Let a stochastic process defines as: $$X(t+1)=A X(t)+B U(t)$$ with: $X(t) \in R^n$, $U(t) \sim N(0,Q_t)$, $Q_t$ semi-positive-definite of size $n \times n$, $X(0) \sim N(0,W_0)$, $A$ of size $n \times n$, $B$ of size $n \times n$.
Is there any condition on $A$ and $B$ to state the existence and the form of the conditional pdf $p_{X_t|X_{t-1}}(x_t|x_{t-1})$?
Thanks in advance.