Equivalence of recurrence and Liouville property for symmetric Dirichlet forms

Given a symmetric Dirichlet form (ℰ,ℱ) on a (nontrivial) finite measure space (𝐸, ℬ,𝑚) with associated Markovian semigroup {𝑇𝑡}𝑡∈(0,∞), we prove that (ℰ,ℱ) is both irreducible and recurrent if and only if there is no nonconstant ℬmeasurable function : → [0,∞] that is ℰexcessive, i.e., such that ≤ 𝑚a. e. for any ∈ (0,∞). We also prove that these conditions are equivalent to the equality {𝑢 ∈ ℱ𝑒 | ℰ(𝑢, 𝑢) = 0} = R1, where ℱ𝑒 denotes the extended Dirichlet space associated with (ℰ,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 ℰexcessiveness in terms of ℱ𝑒 and ℰ, which is valid for any symmetric positivity preserving form.