  We are mainly concerned with equations of the form $-Lu=f(x,u)+\mu$, where
$L$ is an operator associated with a quasi-regular possibly nonsymmetric
Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability
conditions, and $\mu$ is a bounded smooth measure. We prove general results on
existence, uniqueness and regularity of probabilistic solutions, which are
expressed in terms of solutions to backward stochastic differential equations.
Applications include equations with nonsymmetric divergence form operators,
with gradient perturbations of some pseudodifferential operators and equations
with Ornstein-Uhlenbeck type operators in Hilbert spaces. We also briefly
discuss the existence and uniqueness of probabilistic solutions in the case
where $L$ corresponds to a lower bounded semi-Dirichlet form.
