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Papers by Yuri Kondratiev

Fractal and Fractional, 2021

We consider random time changes in Markov processes with killing potentials. We study how random ... more We consider random time changes in Markov processes with killing potentials. We study how random time changes may be introduced in these Markov processes with killing potential and how these changes may influence their time behavior. As applications, we study the parabolic Anderson problem, the non-local Schrödinger operators as well as the generalized Anderson problem.

ukpmc.ac.uk

We introduce a definition of Euclidean Gibbs states corresponding to continuous systemsof quantum... more We introduce a definition of Euclidean Gibbs states corresponding to continuous systemsof quantum particles with Boltzmann statistics. The interaction is described by a stable pair potentialv (x Gamma x0). Assuming that v is an absolutely integrable function, we show ...

Spectral Methods in Infinite-Dimensional Analysis, 1995

ABSTRACT

ABSTRACT We construct the distribution of the infinite dimensional Markov process associated with... more ABSTRACT We construct the distribution of the infinite dimensional Markov process associated with a finite temperature Gibbs state for a quatum mechanical anharmonic crystal. The state is constructed via a cluster expanson technique for an arbitrary fixed temperature and, correspondingly, small enough masses of particles. KEY WORDS: quantum Gibbs state, lattice model, unbounded spin, small mass, cluster expansion AMS Subject Classification: Primary: 60H30. Secondary: 82B31 Fakultat fur Mathematik, Ruhr--Universitat, D 44780 Bochum (Germany). y BiBoS Research Center, D 33615 Bielefeld (Germany) and Institute of Mathematics, 252601 Kiev (Ukraine). z Institute of Information Transmission Problems, Moscow, (Russia). x Institute of Mathematics, 252601 Kiev, (Ukraine). Contents 1 Introduction 2 2 Description of the system and main result 3 3 Cluster expansion. Proof of Theorem 2.2 8 4 Convergence of the cluster expansion 11 1 Introduction The small mass dependence (or "strong quantumness"...

Journal of Functional Analysis, 2007

Functional Analysis and Its Applications, 1995

UDC 517.515 In recent years two approaches to the generalization of the Gaussian infinite-dimensi... more UDC 517.515 In recent years two approaches to the generalization of the Gaussian infinite-dimensional analysis (white noise analysis) to non-Gaussian measures have appeared: one of them is based on spectral theory for families of commuting self-adjoint operators [1, 2] and the other proceeds from biorthogonal expansions [3, 4]. In this note we show that in the one-dimensional model case the second approach can be extensively generalized if the characters of an Ll-hypergroup are used in its construction instead of exponential functions. For the properties of hypergroups applied below see [5-7] (here we use the term "Ll-hypergroup" instead of "hypercomplex system with locally compact base" according to [7]). 1. Consider a commutative Ll-hypergroup H I with locally compact basis Q, nonnegative structure measure, and multiplicative measure din(x) (x e Q); H I is assumed to be normal (Q 9 x ~-~ x* • Q is

Fractional Calculus and Applied Analysis

We give an overview of the concept of random time changes in evolution processes. First of all, w... more We give an overview of the concept of random time changes in evolution processes. First of all, we discuss random times in Markov processes. Secondly, we propose to use the concept of random times for dynamical systems. In both cases did appear fractional evolution equations. In the case of Markov processes we arrive to fractional Kolmogorov equations. For dynamical systems it leads to fractional Liouville equations.

Мiждисциплiнарнi дослiдження складних систем, 2021

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classica... more We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol (m)_n can be extended from a natural number m∈N to the falling factorials (z)_n=z(z-1) (z-n+1) of an argument z from F=R or C, and Stirling numbers of the first and second kinds are the coefficients of the expansions of (z)_n through z^k, k≤ n and vice versa. When taking into account spatial positions of elements in a locally compact Polish space X, we replace N by the space of configurations—discrete Radon measures γ=∑_iδ_x_i on X, where δ_x_i is the Dirac measure with mass at x_i.The spatial falling factorials (γ)_n:=∑_i_1∑_i_2 i_1∑_i_n i_1,..., i_n i_n-1δ_(x_i_1,x_i_2,...,x_i_n) can be naturally extended to mappings M^(1)(X)∋ω (ω)_n∈ M^(n)(X), where M^(n)(X) denotes the space of F-valued, symmetric (for n>2) Radon measures on X^n. There is a natural duality between M^(n)(X) and the space CF^(n)(X) of F-valued, symmetric continuous functions on...

We introduce an infinite-dimensional p-adic affine group and construct its irreducible unitary re... more We introduce an infinite-dimensional p-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.

In this paper we investigate the long time behavior of solutions to fractional in time evolution ... more In this paper we investigate the long time behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The class of subordinators for which asymptotic analysis may be realized is described.

General birth-and-death as well as hopping stochastic dynamics of infinite multicomponent particl... more General birth-and-death as well as hopping stochastic dynamics of infinite multicomponent particle systems in the continuum are considered. We derive the corresponding evolution equations for quasi-observables and correlation functions. We also present sufficient conditions that allows us to consider these equations on suitable Banach spaces.

$Research paper thumbnail of Fractional statistical dynamics and fractional kinetics$

We apply the subordination principle to construct kinetic fractional statistical dynamics in the ... more We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.

We consider general convolutional derivatives and related fractional statistical dynamics of cont... more We consider general convolutional derivatives and related fractional statistical dynamics of continuous interacting particle systems. We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. Conditions for the intermittency property of fractional kinetic dynamics are obtained.

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

Let Γ denote the space of all locally finite subsets (configurations) in R^d. A stochastic dynami... more 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.

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in ... more A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in 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 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.