Eugene Lytvynov

Swansea University | SWAN · Department of Mathematics

Let X be an underlying space with a reference measure \(\sigma \). Let K be an integral operator in \(L^2(X,\sigma )\) with integral kernel K(x, y). A point process \(\mu \) on X is called determinantal with the correlation operator K if the correlation functions of \(\mu \) are given by \(k^{(n)}(x_1,\dots ,x_n)={\text {det}}[K(x_i,x_j)]_{i,j=1,\d...

Multicomponent commutations relations (MCRs) describe plektons, i.e. multicomponent quantum systems with a generalized statistics. In such systems, exchange of quasiparticles is governed by a unitary matrix [Formula: see text] that depends on the position of quasiparticles. For such an exchange to be possible, the matrix must satisfy several condit...

Let $X$ be an underlying space with a reference measure $\sigma$. Let $K$ be an integral operator in $L^2(X,\sigma)$ with integral kernel $K(x,y)$. A point process $\mu$ on $X$ is called determinantal with the correlation operator $K$ if the correlation functions of $\mu$ are given by $k^{(n)}(x_1,\dots,x_n)=\operatorname{det}[K(x_i,x_j)]_{i,j=1,\d...

Let [Formula: see text] be a locally compact Polish space and [Formula: see text] a nonatomic reference measure on [Formula: see text] (typically [Formula: see text] and [Formula: see text] is the Lebesgue measure). Let [Formula: see text] be a [Formula: see text]-matrix-valued kernel that satisfies [Formula: see text]. We say that a point process...

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...

Let $X$ be a locally compact Polish space and $\sigma$ a nonatomic reference measure on $X$ (typically $X=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). Let $X^2\ni(x,y)\mapsto\mathbb K(x,y)\in\mathbb C^{2\times 2}$ be a $2\times 2$-matrix-valued kernel that satisfies $\mathbb K^T(x,y)=\mathbb K(y,x)$. We say that a point process $\mu$ in $X$...

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 $m\in\mathbb N$ to the falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\text{ or }\mathbb C$, and Stirling numbers of the fi...

Let H be a separable Hilbert space and T be a self-adjoint bounded linear operator on H⊗2 with norm ≤1, satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator T determines a T-deformed Fock space ℱ(H)=⊕n=0∞ℱn(H). We start with reviewing and extending the known results about the structure of the n-particle spaces ℱ...

For certain Sheffer sequences (sn)n=0∞ on C, Grabiner (1988) proved that, for each α∈[0,1], the corresponding Sheffer operator zn↦sn(z) extends to a linear self-homeomorphism of Eminα(C), the Fréchet topological space of entire functions of order at most α and minimal type (when the order is equal to α>0). In particular, every function f∈Eminα(C) a...

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending...

Assume that, in a parabolic or hyperbolic equation, the right-hand side is analytic in time and the coefficients are analytic in time at each fixed point of the space. We show that the infinitely differentiable solution to this equation is also analytic in time at each fixed point of the space. This solution is given in the form of the Taylor expan...

We discuss some representations of the anyon commutation relations (ACR) both in the discrete and continuous cases. These non-Fock representations yield, in the vacuum state, gauge-invariant quasi-free states on the ACR algebra. In particular, we extend the construction from [20] to the case where the generator of the one-point function is not nece...

For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $\alpha\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$, the Fr\'echet topological space of entire functions of exponential order $\alpha$...

We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to q-deformed commutation relations with q∈(-1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...

We show that infinitely differentiable solutions to parabolic and hyperbolic equations are analytical in time at each point of the space. These solutions are given in the form of the Taylor expansion with respect to time $t$ with coefficients depending on $x$. The coefficients of the expansion are defined by recursion relations, which are obtained...

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on $X$ that is concentrated on $\mathbb K(X)$. We consider the multiplicative (current) group $C_0(X\to\mathbb R_...

We characterize the Dirichlet-Ferguson measure over a locally compact Polish diffuse probability space as the unique random measure satisfying a Mecke-type identity.

Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such that $\partial_x^+$ is the adjoint of $\partia...

The aim of this paper is to develop umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a se...

Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise.. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau)$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$, where $f^{(i...

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 −1e −s ds on \((0,\infty )\). We study a class of Laplace-type operators in \(L^{2}(\mathb...

