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{{MSC|60J80}}

[[Category:Branching processes]]

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|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====
{|
|valign="top"|{{Ref|S}}|| B.A. Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian) {{MR|0408018}} {{ZBL|0291.60039}}
|}

====Comments====
Additional references can be found in the article [[Branching process|Branching process]].