Limit theorem for non homogeneous by space random walks with branching of particles

We consider a symmetric, irreducible, continuous-time random walk (a Markov process) on the lattice $\mathbb{Z}^d$, $d\in \mathbb{N}$, with the possibility of particle branching at any lattice point. The evolution of the process starts from a single particle. Unlike previous works of the authors, the proof of the limit theorem on mean squared convergence of the normalized number of particles at an arbitrary fixed point of the lattice (at $t\rightarrow\infty$) fixed point of the lattice (at $t\rightarrow\infty$) is carried out without an additional assumption on spatial homogeneity of the random walk.