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 coefficient 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 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 C1+epsilon for some epsilon > 0 and the drift is Holder continuous of order alpha is an element of (2/3; 1) in the first component and order beta is an element of (0, 1) in the second, the solution forms a C-1-stochastic diffeormorphism flow. To prove these results, we present some new characterizations of Holder-Dini space by using the heat semigroup and slowly varying functions.