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Information × Registration Number 0225U003098, (0122U001730) , R & D reports Title Asymptotic properties of branching and evolutionary processes. popup.stage_title Послідовність узгоджених схем зайнятості, еволюційні процеси з імпульсним впливом у випадковому середовищі, випадкові vp-дерева. Head Iksanov Oleksandr М., Доктор фізико-математичних наук Registration Date 20-05-2025 Organization Taras Shevchenko National University of Kyiv popup.description1 Development of the latest methods that will solve open problems related to the asymptotic properties of several classes of branched and related processes, as well as evolutionary processes. popup.description2 The main goal of the work was to develop new methods that will allow solving open problems related to the asymptotic properties of several classes of branching and related processes, as well as evolutionary processes. The specific tasks that were set in the work included, in particular, the study of the convergence rate of the Nerman martingale and the derivative martingale associated with branching processes to their limits; asymptotic analysis of the sequence of coordinated employment schemes in deterministic and random environments; finding a probabilistic representation of solutions to kinetic-type evolutionary equations and their asymptotic analysis; analysis of random VP-trees; proving limit theorems for evolutionary processes with impulse influence, analysis of dynamic models of the spread of epidemics with impulse influence. Idea and main hypotheses. Our approach to analyzing the convergence rate of the Nerman martingale was to discretize it by moving to appropriate stopping lines, which allowed us to use the limit theorems for discrete-time martingales. To analyze the convergence rate of the derivative martingale, a new representation of the sublinear solutions of the Poisson equation on a half-line was found. This allowed us to obtain, under optimal assumptions, a binomial asymptotic expansion of the tail of the distribution of a random variable that is the limit for the derivative martingale. When working with sequences of coordinated employment patterns in a random environment, recent powerful results for branched random walks were used. To construct a probabilistic representation of solutions to kinetic-type evolutionary equations, we constructed a suitable labeled branched random walk in continuous time. The asymptotic behavior of these solutions was analyzed by studying the corresponding Biggins martingale in continuous time. The Harris chain technique was applied in combination with convex geometry methods to analyze random vp-trees. Product Description popup.authors Bohun Vladyslav A. Dovhai Bohdan V. Kotelnykova Valeriia H. Marynych Oleksandr V. Rashytov Bohdan S. Samoylenko Ihor V. Samoilenko Oleksandra I. Trebina Nataliia M. Usar Iryna Ya. popup.nrat_date 2025-05-20 Close