Results 1 
7 of
7
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 ..."
Abstract

Cited by 30 (15 self)
 Add to MetaCart
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;
P.T.: On convergence of generators of equilibrium dynamics of hopping particles to generator of a birthanddeath process in continuum. To appear
 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 birthanddeath process in continuum (or Glauber dynamics), ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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 birthanddeath 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 activityhigh temperature regime. Our result generalizes that of [Random. Oper. Stoch. Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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
A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes
, 2010
"... ..."
Binary jumps in continuum. II. Nonequilibrium process and a Vlasovtype scaling limit
, 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 nonequilibrium dynamics of binary jumps. We prove the existence of an evolut ..."
Abstract
 Add to MetaCart
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 nonequilibrium dynamics of binary jumps. We prove the existence of an evolution of correlation functions on a finite time interval. We also show that a Vlasovtype mesoscopic scaling for such a dynamics leads to a generalized Boltzmann nonlinear equation for the particle density. Key words: continuous system, binary jumps, nonequilibrium dynamics, correlation
Free Kawasaki Dynamics in the Continuum
, 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 ..."
Abstract
 Add to MetaCart
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