  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.
