Stephen J. Wills

Existence of Feller Cocycles on a C*-Algebra

The quantum stochastic differential equation dk t = k t ∘ θ α β d Λ β α ( t ) is considered on a unital C *-algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non-commutative topological space.

Quantum stochastic operator cocycles via associated semigroups

A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation.

Homomorphic Feller Cocycles on a C*-Algebra

When a Fock-adapted Feller cocycle on a C*-algebra is regular, completely positive and contractive, it possesses a stochastic generator that is necessarily completely bounded. Necessary and sufficient conditions are given, in the form of a sequence of identities, for a completely bounded map to generate a weakly multiplicative cocycle. These are derived from a product formula for iterated quantum stochastic integrals. Under two alternative assumptions, one of which covers all previously considered cases, the first identity in the sequence is shown to imply the rest.

Construction of some quantum stochastic operator cocycles by the semigroup method

A new method for the construction of Fock-adapted quantum stochastic operator cocycles is outlined, and its use is illustrated by application to a number of examples arising in physics and probability. The construction uses the Trotter-Kato theorem and a recent characterisation of such cocycles in terms of an associated family of contraction semigroups.

A stochastic Stinespring Theorem

Abstract. Completely positive Markovian cocycles on a von Neumann algebra, adapted to a Fock filtration, are realised as conjugations of

MULTIPLICATIVITY VIA A HAT TRICK

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An algebraic construction of quantum flows with unbounded generators

-homomorphic Markovian cocycles. The conjugating processes are affiliated to the algebra, and are governed by quantum stochastic differential equations whose coefficients evolve according to the

Markovian Cocycles on Operator Algebras Adapted to a Fock Filtration

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Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

-homomorphic process. Some perturbation theory for quantum stochastic flows is developed in order to achieve the above Stinespring decomposition.

Solving quantum stochastic differential equations with unbounded coefficients

where σ is the Fock space shift semigroup and s is the map between matrix spaces M(F;A)b and M(F ;A)b (defined below) induced by js. Regularity for the cocycle means that its associated semigroups are norm continuous. If j is completely positive and contractive then it has a CB stochastic generator in the following sense: there is a completely bounded operator θ from A into the matrix space M(k;A)b such that j satisfies the Evans-Hudson equation djt = jt ◦ θ β dΛα(t), in which [θ β ] is the matrix of components of θ with respect to the basis of k used for defining the matrix of quantum stochastic integrators [Λβ ]. Furthermore, if j is ∗-homomorphic then the quantum Ito formula implies that θ satisfies θ(a∗a) = θ(a)∗(a⊗ Ik) + (a ∗ ⊗ Ik)θ(a) + θ(a) (Ih ⊗ P )θ(a), (0.2)