Let $(X_t,Y_t)$ be a pair of stochastic processes such that
$$
\begin{aligned}
dX_t =& A_t X_t dt + C_t dW_t,\\
dY_t = & H_t X_t dt + K_tdB_t
\end{aligned}
$$
for some non-random matrix-valued functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter.  It's clear that $X_t$ follows a (multidimensional) Ornstein-Uhlembeck process so is distributed according to this [wiki post][1].  However, what is the distribution of $Y_t$?  It's not Gaussian... is it?


  [1]: https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process