 +
with the initial condition  $  F(0; s) = s $
 +
has a unique solution if and only if, for any  $  \epsilon > 0 $,
 +
the integral
  
with the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b0175902.png" /> has a unique solution if and only if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b0175903.png" />, the integral
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$$
 +
\int\limits _ {1 - \epsilon } ^ { 1 }
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{
 +
\frac{dx}{f (x) }
 +
}
 +
$$
  
<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/b017590/b0175904.png" /></td> </tr></table>
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is divergent. In the branching [[Bellman–Harris process|Bellman–Harris process]] the generating function  $  F(t; s) $
 +
of the number of particles is the solution of the non-linear integral equation
  
is divergent. In the branching [[Bellman–Harris process|Bellman–Harris process]] the generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b0175905.png" /> of the number of particles is the solution of the non-linear integral equation
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$$ \tag{* }
 +
F (t;  s)  = \
 +
\int\limits _ { 0 } ^ { t }
 +
h (F (t - u;  s))
 +
dG (u) + s
 +
(1 - G (t)),
 +
$$
  
<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/b017590/b0175906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where  $  G(t) $
 +
is the distribution function of the lifetimes of particles and  $  h(t) $
 +
is the generating function of the number of daughter particles ( "direct descendants" ) of a single particle. If, for given  $  t _ {0} , c _ {1} , c _ {2} > 0 $
 +
and an integer  $  n \geq  1 $,
 +
the inequalities
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b0175907.png" /> is the distribution function of the lifetimes of particles and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b0175908.png" /> is the generating function of the number of daughter particles ( "direct descendants" ) of a single particle. If, for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b0175909.png" /> and an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b01759010.png" />, the inequalities
+
$$
 +
c _ {1} t  ^ {n}  \leq  G (t)
 +
\leq  c _ {2} t  ^ {n}
 +
$$
  
<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/b017590/b01759011.png" /></td> </tr></table>
+
are valid for all  $  0 \leq  t \leq  t _ {0} $,
 +
the solution of equation (*) is unique if and only if the equation
  
are valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b01759012.png" />, the solution of equation (*) is unique if and only if the equation
+
$$
  
<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/b017590/b01759013.png" /></td> </tr></table>
+
\frac{d  ^ {n} \phi }{dt  ^ {n} }
 +
  = \
 +
h ( \phi ) - 1
 +
$$
  
 
with initial conditions
 
with initial conditions
  
<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/b017590/b01759014.png" /></td> </tr></table>
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$$
 +
\phi (0)  = 1,\ \
 +
\phi  ^ {(r)} (0)  = 0,\  r = 1 \dots n - 1,
 +
$$
  
 
has a unique solution
 
has a unique solution
  
<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/b017590/b01759015.png" /></td> </tr></table>
+
$$
 +
0 \leq  \phi (t)  \leq  1.
 +
$$
  
 
For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral
 
For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral
  
<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/b017590/b01759016.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \epsilon 
 +
{
 +
\frac{dx}{x ^ {1-1/n } (1-h(1-x))  ^ {1/n} }
 +
}
 +
$$
  
to diverge for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017590/b01759017.png" />.
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to diverge for any $  \epsilon > 0 $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. [B.A. Sevast'yanov] Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) {{MR|0408018}} {{ZBL|0291.60039}} </TD></TR></table>
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{|
 
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|valign="top"|{{Ref|S}}|| B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) {{MR|0408018}} {{ZBL|0291.60039}}
 
+
|}
  
 
====Comments====
 
====Comments====
 
Additional references can be found in the article [[Branching process|Branching process]].
 
Additional references can be found in the article [[Branching process|Branching process]].

Latest revision as of 06:29, 30 May 2020


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

A property of branching processes ensuring that the number of particles at any moment of time is finite. The problem of the regularity of a branching process is usually reduced to the problem of uniqueness of the solution of some differential or integral equation. For instance, in a continuous-time branching process the differential equation

$$ \frac{\partial F (t; s) }{\partial t } = \ f (F (t; s)) $$

with the initial condition $ F(0; s) = s $ has a unique solution if and only if, for any $ \epsilon > 0 $, the integral

$$ \int\limits _ {1 - \epsilon } ^ { 1 } { \frac{dx}{f (x) } } $$

is divergent. In the branching Bellman–Harris process the generating function $ F(t; s) $ of the number of particles is the solution of the non-linear integral equation

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

where $ G(t) $ is the distribution function of the lifetimes of particles and $ h(t) $ is the generating function of the number of daughter particles ( "direct descendants" ) of a single particle. If, for given $ t _ {0} , c _ {1} , c _ {2} > 0 $ and an integer $ n \geq 1 $, the inequalities

$$ c _ {1} t ^ {n} \leq G (t) \leq c _ {2} t ^ {n} $$

are valid for all $ 0 \leq t \leq t _ {0} $, the solution of equation (*) is unique if and only if the equation

$$ \frac{d ^ {n} \phi }{dt ^ {n} } = \ h ( \phi ) - 1 $$

with initial conditions

$$ \phi (0) = 1,\ \ \phi ^ {(r)} (0) = 0,\ r = 1 \dots n - 1, $$

has a unique solution

$$ 0 \leq \phi (t) \leq 1. $$

For a branching process described by equation (*) to be regular, it is necessary and sufficient for the integral

$$ \int\limits _ { 0 } ^ \epsilon { \frac{dx}{x ^ {1-1/n } (1-h(1-x)) ^ {1/n} } } $$

to diverge for any $ \epsilon > 0 $.

References

[S] B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) MR0408018 Zbl 0291.60039

Comments

Additional references can be found in the article Branching process.

How to Cite This Entry:
Branching processes, regularity of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_processes,_regularity_of&oldid=23589
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article