Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $L^q(L^p)$-theory of stochastic differential equations   In this paper we show the weak differentiability of the unique strong
solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's
derivative formula for the following stochastic differential equation in
$\mathbb R^d$: $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sigma(t,X_t){\rm d} W_t,\ \
X_0=x\in\mathbb R^d, $$ where $\sigma$ is bounded, uniformly continuous and
nondegenerate, $\nabla\sigma\in \widetilde{\mathbb L}^{p_1}_{q_1}$ and $b\in
\widetilde{\mathbb L}^{p_2}_{q_2}$ for some $p_i,q_i\in[2,\infty)$ with
$\frac{d}{p_i}+\frac{2}{q_i}<1$, $i=1,2$, where $\widetilde{\mathbb
L}^{p_i}_{q_i}, i=1,2$ are some localized spaces. Moreover, in the endpoint
case $b\in \widetilde{\mathbb L}^{d; {\rm uni}}_\infty$, we also show the weak
well-posedness.

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The categories of this paper are: astro-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph