Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

TL;DR

This paper develops stochastic integrals on infinite-dimensional nilpotent Lie groups, enabling the definition of Brownian motion in this setting and establishing key measure-theoretic and functional inequalities.

Contribution

It introduces a novel class of stochastic integrals on abstract Wiener spaces for nilpotent Lie groups, extending stochastic analysis to infinite-dimensional non-commutative structures.

Findings

Cameron--Martin quasi-invariance for heat kernel measures

Radon--Nikodym derivative estimates

Log Sobolev inequality in the infinite-dimensional setting

Abstract

We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.