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.



When does a stochastic process have its sample paths a.s. in the reproducing kernel hilbert space (RKHS) induced by its covariance function?
If $T$ is an infinite set and the Gaussian measure is non-degenerate, then the RKHS (Cameron Martin space, in Bogachev's language) is infinite dimensional; hence $\operatorname{Prob}(X \in \mathcal H(c)) = 0$.
If $T$ is a finite set, then $\operatorname{Prob}(X \in \mathcal H(c)) = 1$.


For both results, see Bogachev, *Gaussian Measures*, Theorem 2.4.7.


As an example, think about Brownian motion; sample paths are nowhere differentiable with probability 1, whereas the RKHS consists of differentiable 'paths'.


For more information on Gaussian series, including Karhunen-Loeve expansion, see Bogachev, Section 3.5.


