-dimensional continuum fields

2interpreted as a local perturbation from Wiener measure; that is, we consider Brown-ian motion under the presence of potentials ϕ and ψ. The corresponding measure on the path space C(R;Rd) is formulated as a Gibbs measure relative to Brownian motion [17], and defined through the so-called DLR equation (see Definition 2.1). Similarly to the lattice field case, such a measure appears as equilibrium states of random time-evolutions of the fields described by Ginzburg-Landau equations (see [8, 9, 7]); [11] also studies the dynamics by using the Dirichlet form theory. In the case ψ = 0, the corresponding Gibbs measure can be realized as a P (φ)1-stationary Markov process and it has been fairly understood (see, e.g., [20, 22]). On the other hand, in the non-Markovian case ψ = 0, the first mathematical treatment was done by [17]; their motivation was originated from quantum field theory, particularly from the Euclidean quantization of a certain Hamiltonian operator called Nelson’s scalar field model [16]: its physical context is a quantum particle in Rd governed by the Schrödinger operator H0 = −(1/2)∆+ϕ and coupled to a free Bose field. It has been known [16] that, when the coupling is restricted to the time interval [−T, T], the Euclidean quantization involves the following types of measures: Z−1T ex