Restricted Markov uniqueness for the stochastic quantization of $P(\Phi)_2$ and its applications

In this paper we obtain restricted Markov uniqueness of the generator and uniqueness of probabilistically weak solutions for the stochastic quantization problem in both the finite and infinite volume case by clarifying the precise relation between the solutions to the stochastic quantization problem obtained by the Dirichlet form approach and those obtained in [DD03] and in [MW15]. We prove that the solution $X-Z$, where $X$ is obtained by the Dirichlet form approach in [AR91] and $Z$ is the corresponding O-U process, satisfies the corresponding shifted equation (see (1.4) below). Moreover, we obtain that the infinite volume $P(\Phi)_2$ quantum field is an invariant measure for the $X_0=Y+Z$, where $Y$ is the unique solution to the shifted equation.