Ornstein-Uhlenbeck likelihood relative to Wiener measure ($L_t=\exp \left(-\mu {\int }_0^t{\omega }_{s}d{\omega }_s-\frac{{\mu }^2}{2}{\int }_0^t{\omega }_s^{2}ds\right)$)

On each finite horizon the displayed Radon-Nikodym derivative changes Wiener measure into the law of $dX=dB-\mu Xdt$. Stop at coordinate magnitude n so the Novikov condition applies. Under each stopped tilted measure the coordinate solves the drifted equation, and the Gronwall inequality with the Doob L2 maximal inequality bounds its maximal second moment uniformly in n. The probability of reaching n therefore tends to zero; this proves that the unstopped density has expectation one. No unrestricted large-time Novikov estimate is needed.