Let $W_p = (W^{1},\dots,W^p)$ be a $p$-dimensional Wiener process with covariance matrix $\Sigma$.
It is well known that if $t_1 \leq t_2$ then $\mbox{Cov} ( W^i_{t_1}, W^i_{t_2}) = t_1$ so it's easy to find the covariance structure between a single Wiener process between time-points.
Now let $w^x$ and $w^y$ be $p$-dimensional vectors and say I construct the Gaussian processes $X_t = \langle w^x, W^p\rangle $ (the standard euclidean inner product) and $\langle w^y , W^p \rangle$ and I want to find the simultanous distribution of, say $X_{T_1}$ and $Y_{T_2}$ ? I've been messing around with it a bit, it would seem like it could easily be done using linear transformation of gaussians, but I keep hitting the problem that the dimension of the problem is too high.