Ergodicity Of Fractional Burgers Equation With Degenerate Noise On A Bounded Domain

This thesis focuses on the fractional Burgers equation with degenerate noise in bounded intervals,especially the uniqueness of its invariant measure.Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise,the study of the system becomes more complicated.In order to get over the difficulties caused by the fractional Laplacian operator,the usual Hilbert space does not fit the system,hence we introduce an appropriate weighted function to construct weighted spaces to study it.Meanwhile,we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble,since the corresponding Malliavin matrix is not invertible caused by the degenerate noise.We finally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.Furthermore,the exponential ergodicity of the system is obtained.In addition,ergodicity of stochastic quasigeophysical flows equation with multiplicative degenerate noise is proved.