In this article we prove new results concerning thestructure and the stability properties of the global attractor associatedwith a class of nonlinear parabolic stochastic partial differential equationsdriven by a standard multidimensional Brownian motion.We first use monotonicity methodsto prove that the random fields either stabilize exponentially rapidly withprobability one around one of the two equilibrium states, or that they set outto oscillate between them. In the first case we can also compute exactly thecorresponding Lyapunov exponents.The last case of our analysis reveals a phenomenon of exchange of stabilitybetween the two components of the global attractor. In order to prove thisasymptotic property, we show an exponential decay estimate between the randomfield and its spatial average under an additional uniform ellipticityhypothesis.
