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Branching Random Walks with Immigration. Lyapunov Stability

We consider a continuous-time symmetric branching random walk on multidimensional lattices with immigration and infinite number of initial particles. We assume that at every lattice point a process of birth and death of particles is described by a Bienayme-Galton- Watson branching process. The assumption on immigration of particles is that a new particle can appear at every lattice point from the outside. The subject of the study is a limit distribution of the particle field on the lattice. The differential equations for correlation functions for the number of particles at an arbitrary time moment at the fixed points on the lattice are obtained. We study the asymptotic behaviour for the first two moments of a number of particles at every lattice point and Lyapunov stability of the process. The Lyapunov stability is considered under the assumptions that birth, death and immigration intensities depend on the lattice points.