  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.
