Equivalence of recurrence and Liouville property for symmetric Dirichlet forms

• Published in: arXiv.org 2017-10
• Main Author: Kajino, Naotaka
• Format: Article
• Language: English
• Subjects: Dirichlet problem; Equivalence; Markov processes
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1703.08943

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 \emph{\(\mathcal{E}\)-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 \(\{u\in\mathcal{F}_{e}\mid \mathcal{E}(u,u)=0\}=\mathbb{R}\mathbf{1}\), 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.