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 $\mathcal{E}$-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.
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