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We present, starting from a set of canonical axioms, a complete classification of the notions of non-commutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schürmann and is based on a fundamental paper by R. Speicher.

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In a central lemma we characterize 'generating functions' of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to 'unital, associative universal products' on this category, which again define a notion of non-commutative stochastic independence. Using the central lemma, we can prove the existence of cumulants and of 'cumulant Lie algebras' for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in N. Murakis classification, c-free independence of M. Bożejko and R. Speicher, the indented product of T. Hasebe and the bi-free independence of D. Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.

The aim of this article is to characterize unitary increment process by a quantum stochastic integral representation on symmetric Fock space. Under certain assumptions we have proved its unitary equivalence to a Hudson-Parthasarathy flow. §1. IntroductionIn the framework of the theory of quantum stochastic calculus developed by pioneering work of Hudson and Parthasarathy [6], quantum stochastic differential equations (qsde) of the form