Integrability on the Abstract Wiener Space of Double Sequences and Fernique Theorem

The integrability of a function defined on the abstract Wiener space of double Fourier coefficients is explored. The abstract Wiener space is also a Hilbert space. We define an orthonormal system of the Hilbert space to establish a measure and integration on the abstract Wiener space. We examine the integrability of a function e α · 2 defined on the abstract Wiener space for Fernique theorem. With respect to the abstract Wiener measure, the integral of the function turns out to be convergent for α < 1 / 2 . The result provides a wider choice of the constant α than that of Fernique.

Intermediate spaces, Gaussian probabilities and exponential tightness

We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS, thus extending what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result, it is very simple to deduce a result of exponential tightness for Gaussian probabilities.