# Quantum Stochastic Convolution Cocycles II

@article{Lindsay2008QuantumSC, title={Quantum Stochastic Convolution Cocycles II}, author={J. Martin Lindsay and Adam G. Skalski}, journal={Communications in Mathematical Physics}, year={2008}, volume={280}, pages={575-610} }

Schürmann’s theory of quantum Lévy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*-hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic…

## 18 Citations

Quantum stochastic convolution cocycles III

- Mathematics
- 2012

Every Markov-regular quantum Lévy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of…

Sesquilinear quantum stochastic analysis in Banach space

- Mathematics
- 2014

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem…

Transformation of quantum Lévy processes on Hopf algebras

- Physics
- 2010

A quantum Lévy process is given by its generator, a conditionally positive linear functional on the underlying Hopf algebra or bialgebra. A transformation between two bialgebras, in the sense of this…

Quantum Wiener chaos

- Mathematics
- 2017

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…

Quantum random walk approximation in Banach algebra

- Mathematics
- 2015

Abstract Belton's discrete approximation scheme is extended to Banach-algebra-valued sesquilinear quantum stochastic cocycles, through the dyadic discretisation of time. Approximation results for…

Convolution semigroups of states

- Mathematics
- 2009

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of…

On quantum stochastic differential equations

- Mathematics, Physics
- 2007

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given…

Quantum Feynman-Kac perturbations

- Computer Science, MathematicsJ. Lond. Math. Soc.
- 2014

We develop fully noncommutative Feynman-Kac formulae by employing quantum stochastic processes. To this end we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by…

Lévy Processes on Quantum Permutation Groups

- Mathematics
- 2016

We describe basic motivations behind quantum or noncommutative probability, introduce quantum Levy processes on compact quantum groups, and discuss several aspects of the study of the latter in the…

Completely positive quantum stochastic convolution cocycles and their dilations

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2007

Abstract Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C*-hyperbialgebra are characterised. The characterisation is used to obtain…

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