InforLorV4, Main, Exploration, bibRecord, 009798

Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

Identifieur interne : 009798 ( Main/Exploration ); précédent : 009797; suivant : 009799

Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

Auteurs : Uwe Franz [France] ; Remi Leandre [France] ; Rene Schott [France]

Source :

RBID : Hal:hal-00470218

Abstract

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over $\eufrak{h}$. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For $\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integral for adapted integrable processes and with the non-causal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.

Url:

https://hal.archives-ouvertes.fr/hal-00470218

Affiliations:

Links toward previous steps (curation, corpus...)

to stream Hal, to step Corpus: 003049
to stream Hal, to step Curation: 003049
to stream Hal, to step Checkpoint: 005F91
to stream Main, to step Merge: 009D95
to stream Main, to step Curation: 009798

Le document en format XML

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<front><div type="abstract" xml:lang="en">A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over $\eufrak{h}$. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For $\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integral for adapted integrable processes and with the non-causal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.</div>
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Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

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Abstract

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