On the large-time dynamics of a class of parabolic equations subjected to homogeneous white noise: Itô's case

Abstract

We present new results concerning the long-time behavior of random fields that are solutions in some variational sense to a class of semilinear parabolic equations subjected to homogeneous white noise. Our main results state that these random fields eventually converge to a random global attractor. This attractor is represented by a single random variable which, with probability one, takes on its values in a two element-set consisting of spatially and temporally homogeneous asymptotic states. We can then completely elucidate the behavior of the random fields around it.