ИСТИНА

A multitude of publications devoted to different models of branching random walk have appeared within the last decade. Any model of branching random walk (BRW) comprises two random mechanisms. The first one accounts for splitting and death of particles and the second one for their random moving in space. Various combinations of the two arrangements lead to different models of BRW. These models are not only of theoretical interest but also have numerous applications in biology, chemical kinetics, statistical physics, homopolymers theory, queueing theory, etc. A special place in the theory of BRW is occupied by catalytic models. The term “catalytic” means that there are several points of the space where the catalysts are located and just these catalysts make a particle reaching such a location either split or die there. At a point without a catalyst a particle may walk only. Accordingly, the points containing catalysts are called points of catalysis or sources of branching and death of particles. The case of a single source of branching was investigated in many papers. The description of the spread of particles population in catalytic BRW with an arbitrary finite number of branching sources on Zd was initiated in papers S.A.Molchanov, E.B.Yarovaya (2012) and Ph.Carmona, Y.Hu (2014) and the study was accomplished in a series of papers by the author. The case of an infinite set of branching sources having periodical structure, i.e. BRW on periodic graphs, was treated for the first time in M.V.Platonova, K.S.Ryadovkin (2018, 2019). Those papers describe the asymptotic behavior, with respect to growing time, of means of some functionals related to BRW on periodic graphs. In the present talk, in contrast to M.V.Platonova, K.S.Ryadovkin (2018, 2019), we consider the spread almost sure of particles population in BRW on periodic graphs and study the asymptotic behavior (as time t goes to infinity) of the normalized cloud of particles existing at time t. We stipulate that the regime of branching is supercritical and the jumps of a random walk have light tails. Under these assumptions the instant positions of all the particles at time t are normalized by factor 1/t before letting t → ∞. The corresponding set of the normalized particles positions at time t is denoted by P_t. We establish that ∆(P_t,P) → 0 a.s. on event S of population non-degeneracy, as t → ∞, where ∆(D,F) is the Hausdorff distance between sets D and F belonging to Rd. The limiting set P⊂Rd is called the asymptotic shape of the BRW. We prove that, in the mentioned case of periodical structure, the arising set P is compact and convex. Moreover, we also provide an explicit formula to describe it in terms of the level sets of the Perron roots of a family of some parametric quasi-nonnegative matrices. Each element of such a matrix is the Laplace transform of the intensity measure of a specified point process. This means that the particles population spreads asymptotically linearly in time and the shape of random cloud of particles is approximated by tP as t → ∞. Our main result also shows that, with probability one, the cloud of normalized particles not only becomes asymptotically close to the boundary of P but fills the whole set P. The assumption of supercriticality guarantees the positive probability of the event S of the population survival. On the opposite event the problem of the rate of the population spread is meaningless because of the population degeneracy. The assumption of light tails of the random walk jumps leads to the indicated normalizing factor. We also tackl other conditions concerning the behavior of random walk tails (e.g., the tails are regularly varying or the jump has a semi-exponential distribution). Such conditions, effecting the population spread, were employed by us for BRW with finitely many sources of branching. Note that in papers M.V.Platonova, K.S.Ryadovkin (2018, 2019) the authors apply spectral theory of operators to analyze BRW on periodic graphs. For this reason they additionally assume the symmetry of the random walk. However, we use other methods based on consideration of our BRW in the scope of a general BRW with finitely many types of particles. This approach allows to avoid additional assumption on symmetry of the random walk and permits to consider the random walk having, for example, a drift. Our proof introduces specified auxiliary stochastic process and essentially relies on paper Biggins (1997) devoted to general BRW.