Strong Uniqueness for Certain Infinite Dimensional Dirichlet Operators and Applications to Stochastic Quantization

Strong and Markov uniqueness problems in L 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-setting is discussed. As a direct application essential self–adjointness and strong uniqueness in L 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 ⊂ R.