On L^p Liouville theorems for Dirichlet forms

2021/08/26 by Hua, Bobo, Keller, Matthias, Lenz, Daniel +1 · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2108.11678

Abstract

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the L^p generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain L^p growth criteria must be constant. As consequence we give an integral criterion for recurrence.

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• A remark on subharmonicity for symmetric Dirichlet forms

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