Working paper
Intrinsic scales for highdimensional Levydriven models with nonMarkovian synchronizing updates
We consider Ncomponent synchronization models defined in terms of stochastic particle systems with special interaction. For general (nonsymmetric) Markov models we discuss phenomenon of the long time stochastic synchronization. We study behavior of the system in different limit situations related to appropriate changes of variables and scalings. For N = 2 limit distributions are found explicitly.
In this paper, we introduce a principally new method for modelling the dependence structure between two L{\'e}vy processes. The proposed method is based on some special properties of the timechanged Levy processes and can be viewed as an reasonable alternative to the copula approach.

We consider a basic stochastic particle system consisting of N identical particles with isotropic kparticle synchronization, ${k\ge 2}$. In the limit when both the number of particles N and the time $t=t(N)$ grow to infinity we study an asymptotic behavior of a coordinate spread of the particle system. We describe three time stages of $t(N)$ for which a qualitative behavior of the system is completely different. Moreover, we discuss the case when a spread of the initial configuration depends on N and increases to infinity as $N\rightarrow\infty$.
We study the Markov exclusion process for a particle system with a local interaction in the integer strip. This process models the exchange of velocities and particlehole exchange of the liquid molecules. It is shown that the mean velocity profile corresponds to the behavior which is characteristic for incompressible viscous liquid. We prove the existence of phase transition between laminar and turbulent profiles.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnitedimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasisolutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasisolutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasisolutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasisolutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents stateofthe art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.