Finkelshtein D. Stochastic dynamics of continuous systems

The thesis is devoted to the investigation of stochastic dynamics of continuous systems - locally finite subsets of the Euclidean space. Birth-and-death and jump dynamics are considered. Systems of evolution equations for the correlation functions of all orders are derived for general intensities of birth, death, and jumps. There are found the sufficient conditions on the birth-and-death intensities under which the solutions to the corresponding evolution equations for the correlation functions exist and are unique in spaces of bounded functions on finite and infinite time intervals. There is studied a convergence of rescaled correlation function dynamics in the mean field limit, and there is considered the corresponding non-local kinetic equations. The existence for the evolution of probability measures in the non-equilibrium Glauber dynamics and in the dynamics of point pairs jumps is shown. There is constructed the limiting evolution of the equilibrium Kawasaki dynamics with Kac potential. There are developed mathematical tools for an extension of the obtained results for the case of continuous systems whose elements might belong to different types.