Intermediate spaces, Gaussian probabilities and exponential tightness

TL;DR

This paper establishes the existence of an intermediate Banach space with full measure and compact embedding for Gaussian measures on Banach spaces, extending known results and deriving exponential tightness.

Contribution

It introduces a new intermediate Banach space between the measure space and its RKHS for Gaussian measures, generalizing Wiener measure results.

Findings

Existence of an intermediate Banach space with full measure

Compact embedding of the intermediate space

Derivation of exponential tightness for Gaussian probabilities

Abstract

Let us consider a Gaussian probability on a Banach space. We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS. Such a space has full probability and a compact embedding. This extends what happens with Wiener measure, where the intermediate space can be chosen as a space of H\"older paths. From this result it is very simple to deduce a result of exponential tightness for Gaussian probabilities.