An application of a functional inequality to quasi-invariance in infinite dimensions

TL;DR

This paper demonstrates how functional inequalities can establish quasi-invariance of measures in infinite-dimensional spaces, providing new proofs of classical results and extending to geometric examples without relying on traditional reference measures.

Contribution

It introduces a novel approach using functional inequalities to prove quasi-invariance, offering alternative proofs and broader applicability in infinite-dimensional analysis.

Findings

Provides a new proof of the Cameron-Martin theorem using functional inequalities.

Extends quasi-invariance results to geometric settings without reference measures.

Shows the versatility of functional inequalities in infinite-dimensional measure theory.

Abstract

One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron-Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even though the main abstract result concerns quasi-invariance of a measure under a group action on a measure space.