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Quantum stochastic calculus associated with quadratic quantum noises
Journal of Mathematical Physics, 2016We first study a class of fundamental quantum stochastic processes induced by the generators of a six dimensional non-solvable Lie †-algebra consisting of all linear combinations of the generalized Gross Laplacian and its adjoint, annihilation operator, creation operator, conservation, and time, and then we study the quantum stochastic integrals ...
Kalyan B Sinha +2 more
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Quantum and non-causal stochastic calculus
Probability Theory and Related Fields, 1993The quantum stochastic calculus initiated by \textit{R. L. Hudson} and \textit{K. R. Parthasarathy} [Commun. Math. Phys. 93, 301-323 (1984; Zbl 0546.60058)], and the noncausal stochastic calculus originating with the papers of \textit{M. Hitsuda} [Proc. 2nd Japan-USSR Symp. Probab. Theory 2, 111-114 (1972)] and \textit{A. V.
J Martin Lindsay
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Journal of Mathematical Physics, 1985
The physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process (OVSP). In this article, it is shown how the formalism of quantum stochastic calculus of Hudson and Parthasarathy allows, in a simple way, for constructing a large class of OVSP’s that in ...
Alberto Barchielli, Barchielli A
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The physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process (OVSP). In this article, it is shown how the formalism of quantum stochastic calculus of Hudson and Parthasarathy allows, in a simple way, for constructing a large class of OVSP’s that in ...
Alberto Barchielli, Barchielli A
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Journal of Soviet Mathematics, 1991
The main aim of this paper is to introduce the reader into the quantum stochastic calculus in the symmetric Fock space from the stochastic processes point of view. The author discusses the quantum Itô formula, applications to probabilistic representations of solutions of differential equations, and applications to extensions of dynamical semigroups ...
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The main aim of this paper is to introduce the reader into the quantum stochastic calculus in the symmetric Fock space from the stochastic processes point of view. The author discusses the quantum Itô formula, applications to probabilistic representations of solutions of differential equations, and applications to extensions of dynamical semigroups ...
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1986
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Ito’s formula are described.
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The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Ito’s formula are described.
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Wick Calculus of Generalized Operators and its Applications to Quantum Stochastic Calculus
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1998A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated.
Huang, Zhiyuan, Luo, Shunlong
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Classical Stochastic Processes from Quantum Stochastic Calculus
Journal of Mathematical Sciences, 2001Using multidimensional quantum stochastic calculus, the author constructs a (weak) one-parameter representation \(\tilde{j}_t\), \(t\in {\mathbb R}_+\), of the Lie algebra \(gl(N)\) of \(N\times N\) matrices, which is then extended to a representation \(\tilde{J}_t\), \(t\in {\mathbb R}_+\), of the universal enveloping algebra \({\mathcal U}(gl(N ...
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Malliavin calculus for quantum stochastic processes
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1999The purpose of this note is to show that the idea of the Malliavin calculus can be applied to quantum stochastic processes. The authors prove first a Girsanov formula and by a differentiation of this they obtain an integration by parts formula. Using these, the authors give sufficient conditions for the Wigner density to belong to a Sobolev space of ...
Franz, Uwe +2 more
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Quantum Stochastic Calculus Associated With Quantum Lévy White Noise
Mathematical Methods in the Applied SciencesABSTRACTWe establish the commutation relations of the quantum Lévy operators, consisting of the generalized Lévy Gross Laplacian and its adjoint, annihilation operator, creation operator, and the Lévy number operator. In particular, such quantum Lévy operators span a nonsolvable ‐Lie algebra.
Omar Alzeley +2 more
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Exponential formulae in quantum stochastic calculus
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1996The rigorous definition of time-ordered exponentials, solving quantum linear stochastic differential equations, is extended to Boson and Fermion stochastic calculi with infinitely many degrees of freedom. The relation to the classicalmultiplicative stochastic integral, solving the Doleans exponential equation, is discussed.
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