Invariant Behavior Of Stochastic Evolutionary Equations Driven By Degenerate Noise

In this thesis,it is concerned with three kinds of stochastic evolution equations driven by degenerate noise: atmosphere-ocean equation,reaction-diffusion equation with the fractional Laplace operator in a bounded interval,and quasi-geostrophic flows equation.Their ergodicity is investigated in particular.Firstly,the atmosphere-ocean equation,describing climate and geophysical phenomena,is studied.The equation has complex boundary conditions.Through introducing a transform to deal with the complicated boundary conditions,it is established the existence of the invariant measure by applying the Krylov-Bogoliubov argument.Because of degenerate noise,the corresponding Malliavin matrix is irreversible.It is proposed to combine the strong Feller with the asymptotically strong Feller and the irreducibility to show the uniqueness of the invariant measure,which further derives the ergodicity of the system.Next,the reaction-diffusion equation with fractional Laplace operator is considered.The fractional Laplace operator and degenerate noise make the reaction-diffusion equation more complex.By introducing a suitable weighted space,it establishes the well-posedness,and derives the existence of the invariant measure.Moreover,because the degenerate noise drives the equation,so it is obtained the uniqueness of the invariant measure by combining the asymptotically strong Feller,instead of the strong Feller,and irreducibility.Finally,the stochastic quasi-geostrophic flows equation with degenerate additive noise is researched.Through constructing a new function and using the theory of ergodicity,it is obtained the exponential ergodicity of stochastic quasi-geostrophic flows equations.