Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

Details Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes Malliavin Calculus and Skorohod Integration for Quantum Stochastic Processes

2000-04-13

arxiv.org/abs/math/0004088v1

Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 4, No. 1 (2001) 11-38

Mathematics

Probability

28 pages, amsart style

Scientific paper

10.1142/S0219025701000371

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.

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