Q-Gaussian process
q-Gaussian processes are deformations of the usual Gaussian distribution. There are several different versions of this; here we treat a multivariate deformation, also addressed as q-Gaussian process, arising from free probability theory and corresponding to deformations of the canonical commutation relations. For other deformations of Gaussian distributions, see q-Gaussian distribution and Gaussian q-distribution. History. The q-Gaussian process was formally introduced in a paper by Frisch and Bourret under the name of "parastochastics", and also later by Greenberg as an example of "infinite statistics". It was mathematically established and investigated in papers by  Bozejko and Speicher and by Bozejko, Kümmerer, and Speicher in the context of non-commutative probability. It is given as the distribution of sums of creation and annihilation operators in a q-deformed Fock space. The calculation of moments of those operators is given by a q-deformed version of a Wick formula or Isserlis formula. The specification of a special covariance in the underlying Hilbert space leads to the q-Brownian motion, a special non-commutative version of classical Brownian motion. q-Fock space. In the following formula_1 is fixed. Consider a Hilbert space formula_2. On the algebraic full  Fock space where formula_4 with a norm one vector formula_5, called "vacuum", we define a q-deformed inner product as follows: where formula_7 is the number of inversions of formula_8. The "q-Fock space" is then defined as the completion of the algebraic full Fock space with respect to this inner product For formula_10 the q-inner product is strictly positive. For formula_11 and formula_12 it is positive, but has a kernel, which leads in these cases to the symmetric and anti-symmetric Fock spaces, respectively. For formula_13 we define the "q-creation operator" formula_14, given by Its adjoint (with  respect to the q-inner product), the "q-annihilation operator" formula_16, is given by q-commutation relations. Those operators satisfy the q-commutation relations For formula_19, formula_20, and formula_12 this reduces to the CCR-relations, the Cuntz relations, and the CAR-relations, respectively. With the exception of the case formula_22 the operators formula_23 are bounded. q-Gaussian elements and definition of multivariate q-Gaussian distribution (q-Gaussian process). Operators of the form formula_24 for formula_13 are called "q-Gaussian" (or "q-semicircular") elements. On formula_26 we consider the "vacuum expectation state" formula_27, for formula_28. The "(multivariate) q-Gaussian distribution" or  q-Gaussian process is defined as the non commutative distribution of a collection of q-Gaussians with respect to the vacuum expectation state. For formula_29 the joint distribution of formula_30 with respect to formula_31 can be described in the following way,: for any formula_32 we have where formula_34 denotes the number of crossings of the pair-partition formula_35. This is a q-deformed version of the Wick/Isserlis formula. q-Gaussian distribution in the one-dimensional case. For "p" = 1, the q-Gaussian distribution is a probability measure on the interval formula_36, with analytic formulas for its density. For  the special cases formula_19, formula_20, and formula_12, this reduces to the classical Gaussian distribution, the Wigner semicircle distribution, and the symmetric Bernoulli distribution on formula_40. The determination of the density follows from old results on corresponding orthogonal polynomials. Operator algebraic questions. The von Neumann algebra generated by formula_41, for formula_42 running through an orthonormal system formula_43 of vectors in formula_2, reduces for formula_20 to the famous free group factors formula_46. Understanding the structure of those von Neumann algebras for general q has been a source of many investigations. It is now known, by work of Guionnet and Shlyakhtenko, that at  least for finite I and for small values of q, the von Neumann algebra is isomorphic to the corresponding free group factor.