ИСТИНА

The model of a branching random walk (BRW) with periodic branching sources was studied in a series of papers by M.V.Platonova and K.S.Ryadovkin (see, e.g., [1]). It is a natural extension of BRW with single or several sources of branching. The main feature of these models is a combination of a random walk of a particle on an integer lattice Z^d and its splitting at a source of branching. In our present work the number of the branching sources is infinite and they are located periodically on the lattice. Earlier the propagation of the particles population in supercritical BRW with finite number of sources was investigated (see, e.g., [2]). It was proved in [3] that almost surely the particles population spreads asymptotically linearly in time under condition of light tails of the random walk jump. Here our goal is to obtain the similar results for supercritical BRW with periodic branching sources and light tails of the random walk jump. The proposed approach involves certain results for a space-homogeneous BRW ([4]) where the Crump-Mode-Jagers branching processes were employed and a point process described the locations of the offspring. References [1] M.V.Platonova, and K.S.Ryadovkin. Branching random walks on Zd with periodic branching sources. Theory Probab. Appl., 64(2), 2019, pp.229-248. [2] S.Molchanov, and E.Yarovaya. Branching processes with lattice spatial dynamics and a finite set of particle generation centers. Dokl. Math., 86(2), 2012, pp.638-641. [3] E.Vl.Bulinskaya. Spread of a catalytic branching random walk on a multidimensional lattice. Stoch. Proc. Appl., 128(7), 2018, pp.2325-2340. [4] J.D.Biggins. How fast does a general branching random walk spread? In: K.B.Athreya, P.Jagers (Eds.), Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, vol.84. Springer, New York, 1997, pp.19-40.