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arXiv · 2108.11678

On $L^p$ Liouville theorems for Dirichlet forms

Bobo Hua·Matthias Keller·Daniel Lenz·Marcel Schmidt
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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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Bobo Hua, Matthias Keller, Daniel Lenz, Marcel Schmidt. 2021-08-26. On $L^p$ Liouville theorems for Dirichlet forms. https://arxiv.org/abs/2108.11678

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