A limit theorem of branching processes and continuous state branching processes

TL;DR: In this paper, the authors formulate a stochastic equation based on a Poisson system associated with excursion laws of onedimensional continuous state branching diffusions, which gives an intuitive and cornprehensible description of a class of measure-valued diffusion processes and makes the sample path structure clearly observed.

TL;DR: In this paper, it was shown that the random measure of a branching particle diffusion converges almost surely in the vague topology as $t$ tends to infinity, when the generalized principal eigenvalue of the diffusion operator is finite.

TL;DR: In this article, the authors introduce the concept of local spatial clumping with a set of informal calculations that lead to the prediction that the continuous limit of branching particle systems in dimensions d ≥ 3 will lead to infinitely divisible random measures which are almost surely singular.

TL;DR: A review of the Galton and Watson mathematical model that applies probability theory to the effects of chance on the development of populations is given in this article, followed by a systematic development of branching processes, and a brief description of some of the important applications.