Let $\xi\_t$ be a zero-mean gaussian process on $[0,1]$ with covariance operator $C$. 
I would like to better understand the relation between the covariance operator and the regularity of the trajectories. 


I already know that 



> 
> **Theorem** (Kolmogorov) If there exists $\alpha>1,C\geq 0$ and
>  $\epsilon>0$ such that 
>  $$E[|\xi\_t-\xi\_s|^{\alpha}] \leq C
> > |t-s|^{1+\epsilon}$$ then there
>  exists a modification of the process
>  that is almost surely Hölder-$\delta$
>  for $\delta\in ]0,\epsilon/\alpha[$.
> 
> 
> 


I would like to know if there are other results in this direction (with other spaces than Holder ?) and especially one that relates directly a norm (spectral norm under stationnarity assumption ?) of the covariance operator and the regularities of the trajectories. 




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note: if I remember there is a link between the closure of the cameron martin space and the support of the gaussian measure associated to the process... how can I reformulate this to answer my question ? ) 



Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator
Suppose we already know that the process is continuous with probability one, so that the process takes values in the Banach space $X = C([0,1])$ with distribution $\mathbb P$. The covariance operator $C : X^\* \to X$ is then a map from the dual space $X^\*$ to $X$. The support of the Gaussian measure $\mathbb P$ is then the closure of the image of $X^\*$ under $C$: $$\operatorname{supp} \mathbb P = \overline{ CX^\* }.$$ (This is the main theorem of [[Vakhania 1975](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-57/issue-none/The-topological-support-of-Gaussian-measure-in-Banach-space/nmj/1118795361.full)])


Now let's construct the Cameron-Martin space. The operator $C$ defines an inner product on the dual space $X^\*$ by $$\langle f, g \rangle = f(Cg)$$ for $f, g \in X^\*$. The space $X^\*$ isn't necessarily closed under the topology induced by the inner product, so let $H$ be the Hilbert space completion, and let $\iota : X^\* \hookrightarrow H$ be the inclusion map. Define a map $\iota^\* : H \hookrightarrow X$ first on the dense subspace $\iota X^\* \subseteq H$ by $$\iota^\*(\iota f) = Cf,$$ and extend continuously to all of $H$. Thus the covariance operator factors as $C = \iota^\* \circ \iota$.


The Hilbert space $\iota^\* H$ is a subspace of $X$, and is called the Cameron-Martin space of the process. Interpreting this in the context of the support of the Gaussian measure $\mathbb P$, we have $$\operatorname{supp} \mathbb P = \overline{\iota^\* H},$$ so that the closure of the Cameron-Martin space (with respect to the original norm of $X$) is exactly the support of $\mathbb P$.


I go into these ideas in more detail in Section 2 of my preprint [[LaGatta 2010](https://arxiv.org/abs/1003.0975)].


Suppose your covariance operator is an integral operator with kernel $c(s,t)$, called the covariance function of the process. That is, if $\mu$ is a Radon measure on $[0,1]$, then $$(C\mu)(s) = \int\_0^1 c(s,t) \, d\mu(t).$$


If we write $c\_s(t) = c(s,t)$, then the support of $\mathbb P$ is the closure of the span of the functions $c\_s$ in $C([0,1])$. So to answer your question: if you know that the closure $\overline{\operatorname{span}\{c\_s\}}$ is a space with regularity property $P$, then the process $\xi\_t$ satisfies property $P$ with probability one.


