Abstract:

The generalisation of the notion of Gaussian processes from probability theory is investigated in the context of noncommutative probability theory. A noncommutative Gaussian process is viewed as a linear map from an infinite dimensional (real) Hilbert space into an algebra with involution and a positive normalized functional such that the noncommutative moments of the Gaussian variables depend only on the inner products of the Hilbert space vectors. Such a functional is characterised by a so called positive definite function on pair partitions, as shown by Speicher and Bozejko. The representation space of such processes is a Focklike space over the initial Hilbert space but with general symmetry under 'interchanging the particles' on each level. The Gaussian variables are represented as sums of creation and annihilation operators. Generalised Wick products play an important role in the definition of the algebra associated to a Hilbert space of Gaussian variables. Functors of white noise are analyzed for the class of functions on pair partions having a certain multiplicativity property. The characters of the symmetric group of finitary permutations on the natural numbers are extended to positive definite functions on pair partitions and the corresponding algebras are analyzed. The qproduct of Gaussian processes is defined
