Interacting Fock spaces and Gaussianization of probability measures

We prove that any probability measure on $\mathbb R$, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus (in the non symmetric case) a function of the number operator. A corollary of this is that all the momenta of such a measure are expressible in terms of the Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only non crossing pair partitions (and singletons, in the non symmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call {\it Gaussianization}. Finally we define, in terms of the Jacobi parameters, a new convolution among probability measures which we call {\it universal} because any probability measure (with all moments) is infinitely divisible with respect to this convolution. All these results are extended to the case of many (in fact infinitely many) variables.