On a stochastic model for continuous mass branching population

It is well known that the set of non-negative integers is the state-space of usual branching stochastic processes. However in many applications one may have situations when it is difficult to count the number of individuals in the population, but some non-negative characteristic, such as volume, weight or product produced by the individuals can be measured. To model this kind of situation, branching stochastic processes with continuous state-space are introduced. In this paper two theorems which establish relationship between asymptotic behavior of processes continuous and discrete state-space and with immigration in varying environment will be proved.