Abstract
We discuss general concept of Markov statistical dynamics in the continuum. For a class of spatial birth-and-death models, we develop a perturbative technique for the construction of statistical dynamics. Particular examples of such systems are considered. For the case of Glauber type dynamics in the continuum we describe a Markov chain approximation approach that gives more detailed information about statistical evolution in this model.
Highlights
Dynamics of interacting particle systems appear in several areas of the complex systems theory
Note that Markov processes for continuous systems are considered in the stochastic analysis as dynamical point processes [43,44,46] and they appear even in the representation theory of big groups [10,11,12,13,14]
As an essential technical step, we consider related pre-dual evolution chains of equations on the so-called quasi-observables. As it will be shown in the paper, such hierarchical equations may be analyzed in the framework of semigroup theory with the use of powerful techniques of perturbation theory for the semigroup generators, etc
Summary
Dynamics of interacting particle systems appear in several areas of the complex systems theory. A central object now is an evolution of states of the system that will be defined by mean of the Fokker–Planck equation This evolution equation w.r.t. probability measures on (Rd ) may be reformulated as a hierarchical chain of equations for correlation functions of considered measures. As an essential technical step, we consider related pre-dual evolution chains of equations on the so-called quasi-observables As it will be shown in the paper, such hierarchical equations may be analyzed in the framework of semigroup theory with the use of powerful techniques of perturbation theory for the semigroup generators, etc. The solution to the hierarchical equation for correlation functions may be obtained as the limit of the corresponding object for the Markov chain dynamics This limiting evolution generates the state dynamics. This paper is based on a series of our previous works [26,28,29,30,34,53], but certain results and constructions are detailed and generalized, in particular, in more complete analysis of the dual dynamics on correlation functions
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