Given a stochastic process \begin{equation} x_{k+1}=Ax_k+Gw_k, \end{equation} where $w_k\in \Re^r$ is a zero mean white Gaussian noise, such that $w_k\sim\mathcal{N}(0,Q)$, $x_k\in \Re^n$, $A\in\Re^{n\times n}$ and $G\in\Re^{n\times r}$ are two constant full rank matrices.

I think, if $r=n$ and $G$ is nonsingular, then the probability density function $p(x_{k+1}|x_k)$ is given by \begin{equation} p(x_{k+1}|x_k)=p(x_{k+1}-Ax_k)=p(Gw_k)\propto p(w_k), \end{equation} where \begin{equation} p(w_k)=\frac{1}{\sqrt{(2\pi)^r |Q|}}e^{-\frac{1}{2}w_k^\top Q^{-1} w_k}. \end{equation}

Now, my question is that if $r<n$ and $G$ is full colcumn rank constant matrix, what is the probability density function $p(x_{k+1}|x_k)$??

Many thanks




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