Yu. G. Kondratiev
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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.
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
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.
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. . 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.
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.
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.
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.
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.
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.
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.
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.
Methods Funct. Anal. Topology 11 (2005), no. 2, 126-155
Methods Funct. Anal. Topology 10 (2004), no. 3, 33-43
Methods Funct. Anal. Topology 9 (2003), no. 1, 9-58
Methods Funct. Anal. Topology 8 (2002), no. 4, 15-48
Methods Funct. Anal. Topology 6 (2000), no. 4, 14-25
Methods Funct. Anal. Topology 5 (1999), no. 4, 87-100
Methods Funct. Anal. Topology 5 (1999), no. 1, 29-64
Methods Funct. Anal. Topology 4 (1998), no. 4, 50-81
Methods Funct. Anal. Topology 4 (1998), no. 2, 1-15
Methods Funct. Anal. Topology 4 (1998), no. 1, 32-58
Methods Funct. Anal. Topology 3 (1997), no. 3, 28-61
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
Methods Funct. Anal. Topology 2 (1996), no. 3, 1-35
Methods Funct. Anal. Topology 2 (1996), no. 2, 1-50
Methods Funct. Anal. Topology 1 (1995), no. 1, 3-27