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Diffusion approximation for equilibrium Kawasaki dynamics in continuum. Stochastic Process. (2008)

by Y G Kondratiev, O L Kutiviy, E W Lytvynov
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Markov evolutions and hierarchical equations in the continuum II. Multicomponent systems

by Dmitri L. Finkelshtein, Yuri G. Kondratiev, Maria João Oliveira - 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 ..."
Abstract - Cited by 30 (15 self) - Add to MetaCart
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;

P.T.: On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum. To appear

by E Lytvynov , P T Polara - In: Proceedings of the International Conference on Infinite Particle Systems, 8–11 October 2006, Kazimierz Dolny
"... We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the L 2 -norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Random. Oper. Stoch. Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics.
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...continuum (Glauber dynamics), i.e., dynamics where there is no motion of particles, but rather particles disappear (die) or appear (are born) at random, see, e.g., [1,6,8,11,13,14,19,23]; • hopping particles (Kawasaki dynamics), i.e., dynamics where each particle randomly hops over the space [13]. ∗The first named author acknowledges the financial support of the Royal Society 2007/R2 Conference Grant. c© E.Lytvynov, P.T.Polara 223 E.Lytvynov, P.T.Polara In order to profoundly understand these dynamics, it is important to see how they are related to each other. For example, in the recent paper [10], it was shown that a typical diffusion dynamics can be derived through a diffusive scaling limit of a corresponding Kawasaki dynamics. In [3], it was proved that a special Glauber dynamics can be derived through a scaling limit of Kawasaki dynamics. Furthermore, it was conjectured in [3] that such a result holds, in fact, for a wide class of birth-and-death dynamics (dynamics of hopping particles, respectively), which are indexed by a parameter s ∈ [0, 1]. (Note that the result of [3] corresponds to the choice of parameter s = 0.) The purpose of this work is to show that the conjecture of [3]...

Binary jumps in continuum. I. Equilibrium processes and their scaling limits

by Dmitri L. Finkelshtein, Yuri G. Kondratiev, Oleksandr V. Kutoviy, Eugene Lytvynov
"... 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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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 birth-and-death 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
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...ss on Γ in which particles hop over Rd so that, at each jump time, only one particle changes its position. For a study of equilibrium Kawasaki dynamics in continuum, we refer the reader to the papers =-=[5, 7, 9, 12, 14, 16]-=- and the references therein. In this paper, we will study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. In several cases, an equi...

A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes

by Guanhua Li, Eugene Lytvynov , 2010
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Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit

by Dmitri Finkelshtein, Yuri Kondratiev Oleks, R Kutoviy, Eugene Lytvynov , 2011
"... 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. We discuss a non-equilibrium dynamics of binary jumps. We prove the exis-tence of an evolut ..."
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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. We discuss a non-equilibrium dynamics of binary jumps. We prove the exis-tence of an evolution of correlation functions on a finite time interval. We also show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the particle density. Key words: continuous system, binary jumps, non-equilibrium dynamics, correlation

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by Dmitri L. Finkelshtein, Yuri G. Kondratiev, Oleksandr V. Kutoviy, Eugene Lytvynov
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...ss on Γ in which particles hop over Rd so that, at each jump time, only one particle changes its position. For a study of equilibrium Kawasaki dynamics in continuum, we refer the reader to the papers =-=[8, 10, 12, 15, 17, 19]-=- and the references therein. In this paper, we will study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. In several cases, an equi...

Free Kawasaki Dynamics in the Continuum

by L. Streit , 2009
"... The Ising model with spins ±1 on the sites of a lattice, in its interpretation as a “lattice gas ” is paradigmatic for models of discrete configurations where a particle is present resp. absent at the site. Sin flips are interpreted as birth resp. death of a particle at the site. Processes with inde ..."
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The Ising model with spins ±1 on the sites of a lattice, in its interpretation as a “lattice gas ” is paradigmatic for models of discrete configurations where a particle is present resp. absent at the site. Sin flips are interpreted as birth resp. death of a particle at the site. Processes with independent births and deaths
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