Let $H \subset E$ be the Cameron-Martin space of a Gaussian measure $\mu$ on a separable Banach space $E$. The Cameron-Martin theorem states that for all $h \in E$ we have $h \in H$ if and only if $\mu(\cdot + h)$ is equivalent to $\mu$, and then the Radon-Nikodym derivative is $$ \frac{d\mu(\cdot + h)}{d\mu}(x) = e^{- \frac12 \|h\|_H + \langle h, x \rangle^\sim} \,, $$ where $\langle h, x \rangle^\sim$ is the Paler-Wiener integral, i.e. the unique continuous extension of the canonical embedding $E^\ast \hookrightarrow L^2(E, \mu; \mathbb{R})$ to the whole of $H$.
I have been told that there exists a version (special case) of Girsanov's theorem which is a generalization of the Cameron-Martin theorem to the case of random shifts $h$. I have also seen Girsanov's theorem being used in this fashion. Yet I am completely unable to find a reference for this fact. Usually reference is given to a classical version of Girsanov's theorem involving Brownian motion, martingales, Itô integrals and so on. In particular, involving time. As I understand there should exist a version of the Cameron-Martin theorem with random shifts which does not involve differentiating between time and space in any way, instead purely using the language of Gaussian Measures on a separable Banach space.
My question is whether this does really exist and if so I would be very happy to know a reference :).
I know reference requests are not exactly high quality questions, but I think creating some clarification on Girsanov vs Cameron-Martin and a link to a reference as a findable page on Google would tremendously help others in my situation in the future. Thanks!
