Degenerate SDE with holder-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 coeffcient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e., the second component) and locally Holder-Dini continuous of order in the first component. Moreover, the weak uniqueness is proved under weaker conditions on the noise coeffcient. Furthermore, if the noise coeffcient is for some 0 and the drift is Holder continuous of order in the first component and order in the second, the solution forms a-stochastic diffeormorphism ow. To prove these results, we present some new characterizations of Holder-Dini space by using the heat semigroup and slowly varying functions.