On Large Deviations of Gaussian Measures in Banach Spaces

Abstract

Throughout this paper E will denote a separable Banach space and γ will be a full centered Gaussian measure on E. Define an operator S: E’ → E by S f = ∫ x f(x)γ(dx), and a scalar product <, >_γ on SE’ by

$$ <Sf,Sg>_\gamma =\int f(x)g(x)\gamma(dx) $$

The completion of S E’ with respect to the norm ∥x;∥_γ = √<x,x >_γ is denoted by H_γ and called the reproducing kernel Hilbert space of γ. Since

$$||Sf||\leq _{||g||E^{*}\leq1}^<Superscript>p</Superscript>[\int g^{2}(x)\gamma(dx)]^{1/2}\cdot ||Sf||_{\gamma} $$

Hγ can and will be viewed as a subset of E. For details on the construction and properties of the reproducing kernel Hilbert space we refer the reader to [2] or [5].