Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quasi-invariance of the stochastic flow associated to It\^o's SDE with
  singular time-dependent drift   In this paper we consider the It\^o SDE
  $$d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in {\mathbb R}^d,$$ 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 $p\geq 2, q>2$ and $\frac dp +\frac 2q<1$. In 2005, Krylov and
R\"ockner \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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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: nlin