A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock scale
Abstract
A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and basis form in terms of Malliavin derivative on a projective Fock scale, and their uniform continuity and QS differentiability with respect to the inductive limit convergence is proved. A new form of QS calculus based on an inductive *-algebraic structure in an indefinite space is developed and a nonadaptive generalization of the QS Ito formula for its representation in Fock space is derived. The problem of solution of general QS evolution equations in a Hilbert space is solved in terms of the constructed operator representation of chronological products, defined in the indefinite space, and the unitary and *-homomorphism property respectively for operators and maps of these solutions, corresponding to the pseudounitary and *-homomorphism property of the QS integrable generators, is proved.
Keywords
Cite
@article{arxiv.math-ph/0512076,
title = {A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock scale},
author = {V. P. Belavkin},
journal= {arXiv preprint arXiv:math-ph/0512076},
year = {2007}
}
Comments
28 pages. See also related papers at http://www.maths.nott.ac.uk/personal/vpb/research/ana_cal.html and http://www.maths.nott.ac.uk/personal/vpb/research/cha_noi.html