A nonadapted stochastic calculus and non stationary evolution in Fock scale

Abstract

A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of Malliavin derivative on a projective Fock space, and their uniform continuity with respect to the inductive limite 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 isometry and *-homomorphism property respectively for operators and maps of these solutions, corresponding to the peseudounitary and *-homomorphism property of the QS integrable generators is proved.