Attractive polymer models for two- and three-dimensional Brownian motion

We show the existence of a weakly self-attractive Brownian motion in dimensions two and three. In other words, we show the existence of a “polymer measure” that is formally defined by P(dΩ) = L−1 exp {λ∫∫0 ⩽ s < t ⩽ 1 δ(Ω(t) − Ω(s)) ds dt}P(dΩ), where P is the standard Wiener measure in dimensions two or three, δ is the Dirac delta function at 0, L is a renormalizing constant and λ is a positive constant.