On the asymptotic behavior of the average value of functionals of a random field of particles defined by a branching random walk

Consider a homogeneous Markov process with continuous time on the phase space $\mathbb Z_+ = \{0, 1, 2, \dots\}$, which we interpret as the movement of a particle. The particle can only transition to neighboring points in $\mathbb Z_+$, meaning that with each change in position, its coordinate changes by one unit. The process is equipped with a branching mechanism. Branching sources can be located at each point in $\mathbb Z_+$. At the moment of branching, new particles appear at the branching point and then evolve independently of each other (and of the other particles) according to the same laws as the initial particle. At each time $t$, we have a random field on $\mathbb Z_+$ consisting of particles present in the system at that moment. Functionals of this field $\sum_{(m_j, m_k)}\Phi (m_j, m_k)$ are considered, where the sum is taken over all ordered pairs $(m_j, m_k)$ of different particles in the field. The asymptotic behavior of the average value of this functional as $t \to +\infty$ is studied.