 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

For such $h$, the exponential is a uniformly integrable stochastic exponential. Under the measure defined by this density, the Girsanov theorem makes $X_t-{\int }_0^t{\dot{h}}_{s}ds=X_t-h(t)$ 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 ${\int }_0^{\infty }{\dot{h}}_s^{2}ds=\infty$, 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.