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Analysis on local Dirichlet spaces. I: Recurrence, conservativeness and $$L^ p$$-Liouville properties. (English) Zbl 0806.53041
Local Dirichlet spaces are used as an appropriate frame to unify and extend various famous results from differential geometry obtained originally by Yau, Cheng/Yau, Karp, Karp/Li, Grigor’yan. Sharp conditions for recurrence as well as for conservativeness and for exponential instability are given. These conditions are in terms of the volume growth $$r \mapsto m(B_ r(x))$$ of metric balls $$B_ r(x) \subset X$$ which are defined intrinsically by the respective Dirichlet form on $$L^ 2(X,m)$$. Also $$L^ p$$-growth conditions for nonnegative sub- or supersolutions on $$X$$ are derived. In particular, $$L^ p$$-Liouville theorems are obtained.

##### MSC:
 31C25 Dirichlet forms 60J45 Probabilistic potential theory 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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