Abstract
Although the boundary behavior of harmonic functions is an old subject (Fatou’s theorem was proved in 1906), interesting results are still being obtained today. In this article we discuss some recent results concerning harmonic functions, heat kernels, and related topics that have been obtained using Brownian motion. In the following sections we will discuss the heat kernels for the Neumann Laplacian, the boundary Harnack principle, the Martin boundary, conditional lifetimes, and the conditional gauge theorem.
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