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Yuri G. Kondratiev

Publications and source records attributed to Yuri G. Kondratiev.

At least 19 recordsLinked to original sources
arXiv2023

Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems

An infinite particle system of independent jumping particles in infinite volume is considered. Their construction is recalled,further properties are derived, the relation with hierarchical equations, Poissonian analysis, and second quantization are discussed. The hydrodynamic limit for a general initial distribution satisfying a mixing condition is derived. The long time asymptotic is computed under an extra assumption. The relation with constructions based on infinite volume limits is discussed.

math.PR↗
arXiv2021

Cesaro Limits for Fractional Dynamics

We study the asymptotic behavior of random time changes of dynamical systems. As random time changes we propose three classes which exhibits different patterns of asymptotic decays. The subordination principle may be applied to study the asymptotic behavior of the random time dynamical systems. It turns out that for the special case of stable subordinators explicit expressions for the subordination are known and its asymptotic behavior are derived. For more general classes of random time changes explicit calculations are essentially more complicated and we reduce our study to the asymptotic behavior of the corresponding Cesaro limit.

math.PR↗
arXiv2020

Green Measures for Markov Processes

In this paper we study Green measures of certain classes of Markov processes. In particular Brownian motion and processes with jump generators with different tails. The Green measures are represented as a sum of a singular and a regular part given in terms of the jump generator. The main technical question is to find a bound for the regular part.

math.PR↗
arXiv2019

Heat Kernel Estimates for Fractional Heat Equation

We study the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.

math-ph↗
arXiv2015

Equilibrium diffusion on the cone of discrete Radon measures

Let $\mathbb K(\mathbb R^d)$ denote the cone of discrete Radon measures on $\mathbb R^d$. There is a natural differentiation on $\mathbb K(\mathbb R^d)$: for a differentiable function $F:\mathbb K(\mathbb R^d)\to\mathbb R$, one defines its gradient $\nabla^{\mathbb K} F $ as a vector field which assigns to each $η\in \mathbb K(\mathbb R^d)$ an element of a tangent space $T_η(\mathbb K(\mathbb R^d))$ to $\mathbb K(\mathbb R^d)$ at point $η$. Let $ϕ:\mathbb R^d\times\mathbb R^d\to\mathbb R$ be a potential of pair interaction, and let $μ$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $\mathbb R^d$. In particular, $μ$ is a probability measure on $\mathbb K(\mathbb R^d)$ such that the set of atoms of a discrete measure $η\in\mathbb K(\mathbb R^d)$ is $μ$-a.s.\ dense in $\mathbb R^d$. We consider the corresponding Dirichlet form $$ \mathscr E^{\mathbb K}(F,G)=\int_{\mathbb K(\mathbb R^d)}\langle\nabla^{\mathbb K} F(η), \nabla^{\mathbb K} G(η)\rangle_{T_η(\mathbb K)}\,dμ(η). $$ Integrating by parts with respect to the measure $μ$, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If $d\ge2$, there exists a conservative diffusion process on $\mathbb K(\mathbb R^d)$ which is properly associated with the Dirichlet form $\mathscr E^{\mathbb K}$.

math.PR↗
arXiv2014

Fractional contact model in the continuum

We consider the evolution of correlation functions in a non-Markov version of the contact model in the continuum. The memory effects are introduced by assuming the fractional evolution equation for the statistical dynamics. This leads to a behavior of time-dependent correlation functions, essentially different from the one known for the standard contact model.

math-ph↗
arXiv2011

Kawasaki dynamics in the continuum via generating functionals evolution

We construct the time evolution of Kawasaki 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, which leads to a local (in time) solution. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.

math-ph↗
arXiv2011

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

Let $Γ$ denote the space of all locally finite subsets (configurations) in $R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $Γ$ in which pairs of particles simultaneously hop over $R^d$. 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.

math.PR↗
arXiv2011

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.

math-ph↗
arXiv2007

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.

