First, let us give the setting.
Let $(\Omega, \Sigma, \mathbf{P})$ be a probability space, let $T$ be some interval of time, and let $X: T \times \Omega \rightarrow S$ be a stochastic process.
By Mean Square continuity I mean:
Given a time $t \in T, X$ is said to be continuous in mean-square at $t$ if $\mathrm{E}\left[\left|X_t\right|^2\right]<+\infty$ and $$ \lim _{s \rightarrow t} \mathbf{E}\left[\left|X_s-X_t\right|^2\right]=0 $$
One can check here also the definitions of continuity with probability 1 and sample continuity.
Q: I would like to know under which conditions for Gaussian Processes, continuity in mean square implies sample continuity.
I know that there exists the Kolmogorov Continuity Criterion that gives a sort of an answer. However, it requires an upper bound for the second moment to be bounded for all $s,t \in T$ where $T$ compact. A version for Gaussian Processes of this theorem is given here below:
Proposition. If ( $W_t: t \in T$ ) is a centered Gaussian process indexed by a compact set $T \subset \mathbb{R}^d$ with $\mathrm{E}\left|W_s-W_t\right|^2 \leq\|s-t\|^{2 \alpha}$, for all $s, t \in T$ and some $\alpha \in(0,1]$, then $W$ possesses a version with continuous sample paths such that $\left|W_s-W_t\right|=O\left(\|s-t\|^\alpha \log (1 / \| s-\right.$ $t \|)$ ), uniformly in $(s, t)$ with $\|s-t\| \rightarrow 0$, almost surely.
Now imagine that we have something of the form, $\forall \: s,t \in T$:
$$ \mathrm{E}\left|W_s-W_t\right|^2 \leq |s-t| + o(|s-t|)$$
Q: Can we still conclude that the paths are sample continuous?
Another way to rephrase the question in an interesting way is by using the following property of Gaussian Processes:
Theorem A Gaussian process $X$ on $T$ has continuous sample paths with probability one if, and only if, it is continuous at each fixed $t \in T$ with probability one; i.e. $$P \left( \lim_{s \rightarrow t} X_s =X_t, \quad \forall t \in T \right) =1$$ if and only if $$P\left(\lim_{s \rightarrow t} X_s=X_t\right)=1, \quad \forall t \in T$$
Q: Therefore it would be enough to know when, for Gaussian Processes, mean square continuity implies continuity with probability 1.
