Deviation, ergodicity and reduction for random dynamical systems

This thesis is about large deviation, ergodicity, and invariant manifold restriction for random dynamical systems. These random dynamical systems are generated by stochastic partial differential equations (SPDEs) or stochastic ordinary differential equations (SODEs), arising from analytical and numerical modeling of geophysical systems.;After introducing some basic concepts in random dynamical systems, the author reviews some literature relevant to the research in this thesis. The research consists of three parts.;First, the author proves the large deviation principle for a stochastic two layer quasi-geostrophic flow model. The proof is based on the Laplace principle and a weak convergence approach. This approach does not require the exponential tightness estimates which are needed in other methods for establishing large deviation principles.;Second, the author considers a mathematical model for a coupled stochastic atmosphere-ocean model. This model consists of the large scale quasi-geostrophic oceanic flow equation and the transport equation for oceanic temperature, coupled with an atmospheric energy balance equation. After reformulating this coupled model as a random dynamical system (cocycle), it is shown that the coupled system has a random attractor, and under further conditions on the physical data and the covariance of the noise, the system is ergodic. Namely, for any observable of this coupled system, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.;Finally, motivated by the need of efficient simulation of high dimensional systems of stochastic ordinary differential equations, arising from numerical projection or truncation (for example, spectral method) of the above two stochastic partial differential equations systems, the author investigates dimension reduction for stochastic dynamical systems. The author has derived an almost sure invariant manifold restriction principle for systems of stochastic differential equations. The restriction of the original stochastic system on this deterministic invariant manifold is still a stochastic system but with lower dimension.