A stochastic pitchfork bifurcation in a reaction-diffusion equation

We study in some detail the structure of the random attractor for the Chafe e-Infante reaction-diffusion equation perturbed by a multiplicative white n oise, du = (Deltau + betau - u(3)) dt + sigmau circle dW(t), x is an element of D subset of R-m. First we prove., for m less than or equal to 5, a lower bound on the dimens ion of the random attractor, which is of the same order in beta as the uppe r bound we derived in an earlier paper. and is the same as that obtained in the deterministic case. Then we show, for m = 1. that as beta passes throu gh lambda (1) (the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infi nite-dimensional setting. Central to our approach is the existence of a ran dom unstable manifold.