  Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on
rigged Hilbert spaces are studied. An analytic approach based on a--priori
estimates is used. The extension of the problem to the $L^p$-setting is
discussed. As a direct application essential self--adjointness and strong
uniqueness in $L^p$ is proved for the generator (with initial domain the
bounded smooth cylinder functions) of the stochastic quantization process for
Euclidean quantum field theory in finite volume $\Lambda \subset \R^2$.
