Branching process with diffusion

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.