Weak existence and uniqueness in law for an additive-noise SDE with bounded drift

If $b:\mathbb{R}\to \mathbb{R}$ is bounded and measurable, then on every finite time intervalhas a weak solution and uniqueness in law. Starting with Wiener measure, the Novikov condition and Girsanov theorem add the drift. Applying the inverse change of measure to any weak solution recovers Wiener measure and identifies its law by the same pathwise density.