  We consider the stochastically forced Burgers equation with an emphasis on
spatially rough driving noise. We show that the law of the process at a fixed
time $t$, conditioned on no explosions, is absolutely continuous with respect
to the stochastic heat equation obtained by removing the nonlinearity from the
equation. This establishes a form of ellipticity in this infinite dimensional
setting. The results follow from a recasting of the Girsanov Theorem to handle
less spatially regular solutions while only proving absolute continuity at a
fixed time and not on path-space. The results are proven by decomposing the
solution into the sum of auxiliary processes which are then shown to be
absolutely continuous in law to a stochastic heat equation. The number of
levels in this decomposition diverges to infinite as we move to the
stochastically forced Burgers equation associated to the KPZ equation, which we
conjecture is just beyond the validity of our results (and certainly the
current proof). The analysis provides insights into the structure of the
solution as we approach the regularity of KPZ. A number of techniques from
singular SPDEs are employed as we are beyond the regime of classical solutions
for much of the paper.
