A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock scale

TL;DR

This paper develops a nonadaptive quantum stochastic calculus framework using Fock space and Malliavin derivatives, deriving a generalized Ito formula and solving QS evolution equations with operator representations.

Contribution

It introduces a nonadaptive QS calculus based on an inductive *-algebraic structure and derives a generalized Ito formula in Fock space, extending quantum stochastic analysis.

Findings

Established a generalized QS integral and differential framework.

Derived a nonadaptive QS Ito formula in Fock space.

Solved QS evolution equations using operator representations.

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.