Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Heat kernels of non-symmetric jump processes: beyond the stable case   Let $J$ be the L\'evy density of a symmetric L\'evy process in $\mathbb{R}^d$
with its L\'evy exponent satisfying a weak lower scaling condition at infinity.
Consider the non-symmetric and non-local operator $$ {\mathcal
L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \mathbb{R}^d:
|z|>\epsilon\}}(f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$ where $\kappa(x,z)$ is
a Borel measurable function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying
$0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and
$|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0,
1)$. We construct the heat kernel $p^\kappa(t, x, y)$ of ${\mathcal L}^\kappa$,
establish its upper bound as well as its fractional derivative and gradient
estimates. Under an additional weak upper scaling condition at infinity, we
also establish a lower bound for the heat kernel $p^\kappa$.

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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: patt-sol
3
The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: q-bio