Quantum Stochastic Integrals as Operators

A. V. Gorbunov

Andrzej Łuczak

Stéphane Attal, J. Martin Lindsay

Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.

R. F. Streater

Mateusz Jurczynski

In this thesis we develop the theory of quantum Wiener integrals on the bosonic Fock space. We study multiple quantum Wiener integrals as an algebra of unbounded operators, investigating its properties, including closedness, common domains and multiplication formulas. We show the applications of the new formalism by providing new proofs to the established theory of quantum stochastic calculus and new conditions for generating quantum stochastic cocycles and quantum stochastic evolutions. The corresponding quasifree case is also studied and the constructions extended to fit in that formalism. We construct the multiple quantum Wiener integral as one operator on a family of operators which we dub operator kernels. This in particular covers the case of quantum stochastic cocycles and evolutions. We show that the family of quantum Wiener integrals forms a WOT-dense algebra of unbounded operators on the bosonic Fock space. We provide more general conditions for an operator kernel to be multiple quantum Wiener integrable, which allows us to treat multiple quantum Wiener integrals as an algebra. We explore the influence of an initial space on the theory. Our setting gives natural conditions for a product of two cocycles (evolutions) to still be a cocycle (an evolution). We apply our theory by solving quantum stochastic differential equations (QSDEs) and by finding more elementary proofs of structure conditions on the generator of a quantum stochastic evolution and of the fundamental estimate in the proof of quantum stochastic Lie–Trotter formula. We also show how our theory unifies and generalises the theory of integral kernels and chaotic representation properties, proving in particular that every Hilbert–Schmidt operator is a quantum Wiener integral.

Un Cig Ji

Based on the quantum white noise theory, we formulate new types of anticipating quantum stochastic integrals by combining the Hitsuda–Skorokhod quantum stochastic integrals and the interactions between the integrands and the integrators. For our purpose, we prove various versions of analytic characterization theorems of symbols of white noise operators as more general cases. Also, several types of quantum (stochastic) gradients are formulated as continuous linear operators which are related to the quantum white noise derivatives and reflecting the interactions with the external noises. Finally, we formulate further systematical study of anticipating quantum stochastic integrals.