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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.

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 Holder-$\delta$ HÃ¶lder-$\delta$ 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.

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 ? )

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# Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator

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.

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 Holder-$\delta$ for $\delta\in ]0,\epsilon/\alpha[$.