Catalytic branching random walk on three-dimensional lattice

E. V. Bulinskaya
Theory of Stochastic Processes
Vol.16 (32), no.2, 2010, pp.23-32
We consider a critical catalytic continuous time branching random walk on the integer lattice under the assumption that the birth and the death of particles occur at a single source of branching located at the origin. For the introduced joint generating function of the number of particles, the differential and integral equations are obtained in case of $\mathbb{Z}^{d}$ with arbitrary $d\in\mathbb{N}$. The limit value of the population survival probability as $t\rightarrow\infty$ is found as a function of the starting point $x\in\mathbb{Z}^{3}$. We establish the asymptotic behavior of the probability that the number of particles at the origin at time $t$ is positive. The Yaglom-type conditional limit theorem for the number of particles at the origin is proved. A joint conditional limit distribution of the number of particles at the source and the number of particles outside of it with finite lifetime is studied as well.