What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?

Question

Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having continuous covariance kernel, the covariance operator having positive eigenvalues, etc.). $Z$ is given by a measure $\mu_Z$ on the Banach space $C(T;\mathbb{R})$ (with the sup norm), and I would really appreciate any results or references for conditions relating to $\mu_Z$ being absolutely continuous to a nondegenerate Gaussian measure on $C(T;\mathbb{R})$ (such as the Wiener measure). In particular, there is a certain Gauss-null subset of $C(T;\mathbb{R})$ (acquired through a Rademacher theorem-type argument) that I want to know when $Z$ avoids with probability one.

If $Z$ is a Gaussian process, then this happens when the covariance operator is strictly positive. There is also a rather straightforward construction mentioned in the second paragraph of this paper that produces some non-Gaussian processes $Z=X+Y$ where $X$ is a nondegenerate Gaussian, $Y$ is some process with the same covariance structure and Fourier coefficients related to the standard normal, and $\mu_Z \ll \mu_X$. Intuitively, it makes sense that smoothing a process via a nondegenerate Gaussian should produce such a path measure, but it would be nice to have a more checkable condition on $Z$ with which I can understand when $\mu_Z$ gives 0 mass to Gauss-null subsets of $C(T;\mathbb{R})$. I would also be interested in understanding, supposing $Z_t$ is a continuous strong solution to an SDE, what conditions on the drift and diffusion of the SDE are sufficient for this property. I am also eager to learn about the case where the path measure is on other separable spaces than $C(T;\mathbb{R})$ (such as $L^2$), but my direct application is as stated.

Any help would be very appreciated. I am a bit new to this side of analysis, so go easy on me :) Thank you for your time, and have a lovely weekend!

Edit: after some thinking in the shower, I realized that Girsanov's theorem actually answers the question for SDEs driven by (nondegenerate) Brownian motion, and if I am not mistaken we should be able to generate most measures equivalent to the Wiener measure in this way. Still very curious about the weaker condition of absolute continuity to the Wiener measure, though.

Answer 1

Let $\mu_0$ be the law of a Brownian motion $B$. Let $\mu$ be any measure equivalent to $\mu_0$. Then by a converse version of Girsanov there exists a progressively measurable $F$ whose sample paths are almost surely in $W_0^{1,2}$ so that $X(t):=B(t)-F(t)$ is a Brownian motion under $\mu$.

If $\mu$ is a probability measure that is merely absolutely continuous with respect to $\mu_0$ with support $S\subset C$ then define $\tilde \mu$ to be any probability measure which agrees up to scaling with $\mu$ on $S$ but is positive outside. Then $\mu$ is just the law of $\tilde \mu$, conditioned to stay in $S$ and $\tilde \mu$ can be described as in the first paragraph.