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.

中文翻译：

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零稳态谐波测量集

在本文中，我们证明了具有适当的亚线性水平增长的任何子集都具有非零平稳谐波测度。另一方面，我们还展示了具有超线性水平增长的任何子集在每个点都将具有 $0$ 的平稳谐波测度。这一结果对于任何未来的静态 DLA 研究都是基础。作为一个应用，我们证明了任何可能的增长率与平稳谐波测度成正比的聚合过程在任何时候都具有非零测度。