Inserted: 24 sep 2019
Last Updated: 24 sep 2019
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.