Quantum stochastic calculus with maximal operator domains

• 成果类型: Article
• 署名作者: Attal, SP; Lindsay, JM
• 署名单位: Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); University of Nottingham
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2004
• 页码: 488-529
• 关键词: ito formula; fermion; EXISTENCE; Algebra; FLOWS

摘要

Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Ito formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery-one closely related to classical Ito calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal-Meyer equations for implicit definition of quantum stochastic integrals.