Stochastic equations with low regularity drifts

Abstract

By using the Itô-Tanaka trick, we prove the unique strong solvability as well as the gradient estimates for stochastic differential equations with irregular drifts in low regularity Lebesgue-Hölder space $L^q(0,T;{\mathcal{C}}_b^{\alpha }({\mathbb{R}}^d))$ with $\alpha \in (0,1)$ and $q\in (2/(1+\alpha ),2$). As applications, we show the unique weak and strong solvability for stochastic transport equations driven by the low regularity drift with $q\in (4/(2+\alpha ),2$) as well as the local Lipschitz estimate for stochastic strong solutions.