# Yu. G. Kondratiev

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Articles: 27

### Markov dynamics on the cone of discrete Radon measures

Methods Funct. Anal. Topology 27 (2021), no. 2, 173-191

We start with a brief overview of the known facts about the spaces of discrete Radon measures those may be considered as generalizations of configuration spaces. Then we study three Markov dynamics on the spaces of discrete Radon measures: analogues of the contact model, of the Bolker--Dieckmann--Law--Pacala model, and of the Glauber-type dynamics. We show how the results obtained previously for the configuration spaces can be modified for the case of the spaces of discrete Radon measures.

Стаття розпочинається з короткого огляду відомих фактів про простори дискретних мір Радона, які можуть розглядатися як узагальнення просторів конфігурацій. Далі розглядаються три марківські динаміки на просторах дискретних мір Радона: аналоги моделі контактів та моделі Болкера--Дікмана--Лоу--Пакали та аналог динаміки типу Глаубера. Показано як результати, отримані для просторів конфігурацій, можуть бути узагальнені для випадки просторів дискретних мір Радона.

### Representations of the Infinite-Dimensional Affine Group

Yuri Kondratiev

Methods Funct. Anal. Topology 26 (2020), no. 4, 348-355

We introduce an infinite-dimensional 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.

Вводиться нескінченновимірна аффінна група і будується її незвідне унітарне представлення. Наш підхід наслідує метод Вершика-Гельфанда-Граєва для групи дифеоморфізмів, з необхідними модифікаціями, пов’язаними з тим, що група не діє на фазовому просторі, але можна визначити її дію на деяких класах функцій.

### Green measures for Markov processes

Methods Funct. Anal. Topology 26 (2020), no. 3, 241-248

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.

Ми вивчаємо міри Ґріна для деяких класів марківських процесів. Зокрема для броунівського руху і стрибкових процесів. Міри Ґріна містять сингулярну і регулярну компоненти. Основна задача полягає в оцінці регулярної частини.

### Weak-coupling limit for ergodic environments

Methods Funct. Anal. Topology 25 (2019), no. 2, 118-133

The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [27]. Our approach is based on the study of forward Kolmogorov equations (a.k.a. Fokker-Planck equations). We describe a system evolving as a Markov process on the space of finite configurations, whereas its rates depend on the actual state of another (equilibrium) process on the space of locally finite configurations. We will show that ergodicity of the environment process implies the averaging principle for the solutions of the coupled Fokker-Planck equations.

### Automorphisms generated by umbral calculus on a nuclear space of entire test functions

Methods Funct. Anal. Topology 24 (2018), no. 4, 339-348

In this paper we show that Sheffer operators, mapping monomials to certain Sheffer polynomial sequences, such as falling and rising factorials, Charlier, and Hermite polynomials extend to continuous automorphisms on the space of entire functions of first order growth and minimal type.

### Fractional kinetics in a spatial ecology model

Methods Funct. Anal. Topology 24 (2018), no. 3, 275-287

In this paper we study the effect of subordination to the solution of a model of spatial ecology in terms of the evolution density. The asymptotic behavior of the subordinated solution for different rates of spatial propagation is studied. The difference between subordinated solutions to non-linear equations with classical time derivative and solutions to non-linear equation with fractional time derivative is discussed.

### Fractional statistical dynamics and fractional kinetics

Methods Funct. Anal. Topology 22 (2016), no. 3, 197-209

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.

### Fractional contact model in the continuum

Methods Funct. Anal. Topology 21 (2015), no. 2, 179–187

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.

### An operator approach to Vlasov scaling for some models of spatial ecology

Methods Funct. Anal. Topology 19 (2013), no. 2, 108-126

We consider Vlasov-type scaling for Markov evolution of birth-and-death type in continuum, which 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. The existence of rescaled and limiting evolutions of correlation functions and convergence to the limiting evolution are shown. The obtained results enable us to derive a non-linear Vlasov-type equation for the density of the limiting system.

### Kawasaki dynamics in the continuum via generating functionals evolution

Methods Funct. Anal. Topology 18 (2012), no. 1, 55-67

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.

### On two-component contact model in continuum with one independent component

Methods Funct. Anal. Topology 14 (2008), no. 3, 209-228

Properties of a contact process in continuum for a system of particles of two types, one which is independent of the other, are considered. We study dynamics of the first and the second order correlation functions, their asymptotics, and the dependence on parameters of the~system.

### Measures on configuration spaces defined by relative energies

Methods Funct. Anal. Topology 11 (2005), no. 2, 126-155

### Existence of Gibbs State for Non-Ideal Gas in $R^d$: the case of pair, long-range interaction

Methods Funct. Anal. Topology 10 (2004), no. 3, 33-43

### Correlation functionals for Gibbs measures and Ruelle bounds

Methods Funct. Anal. Topology 9 (2003), no. 1, 9-58

### Analytic aspects of Poissonian white noise analysis

Methods Funct. Anal. Topology 8 (2002), no. 4, 15-48

### Symmetric differential operators of the second order in Poisson spaces

Methods Funct. Anal. Topology 6 (2000), no. 4, 14-25

### On a spectral representation for correlation measures in configuration space analysis

Methods Funct. Anal. Topology 5 (1999), no. 4, 87-100

### Analysis and geometry on ${\mathbb R}_{+}$-marked configuration space

Methods Funct. Anal. Topology 5 (1999), no. 1, 29-64

### Marked Gibbs measures via cluster expansion

Methods Funct. Anal. Topology 4 (1998), no. 4, 50-81

### Some examples of Dirichlet operators associated with the actions of infinite dimensional Lie groups

Methods Funct. Anal. Topology 4 (1998), no. 2, 1-15

### Differential geometry on compound Poisson space

Methods Funct. Anal. Topology 4 (1998), no. 1, 32-58

### Generalized Appell systems

Methods Funct. Anal. Topology 3 (1997), no. 3, 28-61

### Complex Gaussian analysis and the Bargman-Segal space

Methods Funct. Anal. Topology 3 (1997), no. 2, 46-64

### Euclidean Gibbs states for quantum continuous systems with Boltzmann statistics via cluster expansion

Methods Funct. Anal. Topology 3 (1997), no. 1, 62-81

### Quantum hierarchical model

Methods Funct. Anal. Topology 2 (1996), no. 3, 1-35

### Biorthogonal systems in hypergroups: an extension of non-Gaussian analysis

Methods Funct. Anal. Topology 2 (1996), no. 2, 1-50

### Dirichlet operators semigroups in some Gibbs lattice spin systems

Methods Funct. Anal. Topology 1 (1995), no. 1, 3-27