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.

arXiv comment40 pages

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• Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients
• Asymptotic Log-Harnack Inequality and Applications for Stochastic Systems of Infinite Memory
• Large Deviations Principle for SDEs with Dini Continuous Drifts
• Well-posedness for SDEs driven by different type of noises
• Singular SDEs with critical non-local and non-symmetric Lévy type generator