This thesis is in three parts. All parts are motivated by a desire to gain a better understanding of models of the phenomenon of two-dimensional diffusion limited aggregation, henceforth DLA. The first part proves some generalisations of results relating to Hastings-Levitov with α = 0, henceforth HL(0), another two-dimensional growth process. The s...
Published in Journal of Mathematical Sciences
The paper deals with studying a connection between the Littlewood–Offord problem and estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of Arak’s results (1980) are given. They establish a relationship of the concentration function of the sum and arithmetic structure of sup...
Published in Journal of Mathematical Sciences
The Large Deviation Principle is proved for the conditional probabilities of moderate deviations of weighted empirical bootstrap measures with respect to a fixed empirical measure. Using this LDP for the problem of calculation of moderate deviation probabilities of differentiable statistical functionals, it is shown that the importance sampling bas...
Published in Archive for Rational Mechanics and Analysis
In this paper we study the BV regularity for solutions of certain variational problems in Optimal Transportation. We prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well and we also provide a precise estimate of its BV norm. Of particu...
Published in Doklady Mathematics
A general model of a catalytic branching process (CBP) with any number of catalysis centers in a discrete space is studied. The asymptotic (in time) behavior of the total number of particles and of the local particle numbers is investigated. The problems of finding the global and local extinction probabilities are solved. Necessary and sufficient c...
Published in Automation and Remote Control
We consider a sequence of Markov chains that weakly converge to a diffusion process. We assume that the trend contains a linearly growing component. The usual parametrix method does not apply since the trend is unbounded. We show how to modify the parametrix method in order to get local limit theorems in this case.
Published in Ukrainian Mathematical Journal
We study the limit behavior of a sequence of Markov processes whose distributions outside any neighborhood of a “singular” point are attracted to a certain probability law. In any neighborhood of this point, the limit behavior can be irregular. As an example of application of the general result, we consider a symmetric random walk with unit jumps p...
Published in Journal of Mathematical Sciences
We show that the weak convergence of point measures and (2 + ∈)-moment conditions imply hydrodynamic equations at the limit of infinitely many interacting molecules. The conditions are satisfied whenever the solutions of the classical equations for N interacting molecules obey uniform in N bounds. As an example, we show that this holds when the ini...
Published in Journal of Mathematical Sciences
We study approximation of functions in weighted classes with the help of weighted Liouville type operators generated by the mixed Fourier–Bessel transform. Bibliography: 13 titles.
Published in Doklady Mathematics
