Given a Gaussian process on some topological space $T$, with a continuous covariance kernel
$C(\cdot,\cdot)\colon T\times T\to R$, we can associate a Hilbert space, which is the reproducing kernel Hilbert space of real-valued functions on $T$, with $C$ as kernel function. This contruction is given in, for instance, R J Adler & J E Taylor: "Random Fields and Geometry", and surely a lot of other places. We can suppose the topological space $T$ is separable.
A very rapid review of the construction is: Define an inner product space $H_0$ as consisting of all real-valued functions on $T$ of the form $f(x) = \sum_{i=1}^n a_i C(x_i,x)$,
for real numbers $a_i$ and points in $T$, $x_i$. We can define an inner product on $H_0$ by
$\left\langle\sum a_i C(x_i,\cdot), \sum b_j C(y_j,\cdot)\right\rangle = \sum \sum a_i b_j C(x_i,y_j)$.
Then the reproducing kernel Hilbert space associated with our gaussian process is the completion $H$ of $H_0$.
Now, this strongly suggests (to be usefull, and by the Karhunen-Loéve theorem, which is based on this construction) that sample paths of our Gaussian process belongs to H with probability 1. This must be proved somewhere, but where? Anybody knows a reference?
Best Answer
The question of continuity of a Gaussian process is a rich one with a lot of theory. Let $T$ be a compact index set, and suppose that $X_t$ is a mean-zero Gaussian process with covariance function $c(t,s)$. The continuity properties of the process $X_t$ are entirely determined by the covariance function.
One very simple condition uses the Kolmogorov continuity theorem. Let $d \ge 1$, and suppose that the index set $T$ is a compact subset of $\mathbb R^d$. Suppose that the covariance function $c$ satisfies $$c(t,t) - 2c(t,s) + c(s,s) \le C|t - s|^{d + \beta},$$ for some positive constants $C$ and $\beta$. Then the $d$-dimensional Kolmogorov theorem implies that, with probability one, there exists a continuous version of $X_t$ on $T$.
This is far from the most general sufficient condition for a Gaussian process to be continuous. Since you have a copy of Adler & Taylor handy, take a look at Section 1.3, Boundedness and Continuity.