Dear STAN groups

I am a STAN beginner, so I would like to know whether STAN can handle a hierachical spatio temporal model.

y\left ( s,t \right )=z\left ( x,t \right )\beta+\xi\left ( x,t \right )+\epsilon

\xi\left ( x,t \right )=a\xi\left ( x,t-1 \right )+\omega\left ( x,t \right )

\omega follows a multivariate gaussian distribution with zero mean and a matern defined covriance matrix.

It is clear that this model can be done in INLA (grouped AR1 model), but I intend to run this model through a sampling approach rather than an approximation.

Could STAN handle this model?

More clearly, I do not know how to deal with the second stage of this model, how I multiply the separatable covariance matrices.

Best regards