Let $T$ be a compact metrizable space. Consider a centered second order measurable process $(X_t\colon t\in T)$ with continuous covariance function $c(t,s):= \mathbb{E}X_t X_s$.

Are there any known sufficient conditions ensuring that $t \mapsto X_t$ lies almost surely in the reproducing kernel Hilbert space $\mathcal{H}(c)$ generated by $c$? What about necessary conditions on that matter?

The question is motivated by the Karhunen-Loève representation for $X_t$ (endow $T$ with a strictly positive Borel measure $\nu$ for this to make sense), \begin{equation} \sup_{t\in T} \mathbb{E}\left(X_t - \sum_{j=1}^n\langle X_\cdot , \varphi_j\rangle_{L^2(\nu)}\varphi_j(t)\right)^2\rightarrow 0 \end{equation} and the fact that $\{\sqrt{\lambda_j}\varphi_j\}$ is an orthonormal basis for $\mathcal{H}(c)$ so that one may wonder whether the (formal) series \begin{equation} \sum_{j=1}^\infty a_j\,\sqrt{\lambda_j}\varphi_j, \qquad a_j:=\lambda_j^{-1/2}\langle X_\cdot , \varphi_j\rangle_{L^2(\nu)} \end{equation} is a well defined element of $\mathcal{H}(c)$.

In the above, $(\lambda_j,\varphi_j)$ are the eigenvalues / eigenfunctions as given in Mercer's Theorem.