Equivalence of recurrence and Liouville property for symmetric Dirichlet forms

Abstract

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 $T_{t}u\le u$ $m$-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.

9 pages; the numbering of mathematical statements and references has been modified in accordance with the journal version