Banach Intermediate Spaces for Gaussian Fr\'{e}chet Spaces

TL;DR

This paper demonstrates the existence and construction of Banach intermediate spaces between a separable Fréchet space and its Cameron-Martin space for Gaussian measures, with applications to Wiener measure.

Contribution

It introduces a method to generate Banach intermediate spaces for Gaussian measures on Fréchet spaces and characterizes these spaces through a partial converse.

Findings

Existence of full measure Banach intermediate spaces for Gaussian measures.

A method to generate such intermediate spaces.

Construction of an α-Hölder intermediate space in Wiener space.

Abstract

In this article, we show that every centered Gaussian measure on an infinite dimensional separable Fr\'{e}chet space $X$ over $\mathbb R$ admits some full measure Banach intermediate space between $X$ and its Cameron-Martin space. We provide a way of generating such spaces and, by showing a partial converse, give a characterization of Banach intermediate spaces. Finally, we show an example of constructing an $\alpha$-H\"older intermediate space in the space of continuous functions, $\mathcal C_0[0, 1]$ with the classical Wiener measure.