After a review of how Boson Fock space (of arbitrary multiplicity) may be approximated by a countable Hilbert-space tensor product (known as toy Fock space) it is shown that vacuum-adapted multiple quantum Wiener integrals of bounded operators may be expressed as limits of sums of operators defined on this toy space, with strong convergence on the exponential domain.The vacuum-adapted quantum Itô product formula is derived with the aid of this approximation and a brief pointer is given towards the unbounded case.
We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale.The noncommutative algebra of relatively (exponentially) bounded nonadapted quantum stochastic processes is studied in the kernel form as introduced in [10].The differential Q-adapted formula generalizing Itô product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted Itô formula.
Quantum random walks are constructed on operator spaces with the aid of matrix-space lifting, a type of ampliation intermediate between those provided by spatial and ultraweak tensor products. Using a form of Wiener–Itô decomposition, a Donsker-type theorem is proved, showing that these walks, after suitable scaling, converge in a strong sense to vacuum cocycles: these are vacuum-adapted processes which are Feller cocycles in the sense of Lindsay and Wills. This is employed to give a new proof of the existence of ∗-homomorphic quantum stochastic dilations for completely positive contraction semigroups on von Neumann algebras and separable unital C ∗ algebras. The analogous approximation result is also established within the standard quantum stochastic framework, using the link between the two types of adaptedness. Key words: quantum random walk; repeated interactions; Donsker’s invariance principle; functional central-limit theorem; quantum stochastic dilation; quantum dynamical semigroup; completely positive contraction semigroup; toy Fock space; discrete approximation; Feller cocycle; vacuum adapted. MSC 2000: 81S25 (primary); 46L07, 46L53, 46N50, 47D06, 60F17 (secondary). 1
We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness. This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion. A new type of adaptedness, known as
Ω-adaptedness, is defined. We show that quantum stochastic integrals of
Ω-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded,
Ω-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where
W is unitary and
K self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.
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.
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.