  We study global properties of Dirichlet forms such as uniqueness of the
Dirichlet extension, stochastic completeness and recurrence. We characterize
these properties by means of vanishing of a boundary term in Green's formula
for functions from suitable function spaces and suitable operators arising from
extensions of the underlying form. We first present results in the framework of
general Dirichlet forms on $\sigma$-finite measure spaces. For regular
Dirichlet forms our results can be strengthened as all operators from the
previous considerations turn out to be restrictions of a single operator.
Finally, the results are applied to graphs, weighted manifolds, and metric
graphs, where the operators under investigation can be determined rather
explicitly.