math-ph↗
arXiv2006

Equilibrium Glauber dynamics of continuous particle systems as a scaling limit of Kawasaki dynamics

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 scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we expect that, in the limit, only jumps of ``infinite length'' will survive, i.e., we expect to arrive at a Glauber dynamics in continuum (a birth-and-death process in $\mathbb{R}^d$). We prove that, in the low activity-high temperature regime, the generators of the Kawasaki dynamics converge to the generator of a Glauber dynamics. The convergence is on the set of exponential functions, in the $L^2(μ)$-norm. Furthermore, additionally assuming that the potential of pair interaction is positive, we prove the weak convergence of the finite-dimensional distributions of the processes.

math.PR↗
arXiv2005

A-priori Estimates and Existence for Quantum Gibbs States

We prove a priori estimates and, as sequel, existence of Euclidean Gibbs states for quantum lattice systems. For this purpose we develop a new analytical approach, the main tools of which are: first, a characterization of the Gibbs states in terms of their Radon-Nikodym derivatives under shift transformations as well as in terms of their logarithmic derivatives through integration by parts formulae, and second, the choice of appropriate Lyapunov functionals describing stabilization effects in the system. The latter technique becomes applicable since on the basis of the integration by parts formulae the Gibbs states are characterized as solutions of an infinite system of partial differential equations. Our existence result generalize essentially all previous ones. In particular, superquadratic growth of the interaction potentials is allowed and $N$-particle interactions for $N\in \mathbb{N}\cup \{\infty \}$ are included. We also develop abstract frames both for the necessary single spin space analysis and for the lattice analysis apart from their applications to our concrete models. Both types of general results obtained in these two frames should be also of their own interest in infinite dimensional analysis.

math.PR↗
arXiv2005

N/V-limit for Stochastic Dynamics in Continuous Particle Systems

We provide an $N/V$-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on $\mathbb R^d$, $d \ge 1$. Starting point is an $N$-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset $Λ\subset {\mathbb R}^d$ with finite volume (Lebesgue measure) $V = |Λ| < \infty$. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above $N$-particle dynamic in $Λ$ as $N \to \infty$ and $V \to \infty$ such that $N/V \to ρ$, where $ρ$ is the particle density.

math.PR↗
arXiv2003

Invariant measures for Glauber dynamics of continuous systems

We consider Glauber-type stochastic dynamics of continuous systems \cite{BCC02}, \cite{KL03}, a particular case of spatial birth-and-death processes. The dynamics is defined by a Markov generator in such a way that Gibbs measures of Ruelle type are symmetrizing, and hence invariant for the stochastic dynamics. In this work we show that the converse statement is also true. Namely, all invariant measures satisfying Ruelle bound condition are grand canonical Gibbsian for the potential defining the dynamics. The proof is based on the observation that the well-known Kirkwood-Salsburg equation for correlation functions is indeed an equilibrium equation for the stochastic dynamics.

math-ph↗
arXiv2002

Scaling limit of stochastic dynamics in classical continuous systems

We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on ${\mathbb R}^d, d \ge 1$. The aim is to derive macroscopic quantities from a given micro- or mesoscopic system. The scaling we consider has been investigated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}, and \cite{GP86}, under the assumption that the underlying potential is in $C^3_0$ and positive. We prove that the Dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the Dirichlet form of a generalized Ornstein--Uhlenbeck process. The proof is based on the analysis and geometry on the configuration space which was developed in \cite{AKR98a}, \cite{AKR98b}, and works for general Gibbs measures of Ruelle type. Hence, the underlying potential may have a singularity at the origin, only has to be bounded from below, and may not be compactly supported. Therefore, singular interactions of physical interest are covered, as e.g. the one given by the Lennard--Jones potential, which is studied in the theory of fluids. Furthermore, using the Lyons--Zheng decomposition we give a simple proof for the tightness of the scaled processes. We also prove that the corresponding generators, however, do not converge in the $L^2$-sense. This settles a conjecture formulated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}.

math.PR↗