We consider Fock representations of the $Q$-deformed commutation relations $$\partial_s\partial^\dagger_t=Q(s,t)\partial_t^\dagger\partial_s+\delta(s,t), \quad s,t\in T.$$ Here $T:=\mathbb R^d$ (or more generally $T$ is a locally compact Polish space), and the function $Q:T^2\to \{z\in \mathbb C: |z|\leq 1\}$ satisfies $Q(s,t)=\overline{Q(t,s)}$. I...

Let $\sigma$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $d\sigma(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $\Lambda\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{\sigma(\Lambda)}\,\sigma$ on $\Lambda$ and the number of particles,...

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...

We review several applications of Berezansky's projection spectral theorem to Jacobi fields in a symmetric Fock space, which lead to L\'evy white noise 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 $\eta\in \mathbb K(\mathbb R^d)$ an e...

We consider the infinite-dimensional Lie group $\mathfrak G$ which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold $X$ and the commutative multiplicative group of functions on $X$. The group $\mathfrak G$ naturally acts on the space $\mathbb M(X)$ of Radon measures on $X$. We would like to defi...

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 L\'evy 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),...

Let $X$ be a locally compact Polish space. A random measure on $X$ is a probability measure on the space of all (nonnegative) Radon measures on $X$. Denote by $\mathbb K(X)$ the cone of all Radon measures $\eta$ on $X$ which are of the form $\eta=\sum_{i}s_i\delta_{x_i}$, where, for each $i$, $s_i>0$ and $\delta_{x_i}$ is the Dirac measure at $x_i\...

We extend the result of Nualart and Schoutens on chaotic decomposition of the $L^2$-space of a L\'evy process to the case of a generalized stochastic processes with independent values.

Let be a finite measure on whose Laplace transform is analytic in a neighbourhood of zero. An anyon Lévy white noise on is a certain family of noncommuting operators on the anyon Fock space over , where runs over a space of test functions on , while is interpreted as an operator-valued distribution on . Let be the noncommutative -space generated by...

Let "mu" be a point process on a countable discrete space "X". Under assumption that "mu" is quasi-invariant with respect to any finitary permutation of "X", we describe a general scheme for constructing an equilibrium Kawasaki dynamics for which "mu" is a symmetrizing (and hence invariant) measure. We also exhibit a two-parameter family of point p...

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 fi...

Let $\Gamma$ denote the space of all locally finite subsets (configurations) in $\mathbb R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $\Gamma$ in which pairs of particles simultaneously hop over $\mathbb R^d$. We discuss a non-equilibrium dynamics of binary jumps. We prove the existence of an evolution of correlat...

Let $\Gamma$ 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 $\Gamma$ 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...

Let \({T=\mathbb R^d}\) . Let a function \({QT^2\to\mathbb C}\) satisfy \({Q(s,t)=\overline{Q(t,s)}}\) and \({|Q(s,t)|=1}\). A generalized statistics is described by creation operators \({\partial_t^\dagger}\) and annihilation operators ∂t , \({t\in T}\), which satisfy the Q-commutation relations: \({\partial_s\partial^\dagger_t = Q(s, t)\partial^\...

Let X be a locally compact Polish space and let m be a reference Radon measure on X. Let $\Gamma_X$ denote the configuration space over X, that is, the space of all locally finite subsets of X. A point process on X is a probability measure on $\Gamma_X$. A point process $\mu$ is called determinantal if its correlation functions have the form $k^{(n...

We construct two types of equilibrium dynamics of an infinite particle system in a locally compact metric space $X$ for which a permanental point process is a symmetrizing, and hence invariant measure. The Glauber dynamics is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly hop over $X$. In the case $X...

Let T be an underlying space with a non-atomic measure σ on it. In [Comm. Math. Phys. 292, 99–129 (2009)] the Meixner class of non-commutative generalized stochastic processes with freely independent values, w = (w(t))t Î T{\omega=(\omega(t))_{t\in T}} , was characterized through the continuity of the corresponding orthogonal polynomials. In this p...

Let T be an underlying space with a non-atomic measure σ on it (e.g. \({T=\mathbb R^d}\) and σ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of T, with freely independent values. Such a process (field), ω = ω(t), \({t\in T}\) , is given a rigorous meaning through smea...

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 Levy white noises, as well as as to the structure of quantum white noise on these spaces.

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.

