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(d[Omega]) = L-1 exp {[lambda][integral operator][integral operator]0 [less-than-or-equals, slant] s < t [less-than-or-equals, slant] 1 [delta]([Omega](t) - [Omega](s)) ds dt}P(d[Omega]), where P is the standard Wiener measure in dimensions two or three, [delta] is the Dirac delta function at 0, L is a renormalizing constant and [lambda] is a positive constant.