Enhanced Mean Random Attractors for Nonautonomous Mean Random Dynamical Systems in Product Bochner Spaces

We develop the theory of mean random attractors of nonautonomous mean random dynamical systems proposed by Wang (J. Dynam. Differ. Equ., Proc. Amer. Math. Soc., J. Differ. Equ., 2019) in a general setting. Two types of enhanced mean random attractors are introduced by improving the weak compactness and weak attraction in a space of product of finitely many Bochner spaces uniformly over some infinite time-intervals. Then we establish some theoretical results for the existence, topology structures and relations of these enhanced mean random attractors by carefully analyzing the characters and properties of enhanced \(\varOmega \) $\Omega$ -limit sets under the theory of weak topology. Our theoretical results are expected to be applied to many types of stochastic models. In particular, we apply these general results to a class of locally monotone and generally coercive stochastic PDEs, a fractional stochastic reaction–diffusion equation defined on \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ , and a stochastic Schrödinger lattice system with a delay diffusion term defined on \({\mathbb {Z}}^N\) ${\mathbb{Z}}^N$ .