TY - JOUR

T1 - Isotropic Multidimensional Catalytic Branching Random Walk with Regularly Varying Tails

AU - Булинская, Екатерина Владимировна

N1 - Bulinskaya E. Vl. Isotropic multidimensional catalytic branching random walk with regularly varying tails // Компьютерные исследования и моделирование. - 2019. - Т. 11. - № 6. - С. 1033–1039. Публикация для корректировки.

PY - 2019

Y1 - 2019

N2 - 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

AB - 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

KW - Catalytic branching random walk

KW - Spread of population

UR - http://www.scopus.com/inward/record.url?scp=85081955704&partnerID=8YFLogxK

U2 - 10.20537/2076-7633-2019-11-6-1033-1039

DO - 10.20537/2076-7633-2019-11-6-1033-1039

M3 - Article

VL - 11

SP - 1033

EP - 1039

JO - Computer Research and Modeling

JF - Computer Research and Modeling

SN - 2076-7633

IS - 6

T2 - The 3rd BRICS Mathematics Conference

Y2 - 21 July 2019 through 26 July 2019

ER -