# 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)