Local measures in fock space stochastic calculus and a generalized ito-tanaka formula

1988 ◽  
pp. 213-231
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Fock Space ◽  
Stochastic Calculus ◽  
Tanaka Formula

scholarly journals FILTERED STOCHASTIC CALCULUS

2001 ◽  
Vol 04 (03) ◽  
pp. 309-346 ◽  
Author(s):  
ROMUALD LENCZEWSKI
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Fock Space ◽  
Stochastic Calculus ◽  
Itô Formula ◽  
Uniqueness Of Solutions ◽  
Ito Formula ◽  
Itô Calculus ◽  
Boolean Independence

By introducing a color filtration to the multiplicity space [Formula: see text], we extend the quantum Itô calculus on multiple symmetric Fock space [Formula: see text] to the framework of filtered adapted biprocesses. In this new notion of adaptedness, "classical" time filtration makes the integrands similar to adapted processes, whereas "quantum" color filtration produces their deviations from adaptedness. An important feature of this calculus, which we call filtered stochastic calculus, is that it provides an explicit interpolation between the main types of calculi, regardless of the type of independence, including freeness, Boolean independence (more generally, m-freeness) as well as tensor independence. Moreover, it shows how boson calculus is "deformed" by other noncommutative notions of independence. The corresponding filtered Itô formula is derived. Existence and uniqueness of solutions of a class of stochastic differential equations are established and unitarity conditions are derived.

Stop times in fock space stochastic calculus

1987 ◽  
Vol 75 (3) ◽  
pp. 317-349 ◽  
Author(s):  
K. R. Parthasarathy ◽  
Kalyan B. Sinha
Keyword(s):  
Fock Space ◽  
Stochastic Calculus

QUANTUM STOCHASTIC CALCULUS ON BOOLEAN FOCK SPACE

2004 ◽  
Vol 07 (04) ◽  
pp. 631-650 ◽  
Author(s):  
ANIS BEN GHORBAL ◽  
MICHAEL SCHÜRMANN
Keyword(s):  
Fock Space ◽  
Field Operator ◽  
Stochastic Calculus ◽  
Product Formula ◽  
Stochastic Integration ◽  
Quantum Stochastic Calculus ◽  
Basic Field ◽  
Quantum Dynamical ◽  
Quantum Dynamical Semigroups

In this paper we establish a theory of stochastic integration with respect to the basic field operator processes in the Boolean case. This leads to a Boolean version of quantum Itô's product formula and has applications to the theory of dilations of quantum dynamical semigroups.

scholarly journals The qq-bit (III): Symmetric q-Jordan–Wigner embeddings

2019 ◽  
Vol 22 (01) ◽  
pp. 1850023
Author(s):  
Luigi Accardi ◽  
Yun-Gang Lu
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Random Variable ◽  
Stochastic Calculus ◽  
General Context ◽  
Quantum Stochastic Calculus ◽  
Continuous Version ◽  
Text Mode ◽  
Interacting Fock Space ◽  
Canonical Quantum

We prove that, replacing the left Jordan–Wigner [Formula: see text]-embedding by the symmetric [Formula: see text]-embedding described in Sec. 2, the result of the corresponding central limit theorem changes drastically with respect to those obtained in Ref. 5. In fact, in the former case, for any [Formula: see text], the limit space is precisely the [Formula: see text]-mode Interacting Fock Space (IFS) that realizes the canonical quantum decomposition of the limit classical random variable. In the latter case, this happens if and only if [Formula: see text]. Furthermore, as shown in Sec. 4, the limit classical random variable turns out to coincide with the [Formula: see text]-mode version of the [Formula: see text]-deformed quantum Brownian introduced by Parthasarathy[Formula: see text], and extended to the general context of bi-algebras by Schürman[Formula: see text]. The last section of the paper (Appendix) describes this continuous version in white noise language, leading to a simplification of the original proofs, based on quantum stochastic calculus.

scholarly journals Quantum white noises—White noise approach to quantum stochastic calculus

1993 ◽  
Vol 129 ◽  
pp. 23-42 ◽  
Author(s):  
Zhiyuan Huang
Keyword(s):  
Hilbert Space ◽  
White Noise ◽  
Fock Space ◽  
Stochastic Calculus ◽  
Quantum Stochastic Calculus ◽  
Integrable Functions ◽  
Boson Fock Space ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
White Noises

Let H = L2 (R) be the Hilbert space of all complex-valued square integrable functions defined on R, Ф = Γ(H) be the Boson Fock space over H. For each h ∈ H, denote by ε(h) the corresponding exponential vector:in particular ε(0) is the Fock vacuum.

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