Use of spectral methods to study branching processes with diffusion in a noncompact phase space

The results of investigation of branching processes with diffusion in a noncompact phase space are qualitatively different from the compact case. The reason for this is that the spectrum of the generating operator that describes the evolution of the mean density of particles can be continuous. A random walk on $\mathbf{Z}^d$ with one branching point is considered in the paper. Asymptotic expressions are obtained for the moments of the number of particles at an arbitrary point $x\in\mathbf{Z}^d$ as $t\to\infty$, and a limit theorem for supercritical processes is obtained. The asymptotic behavior of the mathematical expectation of the total number of particles on $\mathbf{Z}^d$ as $t\to\infty$ is investigated.