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A model of a branching process in which the reproducing particles diffuse in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175701.png" />. Let the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175702.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175703.png" />-dimensional, with an absorbing boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175704.png" />, and let the particles in the domain itself execute mutually independent Brownian motions. Each particle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175705.png" /> is independently converted, within a time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175706.png" />, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175707.png" /> particles with a probability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b0175709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757010.png" />. Let the daughter particles begin their independent evolution from the point of their genesis. Let
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<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/b017570/b01757011.png" /></td> </tr></table>
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be the generating function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757012.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757013.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757014.png" /> be the number of particles in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757015.png" /> at the moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757016.png" /> if there initially was one particle at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757017.png" />. The generating functional
+
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
  
<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/b017570/b01757018.png" /></td> </tr></table>
+
$$
 +
f (s)  = \
 +
\sum _ {n = 0 } ^  \infty 
 +
p _ {n} s  ^ {n}
 +
$$
 +
 
 +
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
 
satisfies the quasi-linear parabolic 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/b017570/b01757019.png" /></td> </tr></table>
+
$$
 +
\sum _ {i = 1 } ^ { r }
 +
 
 +
\frac{\partial  ^ {2} H }{\partial  x _ {i}  ^ {2} }
 +
+
 +
f (H)  = \
 +
 
 +
\frac{\partial  H }{\partial  t }
 +
 
 +
$$
  
 
with the initial condition
 
with the initial condition
  
<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/b017570/b01757020.png" /></td> </tr></table>
+
$$
 +
H (0, x, s( \cdot ))  = s (x)
 +
$$
  
 
and the boundary condition
 
and the boundary condition
  
<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/b017570/b01757021.png" /></td> </tr></table>
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$$
 +
H (t, x, s ( \cdot ))
 +
\mid  _ {x \rightarrow \partial  G }  = 0 .
 +
$$
 +
 
 +
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
  
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757022.png" /> the eigen values, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757023.png" /> be the eigen function of the problem
+
$$
 +
\sum _ {i = 1 } ^ { r }
  
<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/b017570/b01757024.png" /></td> </tr></table>
+
\frac{\partial  ^ {2} \phi }{\partial  x _ {i}  ^ {2} }
 +
+
 +
\lambda \phi  = 0,\ \
 +
\phi (x) \mid  _ {x \rightarrow \partial  G }  = 0,
 +
$$
  
corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757025.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757026.png" /> the asymptotic relation
+
corresponding to $  \lambda _ {1} $.  
 +
As $  t \rightarrow \infty $
 +
the asymptotic 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/b017570/b01757027.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} \mu _ {x,t} (G)  \approx \
 +
K  e ^ {(a - \lambda _ {1} ) t }
 +
\phi _ {1} (x)
 +
$$
  
holds. For this reason the problem is said to be subcritical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757028.png" />, critical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757029.png" /> and supercritical if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757031.png" />, a branching process with diffusion dies out with probability one, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757032.png" />, both the probability of dying out and the probability of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017570/b01757034.png" /> 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.
+
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====
 
====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>
 
<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====
 
====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


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

[1] B.A. [B.A. Sevast'yanov] Sewastjanow, "Verzweigungsprozesse" , Akad. Wissenschaft. DDR (1974) (Translated from Russian)

Comments

Additional references can be found in the article Branching process.

How to Cite This Entry:
Branching process with diffusion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Branching_process_with_diffusion&oldid=46154
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