In this article we prove new results concerning the long-time behaviour of random fields that are solutions in some generalized sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually converge with probability. one to a global attractor represented by a single random variable whose properties we investigate in detail. We analyze the partial differential equations of this article in light Itô's stochastic calculus and thereby obtain stabilization and stability results which are substantially different from our earlier results concerning their interpretation in the sense of Stratonovitch. In particular, the asymptotic properties of the random fields that we investigate here exhibit no recurrence and oscillatory properties