Free infinite divisibility, fractional convolution powers, and Appell polynomials

In this thesis we study the role of k-divisible non-crossing partitions in Free Probability. First, we consider the combinatorial convolution ∗ in the lattices NC of non-crossing partitions and NC of k-divisible non-crossing partitions. We show that convolving k times with the zeta-function in NC is equivalent to convolving once with the zeta-function in NC. This gives new ways of counting objects like k-equal partitions, k-divisible partitions and k-multichains both in NC and NC. We also consider some statistics of block sizes in k-divisible non-crossing partitions. Second, we introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not a multiple of k. For such k-divisible element x, we derive a formula for the free cumulants of x in terms of the free cumulants of x. For this we use our combinatorial results on the lattice of k-divisible non-crossing partitions. We prove that if a and s are free and s is k-divisible then sps and a are free, where p is any polynomial (in a and s) of degree k − 2 in s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results. Next, we show that free multiplicative convolution between a measure concentrated on the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed. We then concentrate on free additive powers of k-symmetric distributions and prove that μ t is a well defined probability measure, for all t > 1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping μ→ μ. Relations between free multiplicative powers and k-divisible non-crossing partitions are also found and generalized to any product of free random variables. We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.

A determinantal form for Δ h -Appell sequences is proposed and general properties are obtained by using elementary linear algebra tools. As particular cases of Δ h -Appell sequences the sequence of Bernoulli polynomials of second kind and the one of Boole polynomials are considered. A general linear interpolation problem, which generalizes the classical interpolation problem on equidistant points, is proposed. The solution of this problem is expressed by a basis of Δ h -Appell polynomials. Numerical examples which justify theoretical results on the interpolation problem are given.

In this article we study the formal side of operations in free harmonic analysis and examine the emerging general picture of all this. We establish an analytic correspondence of semi-rings between Witt vectors and free probability, by building on previous joint work with Friedrich and McKay (Formal groups, Witt vectors and free probability, 2012. arXiv:1204.6522). In particular, an exponential map, which relates the free additive convolution semigroup on $${\mathbb {R}}$$ with the free multiplicative convolution semigroup on either the unit circle or the positive real axis of compactly supported, freely infinitely divisible probability measures, is derived with complex analytic methods. Then we define several novel operations on these sets, discuss their relation with classically infinitely divisible measures and determine the internal geometry of the spaces involved. Finally, we formalise the structure induced by the various operations we have introduced, in the language of operads and algebraic theories.

k-divisible random variables in free probability

The normal distribution is ⊞-infinitely divisible

Quadratic decomposition of Appell sequences

Eta-diagonal distributions and infinite divisibility for R-diagonals

Let I * and I ⊞ be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici–Pata bijection between I * and I ⊞ . The class type W of symmetric distributions in I ⊞ that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between symmetric distributions in I ⊞ and the free counterpart under Λ of the positive distributions in I * is established. It is shown that the class type W does not include all symmetric distributions in I ⊞ and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I *. Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented.

Infinite divisibility for the free additive convolution was studied in Ref. 20. A complete characterization of [Formula: see text]-infinitely divisible distributions was given, and it was explained in Ref. 21 that this characterization is an analogue of the classical Lévy–Khintchine characterization. In fact, the analogue of the Gaussian distribution appeared even earlier, when the central limit theorem for free additive convolution was proven in Ref. 19. In this paper we define the notion of [Formula: see text]-infinitely divisibility and give the description of infinitely divisible compactly supported probability measures relative to the conditionally free convolution. We also show that the Lévy–Khintchine measures associated with a [Formula: see text]-infinitely divisible distribution μ can be calculated, as in the classical or free case, as a weak limit of measures related with the convolution semigroup generated by (μ, φ) for [Formula: see text]-infinitely divisible.

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I 0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I 0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.

In his article "On the free convolution with a semicircular distribution,"\nBiane found very useful characterizations of the boundary values of the\nimaginary part of the Cauchy-Stieltjes transform of the free additive\nconvolution of a probability measure on the real line with a Wigner\n(semicircular) distribution. Biane's methods were recently extended by Huang to\nmeasures which belong to the partial free convolution semigroups introduced by\nNica and Speicher. This note further extends some of Biane's methods and\nresults to free convolution powers of operator-valued distributions and to free\nconvolutions with operator-valued semicirculars. In addition, it investigates\nproperties of the Julia-Caratheodory derivative of the subordination functions\nassociated to such semigroups, extending certain results from the article\n"Partially Defined Semigroups Relative to Multiplicative Free Convolution" by\nBercovici and the author (reference [7] in the text).\n

Free regular convolution semigroups describe the distribution of free subordinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Levy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions, and we establish an integral identity linking the two semigroups. We provide several explicit examples that illustrate this result.

Formulas for the coefficients of the expansion into a series of Appel polynomials associated with a differential equation of parabolic type are obtained. It has been established that Appel polynomials are involved in the formulas for the expansion of the solution to the Cauchy problem for equations of parabolic type into a series of derivatives of the fundamental solution.A new method for solving the Cauchy problem is proposed, the essence of which is to use seriesexpansion in Appel polynomials. The results generalize the method for solving the heat equationon the real axis by expanding it into a series of Hermite polynomials. The connection betweenthe Fourier transform and series in associated Appel polynomials is studied. The issue of using Hermite polynomials for the Laplace transform has been studied.

Consider a Borel probability measure μ on the real line, and denote by {μt : t≥1} the free additive convolution semigroup defined by Nica and Speicher. We show that the singular part of μt is purely atomic and the density of μt is locally analytic, provided that t > 1. The main ingredient is a global inversion theorem for analytic functions on a half plane.

First we show that the quadratic decomposition of the Appell polynomials with respect to the q-divided difference operator is supplied by two other Appell sequences with respect to a new operator \(\mathcal{M}_{q;q^{-\varepsilon}}\), where e represents a complex parameter different from any negative even integer number. While seeking all the orthogonal polynomial sequences invariant under the action of \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\) (the \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\)-Appell), only the Wall q-polynomials with parameter qe/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.