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Yuri Kondratiev

Publications and source records attributed to Yuri Kondratiev.

At least 91 records · Page 5Linked to original sources
arXiv2011

Glauber Dynamics in Continuum: A Constructive Approach to Evolution of States

The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro- and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.

math-ph↗
arXiv2011

Spectral gap for Glauber type dynamics for a special class of potentials

We consider an equilibrium birth and death type process for a particle system in infinite volume, the latter is described by the space of all locally finite point configurations on $\R^d$. These Glauber type dynamics are Markov processes constructed for pre-given reversible measures. A representation for the "carré du champ" and "second carré du champ" for the associate infinitesimal generators $L$ are calculated in infinite volume and a corresponding coercivity identity is derived. The latter is used to give explicit sufficient conditions for the appearance and bounds for the size of the spectral gap of $L$. These techniques are applied to Glauber dynamics associated to Gibbs measure and conditions are derived extending all previous known results. In the high temperature regime now potentials also with a non-trivial negative part can be treated. Furthermore, a special class of potentials is defined for which the size of the spectral gap is as least as large as for the free system and, surprisingly, is independent of the activity. This type of potentials should not show any phase transition for a given temperature at any activity.

math-ph↗
arXiv2011

Vlasov scaling for the Glauber dynamics in continuum

We consider Vlasov-type scaling for the Glauber dynamics in continuum with a positive integrable potential, and construct rescaled and limiting evolutions of correlation functions. Convergence to the limiting evolution for the positive density system in infinite volume is shown. Chaos preservation property of this evolution gives a possibility to derive a non-linear Vlasov-type equation for the particle density of the limiting system.

math-ph↗
arXiv2010

Gibbs measures of disordered lattice systems with unbounded spins

The Gibbs measures of a spin system on $Z^d$ with unbounded pair interactions $J_{xy} σ(x) σ(y)$ are studied. Here $\langle x, y \rangle \in E $, i.e. $x$ and $y$ are neighbors in $Z^d$. The intensities $J_{xy}$ and the spins $σ(x) , σ(y)$ are arbitrary real. To control their growth we introduce appropriate sets $J_q\subset R^E$ and $S_p\subset R^{Z^d}$ and prove that for every $J = (J_{xy}) \in J_q$: (a) the set of Gibbs measures $G_p(J)= \{μ: solves DLR, μ(S_p)=1\}$ is non-void and weakly compact; (b) each $μ\inG_p(J)$ obeys an integrability estimate, the same for all $μ$. Next we study the case where $J_q$ is equipped with a norm, with the Borel $σ$-field $B(J_q)$, and with a complete probability measure $ν$. We show that the set-valued map $J \mapsto G_p(J)$ is measurable and hence there exist measurable selections $J_q \ni J \mapsto μ(J) \in G_p(J)$, which are random Gibbs measures. We prove that the empirical distributions $N^{-1} \sum_{n=1}^N π_{Δ_n} (\cdot| J, ξ)$, obtained from the local conditional Gibbs measures $π_{Δ_n} (\cdot| J, ξ)$ and from exhausting sequences of $Δ_n \subset Z^d$, have $ν$-a.s. weak limits as $N\rightarrow +\infty$, which are random Gibbs measures. Similarly, we prove the existence of the $ν$-a.s. weak limits of the empirical metastates $N^{-1} \sum_{n=1}^N δ_{π_{Δ_n} (\cdot| J,ξ)}$, which are Aizenman-Wehr metastates. Finally, we prove the existence of the limiting thermodynamic pressure under some further conditions on $ν$.

math-ph↗
arXiv2010

An approximative approach to construction of the Glauber dynamics in continuum

We develop a new approach for the construction of the Glauber dynamics in continuum. Existence of the corresponding strongly continuous contraction semigroup in a proper Banach space is shown. Additionally we present the finite- and infinite-volume approximations of the semigroup by families of bounded linear operators.

math-ph↗
arXiv2010

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.

math-ph↗
arXiv2009

Gibbs random fields with unbounded spins on unbounded degree graphs

Gibbs random fields corresponding to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is non-void and weakly compact, and that they obey uniform exponential integrability estimates. In the second part of the paper, a class of graphs is described in which the mentioned summability is obtained as a consequence of a property, by virtue of which vertices of large degree are located at large distances from each other. The latter is a stronger version of a metric property, introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986). \textrm{Uniqueness of a Gibbs field with a random potential--an elementary approach.}\textit{Theory Probab. Appl.} {\bf 31} 572--589].

math.PR↗
arXiv2008

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.

math-ph↗
arXiv2008

Individual based model with competition in spatial ecology

We analyze an interacting particle system with a Markov evolution of birth-and-death type. We have shown that a local competition mechanism (realized via a density dependent mortality) leads to a globally regular behavior of the population in course of the stochastic evolution.

math-ph↗
arXiv2007

Phase Transitions and Quantum Stabilization in Quantum Anharmonic Crystals

A unified theory of phase transitions and quantum effects in quantum anharmonic crystals is presented. In its framework, the relationship between these two phenomena is analyzed. The theory is based on the representation of the model Gibbs states in terms of path measures (Euclidean Gibbs measures). It covers the case of crystals without translation invariance, as well as the case of asymmetric anharmonic potentials. The results obtained are compared with those known in the literature.

