In this paper, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(t^{2+ε})$. When time is discrete, we also prove a better upper bound of $o(n^{2/3+ε})$, on the maximum height of the aggregate at time $n$. An important tool developed in this paper, is an interface growth process, bounding any process growing according to the stationary harmonic measure. Together with [10] one obtains non zero growth rate for any such process.