ИСТИНА

We study the actual problem of the population spread in a model of catalytic branching random walk (CBRW) on a multidimensional lattice Z^d. The main feature of CBRW is the presence of catalysts located at an arbitrarily finite set of fixed lattice points. Only at the presence of catalysts a particle moving on the lattice in CBRW may produce offspring. In other lattice points the particle performs a random walk without branching. All the new particles behave themselves as independent copies of the parent particle. The main attention is paid to the CBRW with wide conditions of heavy tails of the random walk jumps. Earlier such analysis was implemented for other models of spatial branching processes under assumption of light tails or heavy tails. Other aspects of CBRW with heavy tails were studied in recent paper by A.Rytova, E.Yarovaya (2021). Similar to ordinary branching processes CBRW can be classified as supercritical, critical or subcritical according to the relationship between its characteristics. Only in the supercritical regime both total and local numbers of particles grow unboundedly, as time t tends to infinity. This is the reason why the spread of population is natural to investigate in the supercritical case rather than in critical and subcritical ones. It turns out that different assumptions on the tails of the random walk jump in the CBRW lead to different asymptotic behavior of the population propagation front. So, whenever the tails of the random walk jump are light, i.e. the Cramer condition holds true, the population inhabits new area asymptotically linearly in time and in the time-limit all the population normalized by t forms a convex set in R^d, whose boundary is called the limiting shape of the front. We can observe another picture whenever the random walk jump has a semi-exponential distribution. In this case under additional assumptions the population in the CBRW spreads faster than linearly and the limiting shape of the front is no longer a convex set, but a star-shaped set. In the present talk we concentrate on the assumption that the random walk jump has heavy tails, namely, regularly varying ones. Moreover, we focus on three main cases. In the first one we deal with a one-dimensional lattice and the propagation front behavior is reduced to analysis of the maximum and the minimum of the particles positions. The second one means that the components of the random walk jump are close to be independent and each component has a regularly varying tail with its own parameter. In the third case we consider an isotropic random walk, when the rate of decay of the jump distribution in different directions is given by the same regularly varying function. The common trait of the corresponding results is that the population spreads exponentially-fast, as time t grows to infinity. The differences consist in the limiting shape of the front which can be more or less random contrary to the case of light tails and semi-exponential distribution of the random walk jump. Along with survey we develop the previous results to perform the complete picture of the CBRW propagation in space and time.