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Equilibrium and non-equilibrium stochastic evolutions in continuum


The project is devoted to the study of long-time behavior of large complex systems with interaction between elements. The structure of large systems composed of many or infinite many elements with mutual interaction is ideal for modeling complexity of the real world. The main feature of large complex systems appears as a "collective" behavior of the whole system arising from strong correlations of elements of the systems. The emergence of collective behavior is certainly not restricted to the physical sciences. It is general in nature, from biology to social and economical sciences. For this reason, the study of large complex systems as real world models requires a truly interdisciplinary mind-set. In the project we plan to study equilibrium and non-equilibrium stochastic dynamics for interacting infinite particle systems in continuum. The stochastic dynamics is a Markov process on the configuration space. We will consider a number of birth-and-death (of the Glauber type) stochastic dynamics of continuous particle systems, with different birth and death rates.The main objectives of the project are the following: Objective 1. Equilibrium dynamics. For the equilibrium dynamics, we will study the spectral properties (spectral gap, invariant subspaces) of the generator of a birth and death stochastic dynamics for Gibbs field in the continuous space with a stable pair potential of a general form. Objective 2. Non-equilibrium dynamics. For the non-equilibrium dynamics, we will study some continuous models using statistical approach. We will construct state evolutions using corresponding hierarchical equations for correlation functions, and then we will analyze these equations in the corresponding Banach spaces. 2.1. We plan to study scaling limits of mathematical models for direct aggregation. We plan to construct a scaling limit of a conservative "anti" Kawasaki dynamics, where the jump rates are decreasing in high density regions, and then to study the corresponding dynamical system for the first correlation function of the scaling model and to find solutions describing an aggregation phenomenon. 2.2. We plan to study asymptotical properties of mathematical models for spatial birth-and-death stochastic evolution of mutating genotypes under selection rates and to describe asymptotical regimes for the density of the population in the corresponding system.General methods for the analysis of such systems are the combination of probabilistic methods, methods of stochastic analysis and stochastic processes, mathematical methods of statistical physics, methods of spectral analysis of operators describing evolutions of the systems. (AU)