Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On weak solutions of stochastic differential equations with sharp drift
  coefficients   We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients
in critical Lebesgue space, and prove the existence and uniqueness of weak
solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb
R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily
fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in{\mathbb
R}^d,$$ where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener
process. If $b=b_1+b_2$ such that
$b_1(T-\cdot)\in\mathcal{C}_q^0((0,T];L^p({\mathbb R}^d))$ with $2/q+d/p=1$ for
$p,q\ge1$ and $\|b_1(T-\cdot)\|_{\mathcal{C}_q((0,T];L^p({\mathbb R}^d))}$ is
sufficiently small, and that $b_2$ is bounded and Borel measurable, then there
exits a unique weak solution to the above equation. Furthermore, we obtain the
strong Feller property of the semi-group and existence of density associated
with above SDE. Besides, we extend the classical partial differential equations
(PDEs) results for $L^q(0,T;L^p({\mathbb R}^d))$ coefficients to
$L^\infty_q(0,T;L^p({\mathbb R}^d))$ ones, and derive the Lipschitz regularity
for solutions of second order parabolic PDEs (see Lemma 2.1).

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The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: patt-sol