Belton, Alexander C. R. (2001) *Quantum Ω-semimartingales and stochastic evolutions.* Journal of Functional Analysis, 187 (1). pp. 94-109. ISSN 0022-1236

## Abstract

We explore Ω-adaptedness, a variant of the usual notion of adaptedness found in stochastic calculus. It is shown that the (non-adapted) quantum stochastic integrals of bounded, Ω-adapted processes are themselves bounded and Ω-adapted, a fact that may be deduced from the Bismut–Clark–Ocone formula of Malliavin calculus. An algebra analogous to Attal's class of regular quantum semimartingales is defined, and product and functional Itô formulae are given. We consider quantum stochastic differential equations with bounded, Ω-adapted coefficients that are time dependent and act on the whole Fock space. Solutions to such equations may be used to dilate quantum dynamical semigroups in a manner that generalises, and gives new insight into, that of R. Alicki and M. Fannes (1987, Comm. Math. Phys.108, 353–361); their unitarity condition is seen to be the usual condition of R. L. Hudson and K. R. Parthasarathy (1984, Comm. Math. Phys93, 301–323).

Item Type: | Article |
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Journal or Publication Title: | Journal of Functional Analysis |

Uncontrolled Keywords: | Ω-adaptedness ; quantum semimartingales ; quantum stochastic differential equations ; quantum dynamical semigroups |

Subjects: | Q Science > QA Mathematics |

Departments: | Faculty of Science and Technology > Mathematics and Statistics |

ID Code: | 20987 |

Deposited By: | Dr Alexander Belton |

Deposited On: | 08 Dec 2008 09:16 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 10 Dec 2016 01:19 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/20987 |

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