Abstract
The purpose of this note is to give an expository survey on the notions of p-parabolicity and p-hyperbolicity of metric measure spaces of locally bounded geometry. These notions are extensions of the notions of recurrence and transience to non-linear operators such as the p-Laplacian (with the standard Laplacian or the 2-Laplacian associated with recurrence and transience behaviors). We discuss characterizations of these notions in terms of potential theory and in terms of moduli of families of paths in the metric space.
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Acknowledgements
The author’s research was partially supported by grants from the National Science Foundation (U.S.), DMS# 1500440 and DMS# 1800161. The author thanks the kind referee for helpful suggestions that improved the exposition of this article.
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Shanmugalingam, N. (2021). p-Hyperbolicity of Ends and Families of Paths in Metric Spaces. In: Freiberg, U., Hambly, B., Hinz, M., Winter, S. (eds) Fractal Geometry and Stochastics VI. Progress in Probability, vol 76. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-59649-1_8
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