Quasi-invariance of the stochastic flow associated to Itô's SDE with singular time-dependent drift

Abstract

In this paper we consider the Itô SDE $$d{X}_t=d{W}_t+b(t,X_t)dt,X_0=x\in {\mathbb{R}}^d,$$ where $W_t$ is a $d$-dimensional standard Wiener process and the drift coefficient $b:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ belongs to $L^q(0,T;L^p({\mathbb{R}}^d))$ with $p\ge 2,q>2$ and $\frac{d}{p}+\frac{2}{q}<1$. In 2005, Krylov and Röckner \cite{KR05} proved that the above equation has a unique strong solution $X_t$. Recently it was shown by Fedrizzi and Flandoli \cite{FF13b} that the solution $X_t$ is indeed a stochastic flow of homeomorphisms on ${\mathbb{R}}^d$. We prove in the present work that the Lebesgue measure is quasi-invariant under the flow $X_t$.

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