Semigroup approach to birth-and-death stochastic dynamics in continuum

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 $\mathbb{R}^d$. 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.Comment: 35 page

Markov evolutions and hierarchical equations in the continuum I. One-component systems

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.Comment: 47 page

Vlasov scaling for stochastic dynamics of continuous systems

We describe a general scheme of derivation of 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 correlation functions equations. Several examples of the realization of the proposed approach in particular models are presented.Comment: 23 page

Regulation mechanisms in spatial stochastic development models

The aim of this paper is to analyze different regulation mechanisms in spatial continuous stochastic development models. We describe the density behavior for models with global mortality and local establishment rates. We prove that the local self-regulation via a competition mechanism (density dependent mortality) may suppress a unbounded growth of the averaged density if the competition kernel is superstable.Comment: 19 page

Glauber dynamics in the continuum via generating functionals evolution

We construct the time evolution for states of Glauber dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, leading to a local (in time) solution which, under certain initial conditions, might be extended to a global one. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.Comment: 24 page

Diffusion approximation for equilibrium Kawasaki dynamics in continuum

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb R^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $\mu$ 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, $\phi$, (in particular, admitting a singularity of $\phi$ 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 finite-dimensional distributions of the corresponding equilibrium processes. In particular, if the potential $\phi$ is from $C_{\mathrm b}^3(\mathbb R^d)$ and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi {\it et al.}, J. Math. Phys. 39 (1998) 6509--6536]

Polynomials of Meixner's type in infinite dimensions-Jacobi fields and orthogonality measures

The classical polynomials of Meixner's type--Hermite, Charlier, Laguerre, Meixner, and Meixner--Pollaczek polynomials--are distinguished through a special form of their generating function, which involves the Laplace transform of their orthogonality measure. In this paper, we study analogs of the latter three classes of polynomials in infinite dimensions. We fix as an underlying space a (non-compact) Riemannian manifold $X$ and an intensity measure $\sigma$ on it. We consider a Jacobi field in the extended Fock space over $L^2(X;\sigma)$, whose field operator at a point $x\in X$ is of the form \di_x^\dag+\lambda\di_x^\dag \di_x+\di_x+\di^\dag_x\di_x\di_x, where $\lambda$ is a real parameter. Here, \di_x and \di_x^\dag are, respectively, the annihilation and creation operators at the point $x$. We then realize the field operators as multiplication operators in $L^2({\cal D}';\mu_\lambda)$, where ${\cal D}'$ is the dual of ${\cal D}{:=}C_0^\infty(X)$, and $\mu_\lambda$ is the spectral measure of the Jacobi field. We show that $\mu_\lambda$ is a gamma measure for $|\lambda|=2$, a Pascal measure for $|\lambda|>2$, and a Meixner measure for $|\lambda|<2$. In all the cases, $\mu_\lambda$ is a L\'evy noise measure. The isomorphism between the extended Fock space and $L^2({\cal D}';\mu_\lambda)$ is carried out by infinite-dimensional polynomials of Meixner's type. We find the generating function of these polynomials and using it, we study the action of the operators \di_x and \di_x^\dag in the functional realization