= Solution
The <Cameron-Martin theorem> on Wiener space says that the translated measure $\mathbb P_h(A)=\mathbb P(X+h\in A)$ is equivalent to Wiener measure precisely when $h$ is absolutely continuous, $h(0)=0$, and $\dot h\in L^2(\mathbb R_+)$. In that case
$$
\frac{d\mathbb P_h}{d\mathbb P}
=\exp\left(
\int_0^\infty\dot h_s\,dX_s
-\frac12\int_0^\infty\dot h_s^2\,ds\right).
$$
For such $h$, the exponential is a uniformly integrable <stochastic exponential>. Under the measure defined by this density, the <Girsanov theorem> makes $X_t-\int_0^t\dot h_sds=X_t-h(t)$ a Brownian motion. This identifies the translated law and proves equivalence; replacing $h$ by $-h$ gives the inverse density.
If $h$ fails the Cameron-Martin condition on some finite interval, the finite-horizon theorem gives singularity there. If it belongs locally but $\int_0^\infty\dot h_s^2ds=\infty$, the log likelihood is a Brownian motion run at that diverging energy clock minus half the clock. It tends to $-\infty$ under one measure and to $+\infty$ under the translate, producing disjoint full-measure events. Thus the measures are singular.
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