Free Stochastic Measures via Noncrossing Partitions Ii

Rigorous definitions of all these objects in terms of Riemann sums are given below. These definitions were motivated by [RW97], where corresponding definitions were given for usual Levy processes. There is a number of differences between the classical and the free case. First, the free increments property implies that Stπ = 0 unless π is a noncrossing partition. Second, the point of the analysis of [RW97] was that while we are really interested in the stochastic measures Stπ, these are rather hard to define or to handle. However, by use of Möbius inversion these can be expressed through the Prπ. It is easy to see that if the increments of the process X commute, in the defining expression for Prπ all the terms corresponding to the same class can be collected together, and the result is just a product measure over the classes of the partition, Prπ = ∆B1∆B2 · · ·∆Bk . So in this way stochastic measures St can