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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 37 (16 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;
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 28 (6 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).
On nonequilibrium stochastic dynamics for interacting particle systems in continuum
 Journal of Functional Analysis
"... We propose a general scheme for the construction of Markov stochastic dynamics on configuration spaces in the continuum. An application to the Glaubertype dynamics with competitions is considered. ..."
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Cited by 23 (5 self)
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We propose a general scheme for the construction of Markov stochastic dynamics on configuration spaces in the continuum. An application to the Glaubertype dynamics with competitions is considered.
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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Cited by 7 (5 self)
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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.
Ergodicity of nonequilibrium Glauber dynamics in continuum
 J. Funct. Anal
"... We study asymptotic properties of the continuous Glauber dynamics with unbounded death and constant birth rates. In particular, an information about the spectrum location for the symbol of the Markov generator is obtained. The latter fact is used for the proof of the ergodicity of this process. We ..."
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Cited by 4 (1 self)
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We study asymptotic properties of the continuous Glauber dynamics with unbounded death and constant birth rates. In particular, an information about the spectrum location for the symbol of the Markov generator is obtained. The latter fact is used for the proof of the ergodicity of this process. We show that the speed of convergence to the equilibrium is exponential.
Vlasov scaling for stochastic dynamics of continuous systems
, 2010
"... We describe a general derivation scheme for the Vlasovtype 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 correlation functio ..."
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Cited by 3 (2 self)
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We describe a general derivation scheme for the Vlasovtype 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 correlation functions equations. Several examples of realization of the proposed approach in particular models are presented.
On convergence of generators of equilibrium dynamics of hopping particles to generator of a birthanddeath process in continuum
, 2007
"... ..."
Binary jumps in continuum. I. Equilibrium processes and their scaling limits
"... Let Γ denote the space of all locally finite subsets (configurations) in Rd. A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over Rd. In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure ..."
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Cited by 1 (1 self)
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Let Γ denote the space of all locally finite subsets (configurations) in Rd. A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over Rd. In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. The existence and uniqueness of the corresponding stochastic dynamics are shown. We next prove the main result of this paper: a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. We also study another scaling limit, which leads us to a spatial birthanddeath process in continuum. A remarkable property of the limiting dynamics is that its generator possesses a spectral gap, a property which is hopeless to expect from the initial dynamics of binary jumps. MSC: 60F99, 60J60, 60J75, 60K35