Quasi-invariance for infinite-dimensional Kolmogorov diffusions

TL;DR

This paper establishes quasi-invariance properties of the heat kernel measure for infinite-dimensional Kolmogorov diffusions, using finite-dimensional approximations and uniform functional inequalities.

Contribution

It introduces a novel approach to prove quasi-invariance for infinite-dimensional diffusions by leveraging dimension-independent functional inequalities.

Findings

Proved uniform functional inequalities for finite-dimensional approximations.

Established quasi-invariance of heat kernel measure under initial state changes.

Extended results to infinite-dimensional Kolmogorov diffusions.

Abstract

We prove Cameron-Martin type quasi-invariance results for the heat kernel measure of infinite-dimensional Kolmogorov and related diffusions. We first study quantitative functional inequalities for appropriate finite-dimensional approximations of these diffusions, and we prove these inequalities hold with dimension-independent coefficients. Applying an approach developed by Baudoin, Driver, Gordina, and Melcher previously, these uniform bounds may then be used to prove that the heat kernel measure for certain of these infinite-dimensional diffusions is quasi-invariant under changes of the initial state.