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arXiv · 0711.3787

Free Brownian motion and evolution towards boxplus-infinite divisibility for k-tuples

Serban T. Belinschi·Alexandru Nica
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Abstract

Let D be the space of non-commutative distributions of k-tuples of selfadjoints in a C*-probability space (for a fixed k). We introduce a semigroup of transformations B_t of D, such that every distribution in D evolves under the B_t towards infinite divisibility with respect to free additive convolution. The very good properties of B_t come from some special connections that we put into evidence between free additive convolution and the operation of Boolean convolution. On the other hand we put into evidence a relation between the transformations B_t and free Brownian motion. More precisely, we introduce a transformation Phi of D which converts the free Brownian motion started at an arbitrary distribution m in D into the process B_t (Phi(m)), t>0.

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Serban T. Belinschi, Alexandru Nica. 2007-11-23. Free Brownian motion and evolution towards boxplus-infinite divisibility for k-tuples. https://arxiv.org/abs/0711.3787

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