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 coeffcicient 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 $\ff 2 3$ 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 Hölder continuous of order $å\in (\ff 2 3,1)$ 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 Hölder-Dini space by using the heat semigroup and slowly varying functions.

40 pages

References in corpus (1)

• Degenerate SDEs in Hilbert Spaces with Rough Drifts

Cited by in corpus (5)

• Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli
• Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients
• Singular SDEs with critical non-local and non-symmetric Lévy type generator
• Well-posedness for SDEs driven by different type of noises
• Large Deviations Principle for SDEs with Dini Continuous Drifts