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Quantum stochastic integral representations of Fock space operators

Un Cig Ji, Nobuaki Obata

An (unbounded) operator Ξ on Boson Fock space over L 2(R +) is called regular if it is an admissible white noise operator such that the conditional expectations give rise to a regular quantum martingale. We prove that an admissible white noise operator is regular if and only if it admits a quantum stochastic integral representation.

StochasticsJun 1, 200918 citations

Quasi-free stochastic integral representation theorems over the CCR

Ivan F. Wilde

Abstract It is shown that each vector in the Hilbert space of certain quasi-free representations of the CCR can be written uniquely in terms of quantum stochastic integrals. As a consequence, we obtain general vector-valued and operator-valued boson quantum martingale representation theorems.

Mathematical Proceedings of the Cambridge Philosophical SocietySep 1, 19883 citations

UNIQUENESS OF INTEGRANDS IN QUANTUM STOCHASTIC INTEGRAL

Un Cig Ji, Kalyan B. Sinha

As a general study for uniqueness of integrands in quantum martingale representation, we present a necessary and sufficient condition for uniqueness of integrands in a quantum stochastic integral. Also, several equivalent conditions to the necessary and sufficient condition are studied.

Infinite Dimensional Analysis Quantum Probability and Related TopicsDec 1, 20064 citations

Anticipating Quantum Stochastic Integrals for Basic Quantum Martingales (Mathematical aspects of quantum fields and related topics)

Un Cig Ji

On the other hand, based on the white noise theory [8, 22, 30] introduced by Hida, Kuo & Russek [23] studied anticipating (classical) stochastic integrals by applying the quantum decomposition of a Brownian motion.In this paper, we study some regularity properties of the quantum Hitsuda-Skorohod integrals as anticipating quantum stochastic integrals.Also, motivated by the results in [23], we discuss new types of anticipating quantum stochastic integrals in terms of pointwisely defined quantum white noise derivatives.2 Admissible Generalized Operators Admissible Rigging of Fock SpaceWe now review a construction of admissible rigging of Fock space which provides the basic structure of this paper.Let H=L^{2}(\mathbb{R}_{+}, dt) be the Hilbert space of complex valued Anticipating Quantum Stochastic Integrals for BasicQuantum MartingalesOn the other hand, based on the white noise theory [8, 22, 30] introduced by Hida, Kuo & Russek [23] studied anticipating (classical) stochastic integrals by applying the quantum decomposition of a Brownian motion.In this paper, we study some regularity properties of the quantum Hitsuda-Skorohod integrals as anticipating quantum stochastic integrals.Also, motivated by the results in [23], we discuss new types of anticipating quantum stochastic integrals in terms of pointwisely defined quantum white noise derivatives.2 Admissible Generalized Operators Admissible Rigging of Fock SpaceWe now review a construction of admissible rigging of Fock space which provides the basic structure of this paper.Let H=L^{2}(\mathbb{R}_{+}, dt) be the Hilbert space of complex valued Anticipating Quantum Stochastic Integrals for BasicQuantum Martingales

数理解析研究所講究録Aug 1, 20190 citations

Stochastic integral representation of bounded quantum martingales in Fock space

Kalyanapuram Rangachari Parthasarathy, Kalyan B. Sinha

Journal of Functional AnalysisJun 1, 198663 citations

Stochastic integral representations of quantum martingales on multiple Fock space

Un Cig Ji

arXiv:0707.2144Proceedings of the Indian Academy of Sciences - Section ANov 1, 20061 citation

INTEGRAL REPRESENTATION OF QUANTUM MARTINGALES

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INTEGRAL REPRESENTATION OF QUANTUM MARTINGALES

Abstract

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