Let $W_t$ be a standard Brownian motion, and $X_t$ a measurable adapted process. Girsanov's theorem says that under certain conditions, the Brownian motion with drift $Y_t = W_t - \int_0^t X_s\,ds$ can be a Brownian motion under a certain equivalent probability measure.
I want to apply Girsanov's theorem with $X_t$ an Ornstein-Uhlenbeck process defined by $dX_t = dW_t - X_t dt$, $X_0 = 0$. In this case we would have $Y_t = X_t$, so I would learn that an Ornstein-Uhlenbeck process can be a Brownian motion under an equivalent measure.
The condition needed for Girsanov's theorem to hold is that $$Z_t = \exp\left(\int_0^t X_s\,dW_s - \frac{1}{2} \int_0^t X_s^2\,ds\right)$$ be a martingale.
Is this condition satisfied?
A sufficient condition, due to Novikov, is that $$E \exp\left(\frac{1}{2} \int_0^T X_s^2\,ds\right) < \infty.$$ I can't seem to see how to verify either of these conditions, though the Ornstein-Uhlenbeck process has so many nice properties that one would think something simple would work.
This question came up while studying the solution of the quantum harmonic oscillator via the Feynman-Kac formula. I am trying to understand the "ground state transformation" in terms of Girsanov's formula.
Thanks!