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Quantum stochastic convolution cocycles I.

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Quantum stochastic convolution cocycles I. / Lindsay, J. Martin; Skalski, Adam G.
In: Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Vol. 41, No. 3, 01.06.2005, p. 581-604.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Lindsay, JM & Skalski, AG 2005, 'Quantum stochastic convolution cocycles I.', Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, vol. 41, no. 3, pp. 581-604. https://doi.org/10.1016/j.anihpb.2004.10.002

APA

Lindsay, J. M., & Skalski, A. G. (2005). Quantum stochastic convolution cocycles I. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 41(3), 581-604. https://doi.org/10.1016/j.anihpb.2004.10.002

Vancouver

Lindsay JM, Skalski AG. Quantum stochastic convolution cocycles I. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques. 2005 Jun 1;41(3):581-604. doi: 10.1016/j.anihpb.2004.10.002

Author

Lindsay, J. Martin ; Skalski, Adam G. / Quantum stochastic convolution cocycles I. In: Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques. 2005 ; Vol. 41, No. 3. pp. 581-604.

Bibtex

@article{e394309af035464792df8d72e9ec9c93,
title = "Quantum stochastic convolution cocycles I.",
abstract = "Stochastic convolution cocycles on a coalgebra are obtained by solving quantum stochastic differential equations. We describe a direct approach to solving such QSDE's by iterated quantum stochastic integration of matrix-sum kernels. The cocycles arising this way satisfy a H{\"o}lder condition, and it is shown that conversely every such H{\"o}lder-continuous cocycle is governed by a QSDE. Algebraic structure enjoyed by matrix-sum kernels yields a unital *-algebra of processes which allows easy deduction of homomorphic properties of cocycles on a {\textquoteleft}quantum semigroup{\textquoteright}. This yields a simple proof that every quantum L{\'e}vy process may be realised in Fock space. Finally perturbation of cocycles by Weyl cocycles is shown to be implemented by the action of the corresponding Euclidean group on Sch{\"u}rmann triples.",
keywords = "Noncommutative probability, Quantum group, Stochastic cocycle, Quantum L{\'e}vy process, Quantum stochastic",
author = "Lindsay, {J. Martin} and Skalski, {Adam G.}",
note = "RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics",
year = "2005",
month = jun,
day = "1",
doi = "10.1016/j.anihpb.2004.10.002",
language = "English",
volume = "41",
pages = "581--604",
journal = "Annales de l'Institut Henri Poincar{\'e} (B) Probabilit{\'e}s et Statistiques",
issn = "0246-0203",
publisher = "Institute of Mathematical Statistics",
number = "3",

}

RIS

TY - JOUR

T1 - Quantum stochastic convolution cocycles I.

AU - Lindsay, J. Martin

AU - Skalski, Adam G.

N1 - RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics

PY - 2005/6/1

Y1 - 2005/6/1

N2 - Stochastic convolution cocycles on a coalgebra are obtained by solving quantum stochastic differential equations. We describe a direct approach to solving such QSDE's by iterated quantum stochastic integration of matrix-sum kernels. The cocycles arising this way satisfy a Hölder condition, and it is shown that conversely every such Hölder-continuous cocycle is governed by a QSDE. Algebraic structure enjoyed by matrix-sum kernels yields a unital *-algebra of processes which allows easy deduction of homomorphic properties of cocycles on a ‘quantum semigroup’. This yields a simple proof that every quantum Lévy process may be realised in Fock space. Finally perturbation of cocycles by Weyl cocycles is shown to be implemented by the action of the corresponding Euclidean group on Schürmann triples.

AB - Stochastic convolution cocycles on a coalgebra are obtained by solving quantum stochastic differential equations. We describe a direct approach to solving such QSDE's by iterated quantum stochastic integration of matrix-sum kernels. The cocycles arising this way satisfy a Hölder condition, and it is shown that conversely every such Hölder-continuous cocycle is governed by a QSDE. Algebraic structure enjoyed by matrix-sum kernels yields a unital *-algebra of processes which allows easy deduction of homomorphic properties of cocycles on a ‘quantum semigroup’. This yields a simple proof that every quantum Lévy process may be realised in Fock space. Finally perturbation of cocycles by Weyl cocycles is shown to be implemented by the action of the corresponding Euclidean group on Schürmann triples.

KW - Noncommutative probability

KW - Quantum group

KW - Stochastic cocycle

KW - Quantum Lévy process

KW - Quantum stochastic

U2 - 10.1016/j.anihpb.2004.10.002

DO - 10.1016/j.anihpb.2004.10.002

M3 - Journal article

VL - 41

SP - 581

EP - 604

JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

SN - 0246-0203

IS - 3

ER -