Geodesic
Equivalence of recurrence and Liouville property for symmetric Dirichlet forms
Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 89-98.

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Given a symmetric Dirichlet form $(\mathcal{E},\mathcal{F})$ on a (non-trivial) $\sigma$-finite measure space $(E,\mathcal{B},m)$ with associated Markovian semigroup $\{T_{t}\}_{t\in(0,\infty)}$, we prove that $(\mathcal{E},\mathcal{F})$ is both irreducible and recurrent if and only if there is no non-constant $\mathcal{B}$-measurable function $u:E\to[0,\infty]$ that is $\mathcal{E}$-excessive, i.e., such that $T_{t}u\leq u$ $m$-a.e. for any $t\in(0,\infty)$. We also prove that these conditions are equivalent to the equality $\{u\in\mathcal{F}_{e}\mid \mathcal{E}(u,u)=0\}=\mathbb{R}1$, where $\mathcal{F}_{e}$ denotes the extended Dirichlet space associated with $(\mathcal{E},\mathcal{F})$. The proof is based on simple analytic arguments and requires no additional assumption on the state space or on the form. In the course of the proof we also present a characterization of the $\mathcal{E}$-excessiveness in terms of $\mathcal{F}_{e}$ and $\mathcal{E}$, which is valid for any symmetric positivity preserving form.
Keywords: symmetric Dirichlet forms, symmetric positivity preserving forms, extended Dirichlet space, excessive functions, recurrence, Liouville property. 
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     title = {Equivalence of recurrence and {Liouville} property for symmetric {Dirichlet} forms},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
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N. Kajino. Equivalence of recurrence and Liouville property for symmetric Dirichlet forms. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 89-98. http://geodesic.mathdoc.fr/item/VVGUM_2017_20_3_a7/

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