We show that some classes of birth-and-death processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics)

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,...

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 gener...

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in Rd which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure μ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time....

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 sy...

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 L\'evy white noises, as well as as to the structure of quantum white noise on these spaces.

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.

Let $X$ be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators $a(\Delta)$ indexed by all measurable, relatively compact sets $\Delta$ in $X$ (a quantum stochastic process over $X$). For such a family, we introduce the notion of a correlation measure. We prove that, if the family of...

By definition, a Jacobi field $J=(J(\phi))_{\phi\in H_+}$ is a family of commuting selfadjoint three-diagonal operators in the Fock space $\mathcal F(H)$. The operators $J(\phi)$ are indexed by the vectors of a real Hilbert space $H_+$. The spectral measure $\rho$ of the field $J$ is defined on the space $H_-$ of functionals over $H_+$. The image o...

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1-forms and associated semigroups are considered. Their probabilistic interpretation is given.

The space $\Gamma_X$ of all locally finite configurations in a Riemannian manifold $X$ of infinite volume is considered. The deRham complex of square-integrable differential forms over $\Gamma_X$, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbe...

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, d\sigma)$, and show how to include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We introduce in $\Gama(\Ha)$ operators $a(\xi)=\int_{\R^d} dx \xi(x)a(x)$, $\xi\in S$, with $a(x)=\dig_x+2\dig_x\di_x+1+\di_x...

The paper is devoted to the study of configuration space analysis by using the projective spectral theorem. For a manifold $X$, let $\Gamma_X$, resp.\ $\Gamma_{X,0}$ denote the space of all, resp. finite configurations in $X$. The so-called $K$-transform, introduced by A. Lenard, maps functions on $\Gamma_{X,0}$ into functions on $\Gamma_{X}$ and i...

We carry out analysis and geometry on a marked configuration space $\Omega^M_X$ over a Riemannian manifold $X$ with marks from a space $M$. We suppose that $M$ is a homogeneous space $M$ of a Lie group $G$. As a transformation group $\frak A$ on $\Omega_X^M$ we take the ``lifting'' to $\Omega_X^M$ of the action on $X\times M$ of the semidirect prod...

We carry out analysis and geometry on a marked configuration space $\Omega_X^{R_+}$ over a Riemannian manifold $X$ with marks from the space $R_+$ as a natural generalization of the work {\bf [}{\it J. Func. Anal}. {\bf 154} (1998), 444--500{\bf ]}. As a transformation group $\mathfrak G$ on this space, we take the ``lifting'' to $\Omega_X^{R_+}$ o...

Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given.

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 μ as invariant measure. We study a scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we...

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 Rd, we fix a Gibbs measure μ corresponding to a general pair potential ϕ and activ...

We construct two types of equilibrium dynamics of infinite particle systems in a Riemannian manifold X. These dynamics are analogs of the Glauber, respectively Kawasaki dynamics of lattice spin systems. The Glauber dynamics now is a process where interacting particles randomly appear and disappear, i.e., it is a birth-and-death process in X, while...

In [Yu.M. Berezansky, E. Lytvynov, D. A. Mierzejewski, Ukrainian Math. J. 55 (2003), 853--858 ], the Jacobi field of a L\'evy process was derived. This field consists of commuting self-adjoint operators acting in an extended (interacting) Fock space. However, these operators have a quite complicated structure. In this note, using ideas from [L. Acc...

We review the recent results on the Jacobi field of a (real-valued) L\'evy process defined on a Riemannian manifold. In the case where the L\'evy process is neither Gaussian, nor Poisson, the corresponding Jacobi field acts in an extended Fock space. We also give a unitary equivalent representation of the Jacobi field in a usual Fock space. This re...

We study properties of the semigroup $(e^{-tH})_{t\ge 0}$ on the space $L^ 2(\Gamma_X,\pi)$, where $\Gamma_X$ is the configuration space over a locally compact second countable Hausdorff topological space $X$, $\pi$ is a Poisson measure on $\Gamma_X$, and $H$ is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov...

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 L\'evy process of Meixner's type.

A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Г of all locally finite subsets (configurations) in ℝd and which has a Gibbs measure μ as an invariant measure. We assume that μ corresponds to a symmetric pair potential ϕ(x − y). An important class of stochastic dynamics of a c...

