Spatial Growth of a Branching Process of Particles Living in $\mathbb{R}^d$

Abstract

Consider a branching process in which particles are located in $\mathbb{R}^d$, do not move during their life times, die according to the exponential holding law, and at their deaths give birth to random number of particles which are located at distances from their parents. The total number process is supposed supercritical. We are interested in the number of particles living in a shifted region $D + tc$, denoted by $Z_t(D + tc)$, where $c \in \mathbb{R}^d$ and $D$ is a bounded set of $\mathbb{R}^d$, and observe a.s. convergences of $Z_t(D + tc)/E\lbrack Z_t(D + tc)\rbrack$ as $t \rightarrow \infty$. The result is applied to an associated non-linear evolution equation, which reduces, in a special case, to the equation of a deterministic model of simple epidemics.