math-ph↗
arXiv2004

The semigroup of the Glauber dynamics of a continuous system of free particles

We study properties of the semigroup $(e^{-tH})_{t\ge 0}$ on the space $L^ 2(Γ_X,π)$, where $Γ_X$ is the configuration space over a locally compact second countable Hausdorff topological space $X$, $π$ is a Poisson measure on $Γ_X$, and $H$ is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov semigroup of kernels $(P_t)_{t\ge 0}$ and, using it, we prove the main results of the paper: the Feller property of the semigroup $(P_t)_{t\ge 0}$ with respect to the vague topology on the configuration space $Γ_X$, and the ergodic property of $(P_t)_{t\ge 0}$. Following an idea of D. Surgailis, we also give a direct construction of the Glauber dynamics of a continuous infinite system of free particles. The main point here is that this process can start in every $γ\inΓ_X$, will never leave $Γ_X$ and has cadlag sample paths in $Γ_X$.

math.PR↗
arXiv2003

Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $Γ$ of all locally finite subsets (configurations) in $\Bbb R$ and which has a Gibbs measure $μ$ as an invariant measure. We assume that $μ$ corresponds to a symmetric pair potential $ϕ(x-y)$. An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics--the so-called gradient stochastic dynamics, or interacting Brownian particles--has been investigated. By using the theory of Dirichlet forms, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal E}_μ^Γ$ on $L^2(Γ;μ)$, and under general conditions on the potential $ϕ$, prove its closability. For a potential $ϕ$ having a ``weak'' singularity at zero, we also write down an explicit form of the generator of ${\cal E}_μ^Γ$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\cal E}_μ^Γ$, there exists a diffusion process that is properly associated with it. Finally, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of ${\cal D}{:=}C_0^\infty({\Bbb R})$.

math.PR↗
arXiv2002

The heat semigroup on configuration spaces

In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural ``Riemannian-like'' structure of the configuration space $Γ_X$ over a complete, connected, oriented, and stochastically complete Riemannian manifold $X$ of infinite volume, the heat semigroup $(e^{-tH^Γ})_{t\in\R_+}$ was introduced and studied in [{\it J. Func. Anal.} {\bf 154} (1998), 444--500]. Here, $H^Γ$ is the Dirichlet operator of the Dirichlet form ${\cal E}^Γ$ over the space $L^2(Γ_X,π_m)$, where $π_m$ is the Poisson measure on $Γ_X$ with intensity $m$--the volume measure on $X$. We construct a metric space $Γ_\infty$ that is continuously embedded into $Γ_X$. Under some conditions on the manifold $X$ and we prove that $Γ_\infty$ is a set of full $π_m$ measure. The central results of the paper are two types of Feller properties for the heat semigroup. Next, we give a direct construction of the independent infinite particle process on the manifold $X$, which is a realization of the Brownian motion on the configuration space. The main point here is that we prove that this process can start in every $γ\inΓ_\infty$, will never leave $Γ_\infty$, and has continuous sample path in $Γ_\infty$, provided $\operatorname{dim}X\ge2$. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the $¶_{t,γ}(\cdot)$ above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case $\operatorname{dim}X=1$. Finally, as an easy consequence we get a ``path-wise'' construction of the independent particle process on $Γ_\infty$ from the underlying Brownian motion.

math.PR↗
arXiv1999

Differential Geometry on Compound Poisson Space

In this paper we carry out analysis and geometry for a class of infinite dimensional manifolds, namely, compound configuration spaces as a natural generalization of the work \cite{AKR97}. More precisely a differential geometry is constructed on the compound configuration space $Ω_{X}$ over a Riemannian manifold $X$. This geometry is obtained as a natural lifting of the Riemannian structure on $X$. In particular, the intrinsic gradient, divergence, and Laplace-Beltrami operator are constructed. Therefore the corresponding Dirichlet forms on $L^{2}(Ω_{X})$ can be defined. Each is shown to be associated with a diffusion process on $Ω_{X}$ (so called equilibrium process) which is nothing but the diffusion process on the simple configuration space $Γ_{X}$ together with corresponding marks. As another consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields on $X$ on compound Poisson space. Finally generalizations to the case when the compound Poisson measure is replaced by a marked Poisson measure easily follow from this construction.

math.FA↗
arXiv1999

On a relation between intrinsic and extrinsic Dirichlet forms for interacting particle systems

In this paper we extend the result obtained in \cite{AKR97} (see also \cite {AKR96}) on the representation of the intrinsic pre- Dirichlet form $\mathcal{E}_{π_σ}^Γ$ of the Poisson measure $π_σ$ in terms of the extrinsic one $\mathcal{E}_{π_σ,H_σ^{X}}^{P}$. More precisely, replacing $π_σ$ by a Gibbs measure $μ$ on the configuration space $Γ_{X}$ we derive a relation between the intrinsic pre-Dirichlet form $\mathcal{E}_μ^Γ$ of the measure $μ$ and the extrinsic one $\mathcal{E}_{μ, H_σ^{X}}^{P}$. As a consequence we prove the closability of $\mathcal{E}_μ^Γ$ on $L^{2}(Γ_{X},μ)$ under very general assumptions on the interaction potential of the Gibbs measures $μ$.

math.FA↗
arXiv1999

Generalized Appell Systems

We give a general approach to infinite dimensional non-Gaussian analysis which generalizes the work \cite{KSWY95}. For given measure we construct a family of biorthogonal systems. We study their properties and their Gel'fand triples that they generate. As an example we consider the measures of Poisson type.

math.FA↗
arXiv1999

Analysis on Poisson and Gamma spaces

We study the spaces of Poisson, compound Poisson and Gamma noises as special cases of a general approach to non-Gaussian white noise calculus, see \cite{KSS96}. We use a known unitary isomorphism between Poisson and compound Poisson spaces in order to transport analytic structures from Poisson space to compound Poisson space. Finally we study a Fock type structure of chaos decomposition on Gamma space.

math.FA↗