A Brownian piece in front of a process absolutely continuous with respect to Brownian motion

Burdzy and Scheutzow asked whether, for every process $(X_t)_{t\ge0}$ whose law is equivalent to Wiener measure, one can put a random piece of path in front of $X$ so that the new process is a Brownian motion. We show that the answer is yes, and that absolute continuity suffices: on an extension of the probability space there are a Brownian motion $B$ and an almost surely finite random integer $N\ge0$ with $X_t=B_{N+t}-B_N$ for all $t\ge0$, so that $(B_t)_{0\le t\le N}$ is the required piece. The proof is Rost's filling scheme for the unit shift of two-sided Brownian motion. We give it in the form of a criterion for an invertible measure-preserving map $\theta$ and a target law $fP$: a randomized forward shift $\theta^N$ carries $P$ to $fP$ if and only if $\mathbb E[f\mid\mathcal I]=1$, where $\mathcal I$ is the invariant $\sigma$-field. The time $N$ cannot in general be a stopping time for $B$, and the auxiliary randomness cannot be dropped from the criterion.