  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.
