Let $X$ be a locally convex topological linear space, and $\mathbb P$ be a probability measure on $X$. Denote the mean vector $m \in X$ and covariance operator $k : X^* \to X$. Let $\tau_u : X \to X$ be the translation-by-$u$ operator, and define the translated measure $\mathbb P_u := (\tau_u)_* \mathbb P := \mathbb P \circ \tau_u^{-1}$.

Let $U \subseteq X$ be the Cameron-Martin space of the measure $\mathbb P$, i.e., the Hilbert-space completion of the set $kX^* \subseteq X$.

If $\mathbb P$ is Gaussian, then the Cameron-Martin theorem states that $\mathbb P_u$ is absolutely continuous with respect to $\mathbb P$ if and only if $u \in U$. In that case, the Radon-Nikodym derivative equals $\exp\!\big( \langle u, x\rangle^\sim -\tfrac 1 2 \|u\|^2 \big)$, where $x \mapsto \langle u, x\rangle^\sim$ is the Paley-Wiener integral of $u$.

Suppose that $\mathbb P$ is non-Gaussian, and let $u \in U$. Is $\mathbb P_u$ absolutely continuous with respect to $\mathbb P$? If so, can we write the Radon-Nikodym derivative as $\exp\!\big( \langle u, x\rangle^\sim -\tfrac 1 2 \|u\|^2 + \Phi(u,x) \big)$, for some well-behaved function $\Phi$?