Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we always find a reproducing kernel Hilbert space (RKHS) $\mathcal{H}_r$ with reproducing kernel $r$ that almost surely contains paths of a version of $\mathcal{f}$? If not, can you please provide a counter-example, and would the result hold with stronger regularity assumptions on $k$ and/or compactness of $\mathcal{X}$ ? If so, why?
As a backgound, Lukic & Beder (2001), Theorem 5.1 provides a sufficient condition for the paths of a version of $f$ to lie in $\mathcal{H}_r$, namely that $r>>k$ (i.e. $r$ should dominate $k$, or equivalently $\mathcal{H}_k \subseteq \mathcal{H}_r$, and the dominance operator should be of trace-class type). I'm wondering whether this could be used to conclude.