Models of branching walks and their use in the reliability theory

Application of the branching walk models in the reliability theory was discussed. The results obtained for the models of a symmetric continuous-time branching random walk on $\mathbf Z^d$ with the source of particle birth and death at one of the lattice points were reviewed. Emphasis was made on the survival analysis and study of the branching walk properties depending on the source intensity. It was shown that if $d\ge3$, then under the supercritical branching process at the source the supercritical, critical and even subcritical branching random walk may arise on $\mathbf Z^d$. A classification relying on the asymptotic behavior of the number of particles at an arbitrary lattice point which specifies the phase transitions on lattice dimension for the critical and subcritical branching random walk was presented.