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

$$\begin{aligned} d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in \mathbb {R}^d, \end{aligned}$$

where $W_t$ is a $d$-dimensional standard Wiener process and the drift coefficient $b:[0,T]\times \mathbb {R}^d\rightarrow \mathbb {R}^d$ 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 (Probab Theory Relat Fields 131(2):154–196, 2005) proved that the above equation has a unique strong solution $X_t$. Recently, it was shown by Fedrizzi and Flandoli (Stoch Anal Appl 31:708–736, 2013) 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 $X_t$.