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Markov evolutions and hierarchical equations in the continuum II. Multicomponent systems
 In preparation
, 2007
"... General birthanddeath 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 4 (1 self)
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General birthanddeath 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: Birthanddeath process; Hopping particles; Continuous system;
M.: Equilibrium Kawasaki dynamics of continuous particle systems
 Infin. Dimens. Anal. Quantum Probab. Relat. Top
, 2007
"... We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a prior ..."
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Cited by 3 (2 self)
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We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold X. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over X. We establish conditions on the a priori explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birthanddeath process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).
Diffusion approximation for equilibrium Kawasaki dynamics in continuum
, 2007
"... A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a diffusive limit of such a dynamics, derive ..."
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A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in R d which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure µ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, φ, (in particular, admitting a singularity of φ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of the gradient stochastic dynamics. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finitedimensional distributions of the corresponding equilibrium processes. In particular, if the potential φ is from C 3 b (Rd) and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi et al., J. Math.
unknown title
, 709
"... On convergence of generators of equilibrium dynamics of hopping particles to generator of a birthanddeath process in continuum E. Lytvynov and P.T. Polara ..."
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On convergence of generators of equilibrium dynamics of hopping particles to generator of a birthanddeath process in continuum E. Lytvynov and P.T. Polara