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Quantum stochastic convolution cocycles I.

Research output: Contribution to journalJournal article

Published

Journal publication date1/06/2005
JournalAnnales de L'Institut Henri Poincaré (B) Probabilités et Statistiques
Journal number3
Volume41
Number of pages24
Pages581-604
Original languageEnglish

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 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.

Bibliographic note

RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics