Abstract:
A homogeneous Markov process with continuous time on the phase space $\mathbb{Z}_+ =\{0,1,2,\dots\}$ is considered, which is interpreted as the motion of a particle. The particle can move to neighboring points in $\mathbb{Z}_+$ or make a jump directly $d\in\mathbb{N}$ positions to the right. Such a transition is called a drift. The process is equipped with a branching mechanism. Branching sources can be located at every point of $\mathbb{Z}_+$. At the moment of branching, new particles appear at the branching point and subsequently begin to evolve independently of each other (and of the other particles) according to the same laws as the initial particle. This branching Markov process corresponds to a matrix related to the Jacobi matrix. In terms of the orthogonal polynomials associated with this matrix, formulas are obtained for the average number of particles at an arbitrary fixed point $\mathbb{Z}_+$ at time $t>0$. The results are applied to some specific models, and the asymptotic behavior of the average number of particles at large times is found.
Key words and phrases:Markov branching process, branching random walks, Jacobi matrices, orthogonal polynomials.
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