A separate section of random processes studies laws of reproduction and transformation of particles and it is the theory of branching processes. The basic mathematical assumption distinguishes branching processes among other random processes is the transformation of particles independently from one another. The laws of reproduction and transformation of particles are subject to regularities, in which randomness plays a major role.

This article investigates a homogeneous branching process with one particle type, migration, and continuous time $ \mu (t) $, $ t \in [0, \infty) $. It is assume that there is one particle in the system at the beginning. The process is defined by transient probabilities, determined by the intensities of particle reproduction, immigration, and emigration. Two problems are explored. In the first of these, for this process model, a differential equation for the factorial moment of arbitrary order is found. As a consequence, the form of differential equations for the mathematical expectation of the process, the second and third factorial moments of the process, is given. The result is a mathematical expectation of $ A (t) $, $ t \in [0, \infty) $ branching process, and the appearance of a second factorial moment for the real function of process $ B (t) $, $ t \in [0, \infty ) $, which allows you to explore the process in details depending on the criticality of the process. The second problem investigates the random process $ \rho (t) $, which determines the number of transformations of particles in a system of homogeneous branching process with migration and continuous time up to the time $ t $. The differential equation for the generic function of the process $ \rho (t) $ is obtained. The form of the real function of the process is found and the boundary behavior at $ t \to\infty $ is investigated. It is shown that the centered and normalized process at $ t \to\infty $ in the distribution coincides with the standard to normal distribution.

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