I have a centered Gaussian process $\mathbb G=(G_t)_{t\geq0}$, for which I don't have the covariance function $\Sigma_{s,t}$.
I know that the process is uniquely determined by the mean function, for which we have here $m(t)=0$ and $\Sigma_{s,t}$.
On the other hand a stochastic process is uniquely determined by its finite dimensional distributions, which are for $n\in\mathbb N$, $t^*=({t_1},\ldots,{t_n})\in\mathbb R^n_+$ arbitrary the distributions of the following vector $$ X^n_{t^*}=(X_{t_1},\ldots,X_{t_n}) $$ In my case I know how they are distributed. Each component $X_{t_1}\sim N(0,\sigma^2_{t_1})$ and the joint distribution $X^n_{t^*}\sim N(0,\sigma^2_{\sum_{i=1}^nt_i}) $ while $$ \sigma^2_{t_1}=\int_{\mathbb R} \left(f(t_1)\right)^2(x) \;dP(x) $$ where I omitted some details (actually the integral is about a sum of two Fourier transforms, which are in fact linear). The important thing here I guess is that $f$ is linear.
From this I actually should be able to construct the covariance function but I just couldn't manage it so far. Can someone help me out here?
I was able to fix it:
As a matter of fact, it's straight forward thinking, I simply computed the variance of $$ \sigma^2_{t_1+t_2}=\int_{\mathbb R} \left(f(t_1+t_2)\right)^2 \;dP(x) $$ which gave me all information I was seeking, since this turned out to be (f is linear in $t_i$) $$ \sigma^2_{t_1+t_2}=\sigma^2_{t_1}+\sigma^2_{t_2}+2\int_{\mathbb R} \left(f(t_1)f(t_2)\right)(x) \;dP(x) $$ and because of the fact that $$ Var(X_1+X_2)=\sigma^2_{X_1}+\sigma^2_{X_2}+2Cov(X_1,X_2) $$ we have $$ Cov(X_1,X_2)=\int_{\mathbb R} \left(f(t_1)f(t_2)\right)(x) \;dP(x) $$ and are done.