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A model of a branching process in which the reproducing particles diffuse in some domain  $  G $. 
Let the domain  $  G $
be  $  r $-
dimensional, with an absorbing boundary  $  \partial  G $, 
and let the particles in the domain itself execute mutually independent Brownian motions. Each particle in  $  G $
is independently converted, within a time  $  \Delta t $, 
into  $  n $
particles with a probability of  $  p _ {n} \Delta t + o( \Delta t ) $, 
$  n \neq 1 $, 
$  \Delta t \rightarrow 0 $. 
Let the daughter particles begin their independent evolution from the point of their genesis. Let

$$ 
f (s)  = \ 
\sum _ {n = 0 } ^  \infty  
p _ {n} s  ^ {n}
$$

be the generating function of  $  \{ p _ {n} \} $, 
let  $  p _ {1} = - \sum _ {n \neq 1 }  p _ {n} $, 
and let  $  \mu _ {x,t} (A) $
be the number of particles in a set  $  A \subseteq G $
at the moment  $  t $
if there initially was one particle at the point  $  x \in G $. 
The generating functional

$$ 
H (t;  x, s( \cdot ))  = \ 
{\mathsf E}  \mathop{\rm exp}
\left [ \int\limits _ { G }  \mathop{\rm ln}  s (y)
\mu _ {x,t}  (dy) \right ]
$$

satisfies the quasi-linear parabolic equation

$$ 
\sum _ {i = 1 } ^ { r } 

\frac{\partial   ^ {2} H }{\partial  x _ {i}  ^ {2} }
 +
f (H)  = \ 

\frac{\partial  H }{\partial  t }

$$

with the initial condition

$$ 
H (0, x, s( \cdot ))  =  s (x)
$$

and the boundary condition

$$ 
H (t, x, s ( \cdot ))
\mid  _ {x \rightarrow \partial  G }   =  0 .
$$

Denote by  $  0 < \lambda _ {1} < \lambda _ {2} \leq  \lambda _ {3} \leq  \dots $
the eigen values, and let  $  \phi _ {1} (x) > 0 $
be the eigen function of the problem

$$ 
\sum _ {i = 1 } ^ { r } 

\frac{\partial   ^ {2} \phi }{\partial  x _ {i}  ^ {2} }
 +
\lambda \phi  =  0,\ \ 
\phi (x) \mid  _ {x \rightarrow \partial  G }   =  0,
$$

corresponding to  $  \lambda _ {1} $. 
As  $  t \rightarrow \infty $
the asymptotic relation

$$ 
{\mathsf E} \mu _ {x,t} (G)  \approx \ 
K  e ^ {(a - \lambda _ {1} ) t }
\phi _ {1} (x)
$$

holds. For this reason the problem is said to be subcritical if  $  a < \lambda _ {1} $, 
critical if  $  a = \lambda _ {1} $
and supercritical if  $  a > \lambda _ {1} $. 
If  $  a \leq  \lambda _ {1} $, 
a branching process with diffusion dies out with probability one, while if  $  a > \lambda _ {1} $, 
both the probability of dying out and the probability of the event  $  \mu _ {x,t} (G) \rightarrow \infty $
as  $  t \rightarrow \infty $
will in general be positive. Depending on their criticality, branching processes with diffusion obey limit theorems analogous to those valid for branching processes without diffusion.

====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)</TD></TR></table>

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