Uwe Kähler (CIDMA, Universidade de Aveiro)
In Hida's white noise analysis the exponential function has two jobs. Through the Bochner-Minlos theorem it generates the white noise measure while it also creates the reproducing kernel of the Fock space in countably many variables. Our question is which functions $\varphi$ can take over both roles. For the class ML of entire functions introduced with D. Alpay and P. Cerejeiras, this yields non-Gaussian probability spaces, including grey noise. In recent joint work with P. Jorgensen we go beyond analyticity. We introduce a scale of classes $\text{ML}_r$ and use Schoenberg's theorem on radial positive definite functions on Hilbert space. With these we prove a Rohlin-type disintegration of the measure into Gaussian components and describe the associated integral of Fock spaces. Furthermore, we present a wide range of examples, such as Tsallis and $q$-exponentials, Mittag-Leffler, hypergeometric and Hurwitz-Lerch functions. We will also show some counterexamples in which no polynomial chaos exists.
Poster