# Equivalence of recurrence and Liouville property for symmetric Dirichlet forms

## TL;DR

This paper establishes the equivalence between recurrence, irreducibility, and the Liouville property for symmetric Dirichlet forms, providing a unified analytic framework without extra assumptions.

## Contribution

It proves that recurrence and irreducibility of symmetric Dirichlet forms are equivalent to a Liouville-type property and characterizes $ ext{excessive}$ functions in this context.

## Key findings

- Recurrence and irreducibility are equivalent to the Liouville property.
- The set of zero-energy functions is exactly the constant functions.
- Characterization of $ ext{excessive}$ functions in terms of the extended Dirichlet space.

## 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 $T_{t}u\leq u$ $m$-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.