  In this paper, we prove that any subset with an appropriate sub-linear
horizontal growth has a non-zero stationary harmonic measure. On the other
hand, we also show any subset with super-linear horizontal growth will have a
$0$ stationary harmonic measure at every point. This result is fundamental to
any future study of stationary DLA. As an application we prove that any
possible aggregation process with growth rates proportional to the stationary
harmonic measure has non zero measure at all times.
