ИСТИНА

On the basis of branching processes there arise interesting probabilistic models of evolution of particles when the particles may not only give offspring but also move in a space. The classical examples are closely related branching random walk (BRW) and branching Brownian motion (BBM). Within last decades a lot of publications were devoted to study of the maximal displacement of BRW and BBM (see, e.g., Shi (2015), Mallein (2015), Shiozawa (2019) and Chen, He (2020)). The mechanism of branching and moving combination provides that particles move in a space and randomly produce offsprings scattered in the space according to a given point process. We are interested in an another mechanism, when particles move in a space and produce offsprings only in the presence of catalysts located at fixed points in the space. Such models are called catalytic. There are catalytic branching random walk (CBRW) with a single catalyst (see, e.g., Vatutin, Topchii, Yarovaya (2004)), with finitely many catalysts (see, e.g., Doering, Roberts (2013)) and with periodically located catalysts (see, e.g., Platonova, Ryadovkin (2020)), a branching Brownian motion with a catalyst at the origin (see, e.g., Bocharov (2020)) and their other variants (see, e.g., Khristolyubov, Yarovaya (2019)). The maximal displacement for a supercritical CBRW with finitely many catalysts was studied in Carmona, Hu (2014) in the case when particles move over an integer line $Z$. An important assumption there is also the Cram\'{e}r condition for the walk jump, i.e. the tails of the jump are considered to be light. In Bulinskaya (2018) we extend the results from Carmona, Hu (2014) to the case of lattice $Z^d$, $d\in N$, and investigate the population front -- a counterpart of the maximum in a multidimensional setting. We also solve the same problem under other conditions on the tail behavior of the jump. Namely, in Bulinskaya (2021) we assume that the jump has moderately heavy tails, i.e. has a semi-exponential distribution. In Bulinskaya (2019) and Bulinskaya (2021) we consider heavy tails of the jump separating the case of independent coordinates of the jump and the isotropic case. In these papers we establish strong and weak limit theorems describing the spread of the particle population. In each case we provide normalizing factors for the particle locations leading to existence of non-trivial limits and find the limit shape of the front. We employ various techniques, e.g., analysis of non-linear integral equations derived in our papers, ``many-to-one'' lemma, renewal theory, large deviations theory for random walks, coupling, convex analysis and others. This permits to get a complete picture of the spread of population of a supercritical CBRW on $Z^d$. Maximal displacement in non-supercritical CBRW is treated as well in Bulinskaya (2020).