Rights statement: The original publication is available at www.link.springer.com
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Quantum stochastic convolution cocycles III
AU - Lindsay, J. Martin
AU - Skalski, Adam G.
N1 - The original publication is available at www.link.springer.com
PY - 2012/4
Y1 - 2012/4
N2 - The theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, is extended to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, and automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Working in the universal enveloping von Neumann bialgebra, the stochastic generators of Markov-regular, completely positive, respectively *-homomorphic, quantum stochastic convolution cocycles are characterised. 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 characterised. Applying a recent result of Belton's, we give a thorough treatment of the approximation of quantum stochastic convolution cocycles by quantum random walks.
AB - The theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, is extended to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, and automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Working in the universal enveloping von Neumann bialgebra, the stochastic generators of Markov-regular, completely positive, respectively *-homomorphic, quantum stochastic convolution cocycles are characterised. 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 characterised. Applying a recent result of Belton's, we give a thorough treatment of the approximation of quantum stochastic convolution cocycles by quantum random walks.
U2 - 10.1007/s00208-011-0656-1
DO - 10.1007/s00208-011-0656-1
M3 - Journal article
VL - 352
SP - 779
EP - 804
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 4
ER -