  Let $(W,H,\mu)$ be the classical Wiener space where $H$ is the Cameron-Martin
space which consists of the primitives of the elements of
$L^2([0,1],\,dt)\otimes \R^d$, we denote by $L^2_a(\mu,H)$ the equivalence
classes w.r.t. $dt\times d\mu$ whose Lebesgue densities $s\to\dot{u}(s,w)$ are
almost surely adapted to the canonical Brownian filtration. If $f$ is a Wiener
functional s.t. $\frac{1}{E[e^{-f}]}e^{-f}d\mu$ is of finite relative entropy
w.r.t. $\mu$, we prove that \beaa J_\star&=& \inf\left(E_\mu\left[f\circ
U+\half |u|_H^2\right]: u\in
  L_a^2(\mu,H)\right)\\ &\geq&-\log E_\mu[e^{-f}]=\inf\left(\int_W
fd\ga+H(\ga|\mu):\,\nu\in
  P(W)\right) \eeaa where $P(W)$ is the set of probability measures on
$(W,\calB(W))$ and $H(\ga|\mu)$ is the relative entropy of $\ga$ w.r.t. $\mu$.
We call $f$ a tamed functional if the inequality above can be replaced with
equality, we characterize the class of tamed functionals, which is much larger
than the set of essentially bounded Wiener functionals. We show that for a
tamed functional the minimization problem of l.h.s. has a solution $u_0$ if and
only if $U_0=I_W+u_0$ is almost surely invertible and $$
\frac{dU_0\mu}{d\mu}=\frac{e^{-f}}{E_\mu[e^{-f}]} $$ and then $u_0$ is unique.
To do this is we prove the theorem which says that the relative entropy of
$U_0\mu$ is equal to the energy of $u_0$ if and only if it has a $\mu$-a.s.
left inverse. We use these results to prove the strong existence of the
solutions of stochastic differentail equations with singular (functional)
drifts and also to prove the non-existence of strong solutions of some
stochastic differential equations.
  \noindent {\sl Keywords:} Invertibility, entropy, Girsanov theorem,
variational calculus, Malliavin calculus, large deviations
