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) over cap(d omega) = L(-1) exp{lambda integral integral(0 less than or equal to s < t less than or equal to 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.