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E. Lytvynov

Publications and source records attributed to E. Lytvynov.

At least 19 recordsLinked to original sources
arXiv2014

Laplace operators in gamma analysis

Let $\mathbb K(\mathbb R^d)$ denote the cone of discrete Radon measures on $\mathbb R^d$. The gamma measure $\mathcal G$ is the probability measure on $\mathbb K(\mathbb R^d)$ which is a measure-valued Lévy process with intensity measure $s^{-1}e^{-s}\,ds$ on $(0,\infty)$. We study a class of Laplace-type operators in $L^2(\mathbb K(\mathbb R^d),\mathcal G)$. These operators are defined as generators of certain (local) Dirichlet forms. The main result of the papers is the essential self-adjointness of these operators on a set of `test' cylinder functions on $\mathbb K(\mathbb R^d)$.

math.PR↗
arXiv2010

Meixner class of non-commutative generalized stochastic processes with freely independent values II. The generating function

Let $T$ be an underlying space with a non-atomic measure $σ$ on it. In [{\it Comm.\ Math.\ Phys.}\ {\bf 292} (2009), 99--129] the Meixner class of non-commutative generalized stochastic processes with freely independent values, $ω=(ω(t))_{t\in T}$, was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions $Z=(Z(t))_{t\in T}$ such that $Z(t)$ commutes with $ω(s)$ for any $s,t\in T$. Then a generating function can be understood as $G(Z,ω)=\sum_{n=0}^\infty \int_{T^n}P^{(n)}(ω(t_1),...,ω(t_n))Z(t_1)...Z(t_n)σ(dt_1)...σ(dt_n)$, where $P^{(n)}(ω(t_1),...,ω(t_n))$ is (the kernel of the) $n$-th orthogonal polynomial. We derive an explicit form of $ G(Z,ω)$, which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators $\partial_t$, $t\in T$. In contrast to the classical case, we prove that the operators $\di_t$ related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality.

math.PR↗
arXiv2007

On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the $L^2$-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper. Stochastic Equations], which was proved for a special Glauber (Kawasaki, respectively) dynamics.

math.PR↗
arXiv2007

A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes

We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space $X$, for which certain fermion point processes are invariant. The Glauber dynamics is a birth-and-death process in $X$, while in the case of the Kawasaki dynamics interacting particles randomly hop over $X$. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.

math.PR↗
arXiv2007

Equilibrium Kawasaki dynamics of continuous particle systems

We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold $X$. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over $X$. We establish conditions on the {\it a priori} explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birth-and-death process), and the other leading to a diffusion dynamics of interacting particles (in particular, the gradient stochastic dynamics).

math.PR↗
arXiv2007

Non-equilibrium stochastic dynamics in continuum: The free case

We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process $M$ on a Riemannian manifold $X$. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in $X$ such that, with probability one, infinitely many particles will arrive at this set at some time $t>0$. We assume that $X$ has infinite volume and, for each $α\ge1$, we consider the set $Θ_α$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $α$-th power of the volume of the set. We find quite general conditions on the process $M$ which guarantee that the corresponding infinite particle process can start at each configuration from $Θ_α$, will never leave $Θ_α$, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in $X$), and free Kawasaki dynamics on the configuration space. We also show that if $X=\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.

math.PR↗
arXiv2006

Functional spaces and operators connected with some Lévy noises

We review some recent developments in white noise analysis and quantum probability. We pay a special attention to spaces of test and generalized functionals of some Lévy white noises, as well as as to the structure of quantum white noise on these spaces.

math.PR↗
arXiv2006

A note of spaces of test and generalized functions of Poisson white noise

The paper is devoted to construction and investigation of some riggings of the $L^2$-space of Poisson white noise. A particular attention is paid to the existence of a continuous version of a function from a test space, and to the property of an algebraic structure under pointwise multiplication of functions from a test space.

math.PR↗
arXiv2006

De Rham cohomology of configuration spaces with Poisson measure

The space $Γ_X$ of all locally finite configurations in a Riemannian manifold $X$ of infinite volume is considered. The deRham complex of square-integrable differential forms over $Γ_X$, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra generated by the $L^2$-cohomology of the underlying manifold $X$.

math.PR↗
arXiv2006

Operators of Gamma white noise analysis

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, dσ)$, and show how to include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We introduce in $\Gama(\Ha)$ operators $a(ξ)=\int_{\R^d} dx ξ(x)a(x)$, $ξ\in S$, with $a(x)=\dig_x+2\dig_x\di_x+1+\di_x +\dig_x\di_x\di_x$, where $\dig_x$ and $\di_x$ are the creation and annihilation operators at $x$. We show that $(a(ξ))_{ξ\in S}$ is a family of commuting selfadjoint operators in $\Gama(\Ha)$ and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary $I$ between $\Gama(\Ha)$ and the $L^2$-space $L^2(S',dμ_{\mathrm G})$, where $μ_{\mathrm G}$ is the measure of Gamma white noise with intensity $σ$. The image of $a(ξ)$ under $I$ is the operator of multiplication by $\la\cdot,ξ\ra$, so that $a(ξ)$'s are Gamma field operators. The Fock structure of the Gamma space determined by $I$ coincides with that discovered in {\bf [}{\it Infinite Dimensional Analysis, Quantum Probability and Related Topics} {\bf 1} (1998), 91--117{\bf ]}. We note that $I$ extends in a natural way the multiple stochastic integral (chaos) decomposition of the ``chaotic'' subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.

math.PR↗
arXiv2006

On a spectral representation for correlation measures in configuration space analysis

