Abstract
The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.
Highlights
In the statistical theory of large systems [2], the system states are described as probability measures on the corresponding phase space rather than pointwise, which is typical for the standard theory of dynamical systems
Note that there exists the scale of spaces Gαt ⊂ Gα such that Gt ∈ Gαt for t ∈ [0, T (α, α0)), to Theorem 3.6
We can start in Kα0 with any α0 ∈ R, and obtain that kt ∈ Kαt ⊂ Kα, for any α < α0
Summary
In the statistical theory of large systems [2], the system states are described as probability measures on the corresponding phase space rather than pointwise, which is typical for the standard theory of dynamical systems. We describe the Markov evolution of states in terms of the correlation functions on both microscopic, obtained from (1.5), and mesoscopic levels The latter will be done by means of a nonlinear (kinetic) equation obtained from (1.5) in the Vlasov scaling limit. In Theorem 3.6 we show that, for any α0 ∈ R and any α < α0, there exists T (α0, α) > 0 such that, for any t ∈ [0, T (α0, α)), there exists αt ∈ (α, α0) such that the problem (1.5) with k0 ∈ Kα0 has a classical solution kt ∈ Kαt being the correlation function of a certain μt The latter fact is obtained by means of the corresponding result of [3].
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