Construction of Gibbs measures for 1-dimensional continuum fields

We study 1-dimensional continuum fields of Ginzburg-Landau type under the presence of an external and a long-range pair interaction potentials. The corresponding Gibbs states are formulated as Gibbs measures relative to Brownian motion [17]. In this context we prove the existence of Gibbs measures for a wide class of potentials including a singular external potential as hard-wall ones, as well as a non-convex interaction. Our basic methods are: (i) to derive moment estimates via integration by parts; and (ii) in its finite-volume construction, to represent the hard-wall Gibbs measure on C(ℝℝ ⁺) in terms of a certain rotationally invariant Gibbs measure on C(ℝℝ³).