Strong and Markov uniqueness problems in L2 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 Lp-setting is
discussed. As a direct application essential self--adjointness and strong
uniqueness in Lp 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 Λ⊂R2