Isotropic Multidimensional Catalytic Branching Random Walk with Regularly Varying Tails

Екатерина Владимировна Булинская

Результат исследования: Материалы конференцийтезисырецензирование


The study completes a series of the author's works devoted to the spread of particles population in supercritical catalytic branching random walk (CBRW) on a multidimensional lattice. The CBRW model describes the evolution of a system of particles combining their random movement with branching (reproduction and death) which only occurs at fixed points of the lattice. The set of such catalytic points is assumed to be finite and arbitrary. In the supercritical regime the size of population, initiated by a parent particle, increases exponentially with positive probability. The rate of the spread depends essentially on the distribution tails of the random walk jump. If the jump distribution has “light tails”, the “population front”, formed by the particles most distant from the origin, moves linearly in time and the limiting shape of the front is a convex surface. When the random walk jump has independent coordinates with a semiexponential distribution, the population spreads with a power rate in time and the limiting shape of the front is a star-shape nonconvex surface. So far, for regularly varying tails (“heavy” tails), we have considered the problem of scaled front propagation assuming independence of components of the random walk jump. Now, without this hypothesis, we examine an “isotropic” case, when the rate of decay of the jumps distribution in different directions is given by the same regularly varying function. We specify the probability that, for time going to infinity, the limiting random set formed by appropriately scaled positions of population particles belongs to a set B containing the origin with its neighborhood, in Rd .In contrast to the previous results, the random cloud of particles with normalized positions in the time limit will not concentrate on coordinate axes with probability one
Язык оригиналаанглийский
Число страниц7
СостояниеОпубликовано - 1 янв 2019
СобытиеThe 3rd BRICS Mathematics Conference - Иннополис, Российская Федерация
Продолжительность: 21 июл 201926 июл 2019


КонференцияThe 3rd BRICS Mathematics Conference
СтранаРоссийская Федерация

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