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 $C(T;\mathbb{R})$. 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!