Free Stochastic Measures via Noncrossing Partitions

Abstract

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 Ito^-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. Academic Press

Itô Formula for Free Stochastic Integrals

“…Let X be a free stochastic measure with distribution m. Proof. For part (1), by stationarity, it suffices to prove that…”

“…Our main references for the background in free probability are [9,12]; see also the references in [1]. Let n be a compactly supported…”

“…Noncommutative stochastic processes with freely independent increments, especially the free Brownian motion, have been investigated in a number of sources; see [1,4] and their references. In [1], we started the analysis of such processes, which we also call free stochastic measures, using the combinatorial machinery inspired by the work of Rota and Wallstrom [8]. Starting with a free stochastic process X(t), in that paper we defined a family of multi-dimensional free stochastic measures derived from it, indexed by set partitions.…”

Noncrossing Partitions in Surprising Locations

“…There are two natural conventions for multiplying permutations: functional notation and algebraist notation. We use the algebraic convention throughout so that (1, 2)(1, 3) is (1, 2, 3) rather than (1,3,2). Because of all the symmetries of the objects under consideration this is only of minor importance, but it means that we need to write "prefix" rather than "suffix" in the foregoing definition.…”

“…First defined and studied by Germain Kreweras in 1972 [33], it caught the imagination of combinatorialists beginning in the 1980s [20], [21], [22], [23], [29], [37], [39], [40], [45], and has come to be regarded as one of the standard objects in the field. In recent years it has also played a role in areas as diverse as lowdimensional topology and geometric group theory [9], [12], [13], [31], [32] as well as the noncommutative version of probability [2], [3], [35], [41], [42], [43], [49], [50]. Due no doubt to its recent vintage, it is less well-known to the mathematical community at large than perhaps it deserves to be, but hopefully this short paper will help to remedy this state of affairs.…”