Degenerate SDE with Hölder--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 coefficient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e., the second component) and locally Hölder--Dini continuous of order $\frac{2}{3}$ in the first component. Moreover, the weak uniqueness is proved under weaker conditions on the noise coefficient. Furthermore, if the noise coefficient is $C^{1+\varepsilon}$ for some ${\varepsilon}>0$ and the drift is Hölder continuous of order ${\alpha}{\in} (\frac{2}{3},1)$ in the first component and order ${\beta\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 Hölder--Dini space by using the heat semigroup and slowly varying functions.

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