Free L\'evy Processes on Dual Groups

We give a short introduction to the theory of L\'evy processes on dual groups. As examples we consider L\'evy processes with additive increments and L\'evy processes on the dual affine group.Comment: 12 pages, Extended abstract to be published in Mini-proceedings: Second MaPhySto Conference on ``L\'evy Processes - Theory and Applications,'' January 2002, Aarhus, Denmar

What is Stochastic Independence?

The notion of a tensor product with projections or with inclusions is defined. It is shown that the definition of stochastic independence relies on such a structure and that independence can be defined in an arbitrary category with a tensor product with inclusions or projections. In this context, the classifications of quantum stochastic independence by Muraki, Ben Ghorbal, and Sch\"urmann become classifications of the tensor products with inclusions for the categories of algebraic probability spaces and non-unital algebraic probability spaces. The notion of a reduction of one independence to another is also introduced. As examples the reductions of Fermi independence and boolean, monotone, and anti-monotone independence to tensor independence are presented

Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures

The equivalence of the characteristic function approach and the probabilistic approach to monotone and boolean convolutions is proven for non-compactly supported probability measures. A probabilistically motivated definition of the multiplicative boolean convolution of probability measures on the positive half-line is proposed. Unlike Bercovici's multiplicative boolean convolution it is always defined, but it turns out to be neither commutative nor associative. Finally some relations between free, monotone, and boolean convolutions are discussed.Comment: 32 pages, new Lemma 2.

Boolean convolution of probability measures on the unit circle

We introduce the boolean convolution for probability measures on the unit circle. Roughly speaking, it describes the distribution of the product of two boolean independent unitary random variables. We find an analogue of the characteristic function and determine all infinitely divisible probability measures on the unit circle for the boolean convolution.Comment: 13 pages, to appear in volume 15 of Seminaires et Congre

Multiplicative monotone convolutions

Recently, Bercovici has introduced multiplicative convolutions based on Muraki's monotone independence and shown that these convolution of probability measures correspond to the composition of some function of their Cauchy transforms. We provide a new proof of this fact based on the combinatorics of moments. We also give a new characterisation of the probability measures that can be embedded into continuous monotone convolution semigroups of probability measures on the unit circle and briefly discuss a relation to Galton-Watson processes.Comment: 14 page

Stochastic analysis for obtuse random walks

We present a construction of the basic operators of stochastic analysis (gradient and divergence) for a class of discrete-time normal martingales called obtuse random walks. The approach is based on the chaos representation property and discrete multiple stochastic integrals. We show that these operators satisfy similar identities as in the case of the Bernoulli randoms walks. We prove a Clark-Ocone-type predictable representation formula, obtain two covariance identities and derive a deviation inequality. We close the exposition by an application to option hedging in discrete time.Comment: 26 page

Approximation of quantum Levy processes by quantum random walks

Every quantum Levy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks.Comment: 7 page

On ergodic properties of convolution operators associated with compact quantum groups

Recent results of M.Junge and Q.Xu on the ergodic properties of the averages of kernels in noncommutative L^p-spaces are applied to the analysis of the almost uniform convergence of operators induced by the convolutions on compact quantum groups.Comment: 10 pages, to appear in Colloquium Mathematicum. (v2 corrects the unwieldy text format