Branching Stochastic Processes for Particles Diffusing in a Bounded Domain with Absorbing Boundaries

B. A. Sevast'yanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 3 (1958) no. 2, pp. 121-136

Résumé

Particles diffusing in a multidimensional restricted domain with absorbing boundaries produce independent new particles according to a scheme for branching processes with the probability-generating function $F(z)=\sum_{k=0}^\infty P_{k^{z^k}}$, where $P_k$ is the conditional probability that a particle turns into $k$ particles, if at all. Let $P_n(x,t)$ be the probability that a particle existing at point $x$ after $t$ generations turns into $n$ particles. The probability-generating function $$F(x,t,z)=\sum_n P_n(x,t)z^n$$ satisfies (11). Let $P_0(x)=\lim _{t\to 0}P_0(x,t)$ be the probability of extinction if the initial particle was at point $x$. $P_0(x)$ satisfies a non-linear integral equation of Hammerstein’s type (21). There is a critical value $A_0>1$ for $A= F'$ (1). If $A\leq A_0$, then $P_0(x)\equiv1$; if $A>A_0$, then $P_0(x)<1$. The critical value is $$A_0=\frac{D_{\lambda 1}+c}{c}$$ where $D$ is the coefficient of diffusion, $1/c$ is the mean value for the lifetime of a particle, $\lambda_1$ is the least eigenvalue of the boundary value problem in §1. In §7 the probability distribution for a number of final particles is investigated.