Degenerate SDE with H\"older-Dini Drift and Non-Lipschitz Noise Coefficient
Abstract
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 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+ for some >0 and the drift is H\"older continuous of order ∈ ( 2 3,1) in the first component and order ∈(0,1) in the second, the solution forms a C1-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.