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Markov evolutions and hierarchical equations in the continuum II. Multicomponent systems
- In preparation
, 2007
"... General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Ma ..."
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Cited by 30 (15 self)
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General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications. Keywords: Birth-and-death process; Hopping particles; Continuous system;
Vlasov scaling for stochastic dynamics of continuous systems
, 2010
"... We describe a general derivation scheme for the Vlasov-type equations for Markov evolutions of particle systems in continuum. This scheme is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correla-tion functio ..."
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Cited by 6 (6 self)
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We describe a general derivation scheme for the Vlasov-type equations for Markov evolutions of particle systems in continuum. This scheme is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correla-tion functions equations. Several examples of realization of the proposed approach in particular models are presented.
Glauber Dynamics in Continuum: A Constructive Approach to Evolution of States∗
"... The evolutions of states is described corresponding to the Glauber dy-namics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The mi-croscopic description is based on solving linear equations for correlation functions by ..."
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Cited by 5 (5 self)
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The evolutions of states is described corresponding to the Glauber dy-namics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The mi-croscopic 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 per-formed 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 inhomoge-neous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type. 1
Semigroup approach to birth-and-death stochastic dynamics in continuum
, 2011
"... We describe a general approach to the construction of a state evolution corresponding to the Markov generator of a spatial birth-and-death dynamics in Rd. We present conditions on the birth-and-death intensities which are sufficient for the existence of an evolution as a strongly continuous semigrou ..."
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Cited by 3 (1 self)
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We describe a general approach to the construction of a state evolution corresponding to the Markov generator of a spatial birth-and-death dynamics in Rd. We present conditions on the birth-and-death intensities which are sufficient for the existence of an evolution as a strongly continuous semigroup in a proper Banach space of correlation functions satisfying the Ruelle bound. The convergence of a Vlasov-type scaling for the corresponding stochastic dynamics is considered.
