Several Continuities of a Pullback Random Attractor for Stochastic Non-Autonomous Zakharov Lattice Equations

We study the dynamics of stochastic Zakharov lattice equations driven by multiplicative white noise and time-dependent forces. We first deduce a cocycle (non-autonomous random dynamical system) on the product space of real and complex Hilbert spaces. We then prove the cocycle has a pullback random attractor parameterized by time and sample. We mainly establish several continuities including residual dense continuity, diagonally-invariant continuity, full stochastic continuity and full pre-continuity of the pullback random attractor on the time-sample plane with respect to the Hausdorff metric. The key point in the proof is to verify the time-space-sample continuity of the cocycle and local compactness of the pullback random attractor.