Variance of a Gaussian process

Question

I want to prove that if $Y_t$, $0\leq t\leq 1$ is a zero mean Gaussian process such that there exist $a,b$ with $$\operatorname{Var}(Y_t-Y_s) \leq a|t-s|^b, \;\; s,t\in[0,1]$$ then there exists a version of $Y$ with continuous paths on $[0,1]$. It reminds me of the Kolmogorov continuity theorem, only that instead of expectation we have variance so I tried doing some algebra on this and then using the Kolmogorov continuity theorem but to no avail... Any help would be appreciated!

Answer 1

If $\mathbb E[Y_s]=0$ for each $s\in [0,1]$, then for each $s,t\in [0,1]$, $\mathbb E[Y_t-Y_s]=0$, hence $$\operatorname{Var}(Y_t-Y_s)=\mathbb E\left[(Y_t-Y_s)^2\right]-(\mathbb E[Y_t-Y_s])^2=\mathbb E\left[(Y_t-Y_s)^2\right]$$ and we can use Kolmogorov's continuity theorem.