# Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-00470218
Contributor : Uwe Franz <>
Submitted on : Monday, April 5, 2010 - 10:28:01 AM
Last modification on : Friday, July 6, 2018 - 3:18:04 PM

### Citation

Uwe Franz, Remi Leandre, Rene Schott. Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes. 2000. ⟨hal-00470218⟩

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