The paper is devoted to the study of configuration space analysis by using the projective spectral theorem. For a manifold $X$, let $Γ_X$, resp.\ $Γ_{X,0}$ denote the space of all, resp. finite configurations in $X$. The so-called $K$-transform, introduced by A. Lenard, maps functions on $Γ_{X,0}$ into functions on $Γ_{X}$ and its adjoint $K^*$ maps probability measures on $Γ_X$ into $σ$-finite measures on $Γ_{X,0}$. For a probability measure $μ$ on $Γ_X$, $ρ_μ:=K^*μ$ is called the correlation measure of $μ$. We consider the inverse problem of existence of a probability measure $μ$ whose correlation measure $ρ_μ$ is equal to a given measure $ρ$. We introduce an operation of $\star$-convolution of two functions on $Γ_{X,0}$ and suppose that the measure $ρ$ is $\star$-positive definite, which enables us to introduce the Hilbert space ${\cal H}_ρ$ of functions on $Γ_{X,0}$ with the scalar product $(G^{(1)},G^{(2)})_{{\cal H}_ρ}= \int_{Γ_{X,0}}(G^{(1)}\star\bar G{}^{(2)})(η) ρ(dη)$. Under a condition on the growth of the measure $ρ$ on the $n$-point configuration spaces, we construct the Fourier transform in generalized joint eigenvectors of some special family $A=(A_ϕ)_{ϕ\in\D}$, $\D:=C_0^\infty(X)$, of commuting selfadjoint operators in ${\cal H}_ρ$. We show that this Fourier transform is a unitary between ${\cal H}_ρ$ and the $L^2$-space $L^2(Γ_X,dμ)$, where $μ$ is the spectral measure of $A$. Moreover, this unitary coincides with the $K$-transform, while the measure $ρ$ is the correlation measure of $μ$.

math.PR↗
arXiv2004

The square of white noise as a Jacobi field

We identify the representation of the square of white noise obtained by L. Accardi, U. Franz and M. Skeide in [Comm. Math. Phys. 228 (2002), 123--150] with the Jacobi field of a Lévy process of Meixner's type.

math.PR↗
arXiv2003

Glauber dynamics of continuous particle systems

This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space $Γ$ of all locally finite subsets (configurations) in ${\Bbb R}^d$, we fix a Gibbs measure $μ$ corresponding to a general pair potential $ϕ$ and activity $z>0$. We consider a Dirichlet form $ \cal E$ on $L^2(Γ,μ)$ which corresponds to the generator $H$ of the Glauber dynamics. We prove the existence of a Markov process $\bf M$ on $Γ$ that is properly associated with $\cal E$. In the case of a positive potential $ϕ$ which satisfies $δ{:=}\int_{{\Bbb R}^d}(1-e^{-ϕ(x)}) z dx<1$, we also prove that the generator $H$ has a spectral gap $\ge1-δ$. Furthermore, for any pure Gibbs state $μ$, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result by L. Bertini, N. Cancrini, and F. Cesi.

math.PR↗
arXiv2002

Polynomials of Meixner's type in infinite dimensions-Jacobi fields and orthogonality measures

The classical polynomials of Meixner's type--Hermite, Charlier, Laguerre, Meixner, and Meixner--Pollaczek polynomials--are distinguished through a special form of their generating function, which involves the Laplace transform of their orthogonality measure. In this paper, we study analogs of the latter three classes of polynomials in infinite dimensions. We fix as an underlying space a (non-compact) Riemannian manifold $X$ and an intensity measure $σ$ on it. We consider a Jacobi field in the extended Fock space over $L^2(X;σ)$, whose field operator at a point $x\in X$ is of the form $\di_x^†+λ\di_x^†\di_x+\di_x+\di^†_x\di_x\di_x$, where $λ$ is a real parameter. Here, $\di_x$ and $\di_x^†$ are, respectively, the annihilation and creation operators at the point $x$. We then realize the field operators as multiplication operators in $L^2({\cal D}';μ_λ)$, where ${\cal D}'$ is the dual of ${\cal D}{:=}C_0^\infty(X)$, and $μ_λ$ is the spectral measure of the Jacobi field. We show that $μ_λ$ is a gamma measure for $|λ|=2$, a Pascal measure for $|λ|>2$, and a Meixner measure for $|λ|<2$. In all the cases, $μ_λ$ is a Lévy noise measure. The isomorphism between the extended Fock space and $L^2({\cal D}';μ_λ)$ is carried out by infinite-dimensional polynomials of Meixner's type. We find the generating function of these polynomials and using it, we study the action of the operators $\di_x$ and $\di_x^†$ in the functional realization.

math.CA↗
arXiv2002

Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density

The aim of this paper is to show that fermion and boson random point processes naturally appear from representations of CAR and CCR which correspond to gauge invariant generalized free states (also called quasi-free states). We consider particle density operators $ρ(x)$, $x\in\R^d$, in the representation of CAR describing an infinite free Fermi gas of finite density at both zero and finite temperature, and in the representation of CCR describing an infinite free Bose gas at finite temperature. We prove that the spectral measure of the smeared operators $ρ(f)=\int dx f(x)ρ(x)$ (i.e., the measure $μ$ which allows to realize the $ρ(f)$'s as multiplication operators by $\la\cdot,f\ra$ in $L^2(dμ)$) is a well-known fermion, resp. boson measure on the space of all locally finite configurations in $\R^d$.

math-ph↗
arXiv2002

Orthogonal decompositions for Lévy processes with an application to the gamma, Pacsal, and Meixner processes

It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists in representing any square-integrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of the present paper are to develop the three approaches in the case of a general ($\R$-valued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general case.

math.PR↗