635 research outputs found
Quantum stochastic convolution cocycles II
Schuermann's theory of quantum Levy 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 generators is obtained.
Automatic complete boundedness of a class of derivations is established,
leading to a characterisation of the stochastic generators of *-homomorphic
convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum
Levy process on a compact quantum group are given and, with respect to both of
these, it is shown that an equivalent process on Fock space may be
reconstructed from the generator of the quantum Levy process. In the examples
presented, connection to the algebraic theory is emphasised by a focus on full
compact quantum groups.Comment: 32 pages, expanded introduction and updated references. The revised
version will appear in Communications in Mathematical Physic
Quantum stochastic convolution cocycles III
Every Markov-regular quantum Levy 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 norm-continuous convolution
semigroups on a multiplier C*-bialgebra are then completely characterised.
These results are achieved by extending the theory of quantum Levy processes on
a compact quantum group, and more generally quantum stochastic convolution
cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier
C*-bialgebras. Strict extension results obtained by Kustermans, together with
automatic strictness properties developed here, are exploited to obtain
existence and uniqueness for coalgebraic quantum stochastic differential
equations in this setting. Then, working in the universal enveloping von
Neumann bialgebra, we characterise the stochastic generators of Markov-regular,
*-homomorphic (respectively completely positive and contractive), quantum
stochastic convolution cocycles.Comment: 20 pages; v2 corrects some typos and no longer contains a section on
quantum random walk approximations, which will now appear as a separate
submission. The article will appear in the Mathematische Annale
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