arXiv subjects

Eugene Lytvynov

Publications and source records attributed to Eugene Lytvynov.

45 records · Page 3Linked to original sources

arXiv2006

On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR

Let $X$ be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators $a(Δ)$ indexed by all measurable, relatively compact sets $Δ$ 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 operators possesses a correlation measure which satisfies some condition of growth, then there exists a point process over $X$ having the same correlation measure. Furthermore, the operators $a(Δ)$ can be realized as multiplication operators in the $L^2$-space with respect to this point process. In the proof, we utilize the notion of $\star$-positive definiteness, proposed in [Y. G. Kondratiev and T.\ Kuna, {\it Infin. Dimens. Anal. Quantum Probab. Relat. Top.} {\bf 5} (2002), 201--233]. In particular, our result extends the criterion of existence of a point process from that paper to the case of the topological space $X$, which is a standard underlying space in the theory of point processes. As applications, we discuss particle densities of the quasi-free representations of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like (e.g. para-fermions and para-bosons of order 2) point processes. In particular, we prove that any fermion point process corresponding to a Hermitian kernel may be derived in this way.

math.PR↗

arXiv2005

Lévy processes and Jacobi fields

We review the recent results on the Jacobi field of a (real-valued) Lévy process defined on a Riemannian manifold. In the case where the Lévy 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 representation is inspired by a result by Accardi, Franz, and Skeide (2002).

math.PR↗

arXiv2005

An equivalent representation of the Jacobi field of a Lévy process

In [Yu.M. Berezansky, E. Lytvynov, D. A. Mierzejewski, Ukrainian Math. J. 55 (2003), 853--858 ], the Jacobi field of a Lévy 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. Accardi. U. Franz, M. Skeide, Comm. Math. Phys. 228 (2002), 123--150] and [E. Lytvynov, Infin. Dimen. Anal. Quant. Prob. Rel. Top. 7 (2004), 619--629], we obtain a unitary equivalent representation of the Jacobi field of a Lévy process. In this representation, the operators act in a usual symmetric Fock space and have a much simpler structure.

math.PR↗

arXiv2004

The semigroup of the Glauber dynamics of a continuous system of free particles

We study properties of the semigroup $(e^{-tH})_{t\ge 0}$ on the space $L^ 2(Γ_X,π)$, where $Γ_X$ is the configuration space over a locally compact second countable Hausdorff topological space $X$, $π$ is a Poisson measure on $Γ_X$, and $H$ is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov semigroup of kernels $(P_t)_{t\ge 0}$ and, using it, we prove the main results of the paper: the Feller property of the semigroup $(P_t)_{t\ge 0}$ with respect to the vague topology on the configuration space $Γ_X$, and the ergodic property of $(P_t)_{t\ge 0}$. Following an idea of D. Surgailis, we also give a direct construction of the Glauber dynamics of a continuous infinite system of free particles. The main point here is that this process can start in every $γ\inΓ_X$, will never leave $Γ_X$ and has cadlag sample paths in $Γ_X$.

math.PR↗

arXiv2003

Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $Γ$ of all locally finite subsets (configurations) in $\Bbb R$ 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 classical continuous system is formed by diffusions. Till now, only one type of such dynamics--the so-called gradient stochastic dynamics, or interacting Brownian particles--has been investigated. By using the theory of Dirichlet forms, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal E}_μ^Γ$ on $L^2(Γ;μ)$, and under general conditions on the potential $ϕ$, prove its closability. For a potential $ϕ$ having a ``weak'' singularity at zero, we also write down an explicit form of the generator of ${\cal E}_μ^Γ$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\cal E}_μ^Γ$, there exists a diffusion process that is properly associated with it. Finally, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of ${\cal D}{:=}C_0^\infty({\Bbb R})$.

math.PR↗

arXiv2003

A model of the term structure of interest rates based on Lévy fields

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.

math.PR↗

arXiv2002

The Jacobi field of a Lévy process

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ($\R$-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, Pascal, and Meixner processes as the only Lévy processes whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.

math.PR↗

arXiv2002

Scaling limit of stochastic dynamics in classical continuous systems

We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on ${\mathbb R}^d, d \ge 1$. The aim is to derive macroscopic quantities from a given micro- or mesoscopic system. The scaling we consider has been investigated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}, and \cite{GP86}, under the assumption that the underlying potential is in $C^3_0$ and positive. We prove that the Dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the Dirichlet form of a generalized Ornstein--Uhlenbeck process. The proof is based on the analysis and geometry on the configuration space which was developed in \cite{AKR98a}, \cite{AKR98b}, and works for general Gibbs measures of Ruelle type. Hence, the underlying potential may have a singularity at the origin, only has to be bounded from below, and may not be compactly supported. Therefore, singular interactions of physical interest are covered, as e.g. the one given by the Lennard--Jones potential, which is studied in the theory of fluids. Furthermore, using the Lyons--Zheng decomposition we give a simple proof for the tightness of the scaled processes. We also prove that the corresponding generators, however, do not converge in the $L^2$-sense. This settles a conjecture formulated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}.

math.PR↗

arXiv2002

The heat semigroup on configuration spaces

In this paper, we study properties of the heat semigroup of configuration space analysis. Using a natural ``Riemannian-like'' structure of the configuration space $Γ_X$ over a complete, connected, oriented, and stochastically complete Riemannian manifold $X$ of infinite volume, the heat semigroup $(e^{-tH^Γ})_{t\in\R_+}$ was introduced and studied in [{\it J. Func. Anal.} {\bf 154} (1998), 444--500]. Here, $H^Γ$ is the Dirichlet operator of the Dirichlet form ${\cal E}^Γ$ over the space $L^2(Γ_X,π_m)$, where $π_m$ is the Poisson measure on $Γ_X$ with intensity $m$--the volume measure on $X$. We construct a metric space $Γ_\infty$ that is continuously embedded into $Γ_X$. Under some conditions on the manifold $X$ and we prove that $Γ_\infty$ is a set of full $π_m$ measure. The central results of the paper are two types of Feller properties for the heat semigroup. Next, we give a direct construction of the independent infinite particle process on the manifold $X$, which is a realization of the Brownian motion on the configuration space. The main point here is that we prove that this process can start in every $γ\inΓ_\infty$, will never leave $Γ_\infty$, and has continuous sample path in $Γ_\infty$, provided $\operatorname{dim}X\ge2$. In this case, we also prove that this process is a strong Markov process whose transition probabilities are given by the $¶_{t,γ}(\cdot)$ above. Furthermore, we discuss the necessary changes to be done for constructing the process in the case $\operatorname{dim}X=1$. Finally, as an easy consequence we get a ``path-wise'' construction of the independent particle process on $Γ_\infty$ from the underlying Brownian motion.

math.PR↗