Abstract: This thesis uses both analytic and probabilistic methods to study continuous and discrete problems. The main areas of study are the asymptotic properties of p-harmonic measure, and various aspects of the square root of the Poisson kernel. Fix a domain and a boundary point, subject to certain regularity conditions. Consider the part of the boundary that lies within a disc, centered at the fixed boundary point. It is shown that as the radius of the disc tends to zero, the p-harmonic measure of the boundary set decays as an explicitly given power of the radius. The square root of the Poisson kernel is studied in both continuous and discrete settings. In the continuous case the domain is the unit disc, and a Hardy space related to the square root of the Poisson kernel is defined. The main result is that, as opposed to the classical Hardy space, the positive functions do not admit a characterization in terms of an Orlicz space. Similar results are given also in the discrete case, where the domain is instead a regular tree. Further results in the discrete setting include the construction of a nearest neighbor random walk on the tree with exit distribution determined by powers of the Poisson kernel. The minimally thin sets of these random walks are characterized. Finally, we suggest a generalization of a two-dimensional geometric result – the ring lemma – to three dimensions.
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