An extension of the Heath-Jarrow-Morton model for the development of instantaneous forward interest rates with deterministic coefficients and Gaussian as well as Lévy field noise terms is given. In the special case where the Lévy field is absent, one recovers a model discussed by D.P. Kennedy.

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 $\Gamma$ of all locally finite subsets (configurations) in ${\Bbb R}^d$, we fix a Gibbs measure $\mu$ corresponding to a general pair...

We investigate a scaling limit of gradient stochastic dynamics associated with Gibbs states in classical continuous systems on ${\mathbb R}^d$, $d \ge 1$. The aim is to derive macroscopic quantities from a given microscopic or mesoscopic system. The scaling we consider has been investigated by Brox (in 1980), Rost (in 1981), Spohn (in 1986) and Guo...

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 dimensi...

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ( $$\mathbb{R}$$ -valued) Lévy process on a Riemannian manifold. The support of the measure of jumps in the Lévy–Khintchine representation for the Lévy process is supposed to have an infinite number of points. We characterize the gamma,...

In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural ``Riemannian-like'' structure of the configuration space $\Gamma_X$ over a complete, connected, oriented, and stochastically complete Riemannian manifold $X$ of infinite volume, the heat semigroup $(e^{-tH^\Gamma})_{t\in\R_+}$ was introduced an...

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 L\'evy process. At least three approaches are possible here. The...

Let $\Gamma_X$ denote the space of all locally finite configurations in a complete, stochastically complete, connected, oriented Riemannian manifold $X$, whose volume measure $m$ is infinite. In this paper, we construct and study spaces $L^2_\mu\Omega^n$ of differential $n$-forms over $\Gamma_X$ that are square integrable with respect to a probabil...

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 $\rho(x)$, $x\in\R^d$, in the representation of CAR describing an infinite free Fermi...

We consider a family B of self-adjoint commuting operators b ς = ∫ ς(t) dB t where B t is a quantum compound Poisson process in a Fock space. By using the projection spectral theorem, we construct the Fourier transform in generalized joint eigenvectors of the family B which is unitary between the Fock space and the L 2-space of compound Poisson whi...

The well-known Wick theorem expresses product of Gaussian fields by a sum of their normal products. In the paper, we define, first, a λ, θ-field to be a family of operators of multiplication by a λ, θ-white noise—the time derivative of the corresponding process with independent increments possessing the chaotic representation property. Special case...

We derive white noise calculus for a class of processes with independent increments possessing the chaotic representation property, which includes, in particular, Gaussian and Poisson processes. That is, we consider functionals of a process from this class as functionals of white noise - the time derivative (velocity) of the process. Spaces of test...

By using the projection spectral theorem, we construct the Fourier transform in generalized joint eigenvectors of a Jacobi field whose operators act in Fock space. This transform is actually a generalized Wiener-Itô-Segal isomorphism I X between Fock space and L 2 (ℰ ' ,dμ X ), where ℰ ' is a co-nuclear space and μ X a probability measure on ℰ ' ,...

By using the projection spectral theorem, we construct the Fourier transform corresponding to a family A of generalized field operators, which is unitary between Fock space and an L 2 -space (L μ 2 )≡L 2 (Φ ' ,dμ(x)), where Φ ' is a co-nuclear space and μ the spectral measure of the family A. This transform allows one to construct and study spaces...

Under certain restrictions, it is proved that a family of self-adjoint commuting operatorsA=(A ϕ)ϕεΦ where Φ is a nuclear space, possesses a cyclic vector iff there exists a Hubert spaceH ⊂ Φ′ of full operator-valued measureE, where Φ′ is the space dual to Φ andE is the joint resolution of the identity of the familyA.

We derive white noise calculus for a compound Poisson process. Namely, we consider, on the Schwartz space of tempered distributions, S′, a measure of compound Poisson white noise, μcp, and construct a whole scale of standard nuclear triples (Scp)−x ⊃ L2cp) ≡ L2(S′, dμcp) ⊃(Scpx, x≥ 0, which are obtained as images under some isomorphism of the corre...

. We introduce a definition of Euclidean Gibbs states corresponding to continuous systems of quantum particles with Boltzmann statistics. The interaction is described by a stable pair potential v(x Gamma x 0 ). Assuming that v is an absolutely integrable function, we show that the set of such Euclidean Gibbs states is not empty at least for suffici...