Free stochastic measures via noncrossing partitions

Details Free stochastic measures via noncrossing partitions Free stochastic measures via noncrossing partitions

1999-03-14

arxiv.org/abs/math/9903084v3

Adv. Math. 155 (2000) n. 1, 154-179

Mathematics

Operator Algebras

23 pages, AMS-LaTeX2e; statement of Proposition 3 corrected

Scientific paper

We consider free multiple stochastic measures in the combinatorial framework of the lattice of all diagonals of an n-dimensional space. In this free case, one can restrict the analysis to only the noncrossing diagonals. We give definitions of what free multiple stochastic measures are, and calculate them for the free Poisson and free compound Poisson processes. We also derive general combinatorial It\^{o}-type relationships between free stochastic measures of different orders. These allow us to calculate, for example, free Poisson-Charlier polynomials, which are the orthogonal polynomials with respect to the free Poisson measure.

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