Asymptotic Log-Harnack Inequality for Degenerate SPDEs with Reflection

By constructing a suitable coupling by change of measures, the asymptotic log- Harnack inequality is established for a class of degenerate SPDEs with reflection. This inequality implies the asymptotic heat kernel estimate, the uniqueness of the invariant probability measure, the asymptotic gradient estimate (hence, asymptotically strong Feller property), and the asymptotic irreducibility. As application, the main result is illustrated by d-dimensional degenerate stochastic Navie-Stokes equations with reflection, where the dissipative operator is the Dirichlet Laplacian with a power θ\geq 1 \vee \frac{d+2}{4}, which includes the Laplacian when d \geq 2.