On Some Markov Processes Arising from the Eyre-Hudson Super Lie Algebra Representations

1998 ◽  
Vol 01 (03) ◽  
pp. 485-498
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Brownian Motion ◽  
Markov Processes ◽  
Fock Space ◽  
Linear Manifold ◽  
Free Field ◽  
Natural Manner ◽  
Commutation Relations ◽  
Boson Fock Space ◽  
Lie Algebra Representations ◽  
Super Lie Algebra

It is well-known3,5 that Brownian motion and Poisson process arise naturally from the canonical commutation relations (CCR) of free field operators in a boson Fock space. Eyre and Hudson2 have recently shown how to construct fields of operators in a boson Fock space obeying super Lie commutation relations. We establish the essential self-adjointness of their real and imaginary parts on the domain ∊, the linear manifold generated by all the exponential (coherent) vectors and determine a family of Markov processes which they give rise to in a natural manner. These Markov processes yield examples of Evans–Hudson flows3,5 and Azéma-like martingales.1,4,6

scholarly journals Singular Bogoliubov transformations and inequivalent representations of canonical commutation relations

2019 ◽  
Vol 31 (08) ◽  
pp. 1950026 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Fock Space ◽  
Sufficient Conditions ◽  
Bogoliubov Transformation ◽  
Fock Representation ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Direct Sum ◽  
Boson Fock Space ◽  
Inequivalent Representations ◽  
Bogoliubov Transformations

We introduce a concept of singular Bogoliubov transformation on the abstract boson Fock space and construct a representation of canonical commutation relations (CCRs) which is inequivalent to any direct sum of the Fock representation. Sufficient conditions for the representation to be irreducible are formulated. Moreover, an example of such representations of CCRs is given.

EXOTIC LAPLACIANS AND ASSOCIATED STOCHASTIC PROCESSES

2009 ◽  
Vol 12 (01) ◽  
pp. 1-19 ◽  
Author(s):  
LUIGI ACCARDI ◽  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Explicit Expression ◽  
Separable Hilbert Space ◽  
Fock Space ◽  
Heat Semigroup ◽  
Tempered Distributions ◽  
Infinite Dimensional ◽  
Boson Fock Space

In this paper, we give a decomposition of the space of tempered distributions by the Cesàro norm, and for any [Formula: see text] we construct directly from the exotic trace an infinite dimensional separable Hilbert space Hc,2a-1 on which the exotic trace plays the role as the usual trace. This implies that the Exotic Laplacian coincides with the Volterra–Gross Laplacian in the Boson Fock space Γ(Hc,2a-1) over the Hilbert space Hc,2a-1. Finally we construct the Brownian motion naturally associated to the Exotic Laplacian of order 2a-1 and we find an explicit expression for the associated heat semigroup.

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

scholarly journals LOCALIZATION OF THE NUMBER OF PHOTONS OF GROUND STATES IN NONRELATIVISTIC QED

2003 ◽  
Vol 15 (03) ◽  
pp. 271-312 ◽  
Author(s):  
FUMIO HIROSHIMA
Keyword(s):  
Radiation Field ◽  
Ground State ◽  
Fock Space ◽  
Adjoint Operator ◽  
Ground States ◽  
Electron System ◽  
Number Operator ◽  
Position Variable ◽  
Boson Fock Space ◽  
Quantized Radiation Field

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2 (ℝ3) ⊗ ℱ ≅ L2 (ℝ3; ℱ), where ℱ is the Boson Fock space over L2 (ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to [Formula: see text], where N denotes the number operator of ℱ. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ‖(1 ⊗ Nk/2) ψg (x)‖ℱ ≤ Dk e-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular [Formula: see text] for 0 < β < δ/2 is obtained.

scholarly journals On the Entropy of Fractionally Integrated Gauss–Markov Processes

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2031
Author(s):  
Mario Abundo ◽  
Enrica Pirozzi
Keyword(s):  
Dynamical System ◽  
Stochastic Process ◽  
Brownian Motion ◽  
Markov Process ◽  
Markov Processes ◽  
Numerical Approximation ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Ornstein Uhlenbeck Process ◽  
Application Fields

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

scholarly journals Some Measure-Valued Markov Processes Attached to Occupation Times of Brownian Motion

Bernoulli ◽  
2000 ◽  
Vol 6 (1) ◽  
pp. 63 ◽  
Author(s):  
Catherine Donati-Martin ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Markov Processes ◽  
Occupation Times

scholarly journals A family of inequivalent Weyl representations of canonical commutation relations with applications to quantum field theory

2016 ◽  
Vol 28 (04) ◽  
pp. 1650007 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Fock Space ◽  
Complex Hilbert Space ◽  
Space Region ◽  
Zero Field ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Time Zero ◽  
Conjugate Momentum

We consider a family of irreducible Weyl representations of canonical commutation relations with infinite degrees of freedom on the abstract boson Fock space over a complex Hilbert space. Theorems on equivalence or inequivalence of the representations are established. As a simple application of one of these theorems, the well-known inequivalence of the time-zero field and conjugate momentum for different masses in a quantum scalar field theory is rederived with space dimension [Formula: see text] arbitrary. Also a generalization of representations of the time-zero field and conjugate momentum is presented. Comparison is made with a quantum scalar field in a bounded region in [Formula: see text]. It is shown that, in the case of a bounded space region with [Formula: see text], the representations for different masses turn out to be mutually equivalent.

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