ИСТИНА

Recall that the model of a catalytic branching random walk (CBRW) on an integer lattice combines two mechanisms: particles movement and their splitting in the presence of catalysts located at fixed points of the lattice. In Carmona, Hu (2014) the authors study the maximum M_t of a supercritical CBRW on an integer line at time t. They prove that M_t grows almost surely linearly with a certain rate μ and also analyze the fluctuations M_t – μt, as time tends to infinity. We discuss extension of the results in Carmona, Hu (2014) to the case of multidimensional lattice. In contrast to the one-dimensional case, instead of M_t we consider the propagation front of the particle population in CBRW. Moreover, our approach covers models with an arbitrary finite number of catalysts, whereas Carmona, Hu (2014) mostly allows only the single catalyst. In this regard we employ the results published in papers Bulinskaya (2018) and Bulinskaya (2020). Beside the asymptotic analysis of solution to the derived system of non-linear integral equations, the proofs involve renewal theorems for systems of renewal equations, many-to-one formula, martingale change of measure, large deviation theory, convex analysis and the coupling method. Some new problems of the population evolution are tackled as well.