Stochastic equations with low regularity drifts

TL;DR

This paper establishes the existence, uniqueness, and gradient estimates for stochastic differential equations with irregular, low regularity drifts using the Itô-Tanaka trick, and applies these results to stochastic transport equations.

Contribution

It introduces a novel approach to solve SDEs with low regularity drifts and extends the results to stochastic transport equations with similar irregularities.

Findings

Proves unique strong solvability for SDEs with low regularity drifts.

Provides gradient estimates for solutions of these SDEs.

Shows solvability and Lipschitz estimates for related stochastic transport equations.

Abstract

By using the It\^{o}-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-H\"{o}lder 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.