A Quantum Nonadapted Itô Formula and Stochastic Analysis in Fock Scale

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 Itô 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.