Co-cyclic Attractors And Uniform Attractors Of Non-autonomous Stochastic Dynamic Systems

In this work, we study cocycle and uniform attractors for non-autonomous random dynamical systems (abbrev. NRDS).Firstly, we study cocycle attractors for RDS and NRDS with only a quasi strong-to-weak (abbrev. quasi-S2W) continuity. This continuity is shown inheritable, i.e., if a mapping is quasi-S2W continuous in some space, then so it is automatically in regular spaces. Moreover, by establishing some existence criteria for cocycle attractors we see that the quasi-S2W continuity is adequate to derive the measurability of the attractor. These observations generalize known existence theorems for random attractors on one hand, and enable us to study cocycle attractors in regular spaces without further prov-ing the system's continuity on the other hand. Applying to bi-spatial cocycle attractor theory, we establish an existence theorem indicating that the measurability of bi-spatial attractors is valid in regular space, not only in the basic phase space as it was in the literature.Secondly, for NRDS we compare cocycle attractors with autonomous and non-autonomous attraction universes, and then for autonomous ones we establish some existence criteria and characterization. We also study the upper semi-continuity of cocycle attractors with respect to non-autonomous symbols to find that the upper semi-continuity is equivalent to uniform compactness of the attractor.Thirdly, a (random) uniform attractor theory for NRDS is established. The uni-form attractor is defined as the minimal compact uniformly pullback attracting random set. About the definition we observe that, the uniform pullback attraction of a uniform attractor implies a uniform forward attraction in probability, and implies also an al-most uniform pullback attraction for discrete time-sequences. Though no invariance is required by definition, uniform attractor can have a negative semi-invariance.We further study the existence of uniform attractors and the relationship between uniform and cocycle attractors. To overcome the measurability difficulty, the symbol space is required to be Polish which is shown fulfilled by any locally integrable forcing if the symbol space is defined as the hull of the forcing. For the relationship between u-niform and cocycle attractors we find, that the uniform attractor of a jointly continuous NRDS is composed of states involved in the cocycle attractor on one hand, and can be regarded as the pullback attractor of a corresponding multi-valued (but autonomous) RDS on the other hand. Moreover, uniform attractors for continuous NRDS are shown determined by uniformly attracting nonrandom compact sets.Cocycle and uniform attractors for reaction-diffusion equation, Ginzburg-Landau equation and 2D Navier-Stokes equation with scalar white noise are studied as applications.