  Let $(W,H,\mu)$ be the classical Wiener space, assume that $U=I_W+u$ is an
adapted perturbation of identity where the perturbation $u$ is an equivalence
class w.r.to the Wiener measure. We study several necessary and sufficient
conditions for the almost sure invertibility of such maps. In particular the
subclass of these maps who preserve the Wiener measure are characterized in
terms of the corresponding innovation processes. We give the following
application: let $U$ be invertible and let $\tau$ be stopping time. Define
$U^\tau$ as $I_W+u^\tau$ where $u^\tau$ is given by $$ u^\tau(t,w)=\int_0^t
1_{[0,\tau(w)]}(s)\dot{u}_s(w)ds\,. $$ We prove that $U^\tau$ is also almost
surely invertible. Note that this has immediate applications to stochastic
differential equations.
