The main objective of this paper is to prove the essential self-adjointness of Dirichlet operators in L²(μ) where μ is a Gibbs measure on an infinite volume path space . This operator can be regarded as a perturbation of the Ornstein–Uhlenbeck operator by a nonlinearity and corresponds to a parabolic stochastic partial differential equation (= SPDE, in abbreviation) on . In view of quantum field theory, the solution of this SPDE is called a P()₁-time evolution.