Measurability of Random Attractors for Quasi Strong-to-Weak Continuous Random Dynamical Systems

Abstract

In order to obtain the measurability of a random attractor, the RDS is usually required to be continuous which, however, is hard to verify in many applications. In this paper, we introduce a quasi strong-to-weak (abbrev. quasi-S2W) continuity and establish a new existence theorem for random attractors. It is shown that such continuity is equivalent to the closed-graph property for mappings taking values in weakly compact spaces. Moreover, it is inheritable: if a mapping is quasi-S2W continuous in some space, then so it is automatically in more regular subspaces. Also, a mapping with such continuity must be measurable. These results enable us to study random attractors in regularity spaces without further proving the system’s continuity. In addition, applying the core idea to bi-spatial random attractor theory we establish new existence theorems ensuring that the bi-spatial attractors are measurable in regularity spaces. As an application, for a stochastic reaction–diffusion equation with general conditions we study briefly the random attractor in $$H^1(\mathbb {R}^d)$$ , the $$(L^2(\mathbb {R}^d), H^1(\mathbb {R}^d) )$$ -random attractor and the $$(L^2(\mathbb {R}^d),L^p(\mathbb {R}^d))$$ -random attractor, $$p>2$$ , $$d\in \mathbb {N}$$ .