On sets of zero stationary harmonic measure

In this paper, we study properties of the stationary harmonic measure which are unique to the stationary case. We prove that any subset with an appropriate sub-linear horizontal growth has a non-zero stationary harmonic measure. On the other hand, we show that any subset with at least 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.