Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$.

Given a $C[0, 1] $ valued random variable $F$, define the translation map $T_F: \Omega \to \Omega$ by $T_F (\omega) = \omega + F(\omega)$, and denote the pushforward of $\mathbb P$ under this map by $T_F^{\ast} \, \mathbb P$.

We view $F$ as a stochastic process and say that $F$ is adapted to $W$ if as a stochastic process, $F_t$ is $\mathcal F_t$ adapted. Denote by $W^{1, 2}$ the space of absolutely continuous functions on $[0, 1]$ with derivative in $L^2$.

Question:Is it true that $\mathbb Q$ is equivalent to $\mathbb P$ if and only if $\mathbb Q = T_F ^\ast \, \mathbb P$ for some adapted $F$ such that $F(0) = 0$ and $F \in W^{1, 2}$ almost surely?