Degenerate SDE with H\"older-Dini Drift and Non-Lipschitz Noise Coefficient

The existence-uniqueness and stability of strong solutions are proved for a class of degenerate stochastic differential equations, where the noise coeffcicient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e. the second component) and locally H\"older-Dini continuous of order $\ff 2 3$ in the first component. Moreover, the weak uniqueness is proved under weaker conditions on the noise coefficient. Furthermore, if the noise coefficient is $C^{1+\vv}$ for some $\vv>0$ and the drift is H\"older continuous of order $\aa\in (\ff 2 3,1)$ in the first component and order $\bb\in(0,1)$ in the second, the solution forms a $C^1$-stochastic diffeormorphism flow. To prove these results, we present some new characterizations of H\"older-Dini space by using the heat semigroup and slowly varying functions.