The key result in the paper concerns two transformations, Phi(rho, psi) and B_t(psi) on states on the algebra of non-commutative polynomials, or equivalently on joint distributions of d-tuples of non-commuting operators. These transformations are related to free probability: we show that Phi intertwines the action of B_t and the free convolution with the semigroup {rho_t}. The maps {B_t} were introduced by Belinschi and Nica as a semigroup of transformations such that B_1 is the bijection between infinitely divisible distributions in the Boolean and free probability theories. They proved the intertwining property above for a single-variable version of the map Phi and the particular case of the free heat semigroup. The more general two-variable map Phi comes, not from free probability, but from the theory of two-state algebras, also called the conditionally free probability theory, introduced by Bozejko, Leinert, and Speicher. Orthogonality of the c-free versions of the Appell polynomials, investigated in arXiv:0803.4279, is closely related to the single-variable map Phi. On the other hand, more general free Meixner families behave well under all the transformations above, and provide clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Phi and descriptions of its fixed points and range.

This paper describes the analogs of the Appell polynomial families in the context of algebras with two states, also called the c-free probability theory, introduced by Bozejko, Leinert, and Speicher. This theory includes as two extreme cases the free and Boolean probability theories. We prove recursions, generating functions, and factorization and martingale properties for these polynomials. We characterize the orthogonal c-free Appell polynomials in terms of the map introduced previously by Belinschi and Nica. We also note that the free Appell polynomials are exactly the fixed points of the transformation which takes polynomials of the first kind to polynomials of the second kind, and generalize these notions to higher dimensions. Finally, we describe the c-free version of the Kailath-Segall polynomials, their combinatorics, and Hilbert space representations.

The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the Laha-Lukacs type characterization.

A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi parameters under convolution, the relationship between the Jacobi parameters and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials.

The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-order difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.

Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint distribution of operators on a Fock space.

Free Meixner states are a class of functionals on non-commutative polynomials introduced in [Ans08]. They are characterized by a resolvent-type form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a second-order non-commutative differential equation satisfied by their free cumulant functional. In this paper, we construct an operator model for free Meixner states. By combinatorial methods, we also derive an operator model for their free cumulant functionals. This, in turn, allows us to construct a number of examples. Many of these examples are shown to be trivial, in the sense of being free products of functionals which depend on only a single variable, or rotations of such free products. On the other hand, the multinomial distribution is a free Meixner state and is not a product. Neither is a large class of tracial free Meixner states which are analogous to the simple quadratic exponential families in statistics.

Haiman and Schmitt showed that one can use the antipode S_F of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode S_H of an algebraically free analogue H of F may be used to invert non-commutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for S_F are labeled by trees. By contrast, the noncommuting monomials of S_H are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the Connes-Kreimer Hopf algebra.

Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with a stochastic process and using the stochastic measures machinery introduced by Rota and Wallstrom, we calculate and give an interpretation of linearization coefficients for a number of polynomial families. The processes involved may have independent, freely independent, or q-independent increments. The use of non-commutative stochastic processes extends the range of applications significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier, and Rogers and continuous big q-Hermite polynomials.

We continue the investigation of the Lévy processes on a q-deformed full Fock space started in [Ans01]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras.

This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials when polynomials in non-commuting variables are considered. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and "diagram" formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Lévy processes. For more general free Sheffer families, all the pseudo-orthogonal families are described. Another family investigated are the Kailath-Segall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the q-deformed case.

In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota's finite operator calculus can be modified for the free context.

The objects under investigation are the stochastic integrals with respect to free Lévy processes. We define such integrals for square-integrable integrands, as well as for a certain general class of bounded integrands. Using the product form of the Itô formula, we prove the full functional Itô formula in this context.

We show that for stochastic measures with freely independent increments, the partition-dependent stochastic measures of [Ans00] can be expressed purely in terms of the higher stochastic measures and the higher diagonal measures of the original.

On a q-deformed Fock space, we define multiple q-Lévy processes. Using the partition-dependent stochastic measures derived from such processes, we define partition-dependent cumulants for their joint distributions, and express these in terms of the cumulant functional using the number of restricted crossings of P. Biane. In the single variable case, this allows us to define a q-convolution for a large class of probability measures. We make some comments on the Itô table in this context, and investigate the q-Brownian motion and the q-Poisson process in more detail.

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 Itô-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.

We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator. In classical as well as in free probability theory, we consider two methods giving such a linearization, and interpret the result as a weak form of the CLT. In the classical case the analysis involves dilation operators; in the free case more general composition operators appear.

Abstract: A master field is a limiting object in the 1/N expansion of quantum chromodynamics. I give a general introduction to the terminology of quantum field theory. After explaining why the 1/N expansion is reasonable, I describe Singer's construction of the master field in 2 dimensions. It is followed by a more explicit construction of Gopakumar and Gross (see also Xu). In particular, I explain the relation with the free probability theory, more precisely with free Brownian motion.

Part I gives the results of a computer analysis of the time series given by the Wilshire 5000 stock index. The final model for the logarithm of the series is a linear trend plus an order 1 autoregressive process. A spectral analysis of the series is also performed. Part II concentrates on the distribution of the innovations process. This distribution is well approximated by the t distribution on 3.5 degrees of freedom.

The paper provides a pair of descriptions of the equations of motion on a base of a fiber bundle with a Hamiltonian. These descriptions are shown to be equivalent in a non-canonical way; the equivalence is provided by a connection.

Diffusion-Limited Aggregation (DLA) is a probabilistic model which describes the growth of crystals in a solution. In this project the properties of a simpler model for DLA were investigated. Computer simulations were performed which suggested that the model admits of a complete description. Asymptotic analysis of the model showed sharp dichotomy in the limiting behavior of the process depending on the parameter of the model.