Gaussian process - Hölder continous paths Let $(X_t)_{t\in\mathbb{R}_+}$ be a centered Gaussian process and $\Gamma:\mathbb{R}^2_+\rightarrow\mathbb{R}$ the Covariance function with the following feature
$\Gamma(s,s)+\Gamma(t,t)-2\Gamma(s,t)\leq C(t-s)^\nu$       $~~~~~~~~~(\star)$
for $t\geq s\geq 0 $ with independent Constants $C,\nu >0$.
I got to prove the following statement:
Show that there exists a Version of $(X_t)_{t\in\mathbb{R}_+}$ with Hölder continous paths and identify the Hölder-exponents.
My idea:
Apply the Kolmogorov-Chentsov-Theorem, but i have no clue to use the feature $(\star)$.
The left hand side of $(\star)$ is obviously $\mathbb{Var}[X_t,X_s]$. 
Thanks for any help!
 A: The Kolmogorov-Chentsov theorem states that if for any $T\gt 0$, there exists $\alpha,\beta$ and $C\gt 0$ such that for any $s,t\in [0,T ]$, 
$$ \tag{1} \mathbb E \left |X_t-X_s\right|^ \alpha \leqslant C \left|t-s\right |^ {1+\beta} , $$ 
then there exists a modification of $\left(X_t\right)_{t\in\mathbb R_+}$ whose paths are $\gamma$-Hölder continuous for each $0\lt\gamma\lt\beta/\alpha$.
Let us see which $\alpha$ and $\beta$ satisfy (1). We know that $X_t-X_s$ is Gaussian and centered. Let us denote by $\sigma_{s,t}^2$ its variance. Then $X_t-X_s \sim \sigma_{s,t}N$, where $N$ is standard Gaussian. Therefore, for each positive $\alpha$, 
$$\mathbb E \left |X_t-X_s\right|^ \alpha=\sigma_ {s,t }^{\alpha}\mathbb E \left |N \right |^\alpha.$$
By $( \star)$, $ \sigma_{s,t}\leqslant C^{1/2}  \left|t-s\right |^{  \eta/2} $ hence 
$$\mathbb E \left |X_t-X_s\right|^ \alpha\leqslant C'\left|t-s\right |^{ \alpha \eta/2}$$
for some constant $C'$ independent of $s$ and $t$.
Thus $(1)$ is satisfied for $\beta$ such that $1+\beta= \alpha \eta/2$, that is, $\beta= \alpha \eta/2-1$.
