Random attractor for a damped sine-Gordon equation with white noise

Abstract

We prove the existence of a compact random attractor for the random dynamical system generated by a damped sine-Gordon with white noise. And we obtain a precise estimate of the upper bound of the Hausdorff dimension of the random attractor, which decreases as the damping grows and shows that the dimension is uniformly bounded for the damping. In particular, under certain conditions, the dimension is zero.