Let $k:T \times T \to \mathbb{R}$ be a positive definite kernel on a set $T$. Is it then true that there exists a Gaussian process $\{X_t\}_{t \in T}$ such that $\text{Cov}(X_s,X_t) = k(s,t)$ for all $s,t \in T$?
If yes, then are there any necessary and sufficient conditions on $k$ for the paths of this process to be continuous almost everywhere? (Under the additional assumption that $T$ is a topological/metric space)