Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems.(English)Zbl 1401.37088

Summary: 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, $$p>2$$, $$d\in \mathbb{N}$$.

MSC:

 37L55 Infinite-dimensional random dynamical systems; stochastic equations 37H10 Generation, random and stochastic difference and differential equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35K57 Reaction-diffusion equations
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