On the analytic theory of non-commutative distributions in free probability
Tobias Mai, Universität Des Saarlandes, Universität Des Saarlandes
Non-commutative distributions constitute the backbone of non-commutative probability in general and free probability in particular. In the multivariate case, these objects are mostly treated by combinatorial means, because an analytic description in terms of measures -as one is used to in classical probability -fails due to the underlying noncommutativity. However, the rapidly growing field of "free analysis", which is developed as a counterpart of classical analysis at the highest degree of
... -commutativity, offers already many powerful tools for an analytic treatment of non-commutative distributions. Indeed, during the last years, some significant progress has been made on very fundamental questions in that context. This thesis reports on successful attempts which follow the common strategy that properties of the joint non-commutative distribution µ X 1 ,...,Xn of non-commutative random variables X 1 , . . . , X n can be understood by studying the singlevariable distributions µ f (X 1 ,...,Xn) of f (X 1 , . . . , X n ) for suitable "non-commutative test functions" f . More precisely, we will discuss here the following topics: Computation of analytic distributions and of Brown measures: If X 1 , . . . , X n are freely independent non-commutative random variables, then the single-variable distributions µ X 1 , . . . , µ Xn fully determine their joint non-commutative distribution µ X 1 ,...,Xn and so µ P (X 1 ,...,Xn) for any non-commutative polynomial P . Nevertheless, apart from a few special cases, there was for a long time no general machinery for making this relation explicit. We will explain how the so-called "linearization trick" in several refined versions gives in combination with operator-valued free probability an algorithmic solution to this problem, which applies even more to non-commutative rational expressions and is moreover easily accessible for numerical computations. Depending on the concrete situation, µ P (X 1 ,...,Xn) can be encoded either by the analytic distribution or (at least partially) by the Brown measure of P (X 1 , . . . , X n ). Regularity questions: Free probability has produced some deep quantities like free Fisher information, free entropy, and free entropy dimension that can be attached to families of non-commutative random variables. Despite the lack of a rigorous justification, it was believed that these quantities measure the regularity of the corresponding non-commutative distributions. Following the so-called non-microstates approach, we will give evidence to this by showing that maximality of the free entropy dimension excludes atoms in the distribution of any non-constant self-adjoint polynomial expressions in these variables. Furthermore, we will see that the method of this proof can be generalized such that it also applies to free stochastic calculus. We will use this to exclude atoms in the distribution of any non-constant and self-adjoint element in the finite Wigner chaos (which is the free counterpart of the Wiener-Itô chaos in classical probability theory).