Meixner classes and the square of white noise

The introduction of a new (multiplicative) renormalization procedure leads to a Lie algebra for the square of white noise which turns out to be a current algebra on the Lie algebra $sl(2, \R)$. All the representations of this algebra enjoying a certain irreducibility property are constructed. A one parameter class of classical processes is defined in terms of the generators. The vacuum distributions of these processes are identified with the three exceptional (i.e. non Gaussian or Poisson) classes in the Meixner classification. This class of distributions has been recently studied by several authors in connection with different problems arising in decision theory, mathematical finance, quantum field theory, classical probability, $\dots$. In the last section of the paper these developments will be quickly reviewed.