Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Heat kernel estimates for $\Delta+\Delta^{\alpha/2}$ under gradient
  perturbation   For $d \ge 2$, $\alpha \in (0,2)$ and $M > 0$, we consider the gradient
perturbation of a family of nonlocal operators
$\{\Delta+a^\alpha\Delta^{\alpha/2}, a\in (0,M]\}$. We establish the existence
and uniqueness of the fundamental solution $p(t, x, y)$ for \begin{equation*}
\mathcal{L}^{a,b} = \Delta+a^\alpha\Delta^{\alpha/2} + b\cdot \nabla,
\end{equation*} where $b$ is in Kato class $\mathbb{K}_{d,1}$ on
$\mathbb{R}^d$. We show that $p(t, x, y)$ is jointly continuous and derive its
sharp two-sided estimates. The kernel $p(t, x, y)$ determines a conservative
Feller process $X$. We further show that the law of $X$ is the unique solution
of the martingale problem for $(\mathcal{L}^{a,b}, C^\infty_c (\mathbb{R}^d)$
and $X$ can be represented as $$ X_t = X_0 + Z^a_t + \int_0^t b(X_s) ds, \qquad
t\geq 0, $$ where $Z^a_t= B_t +aY_t$ for a Brownian motion $B$ and an
independent isotropic $\alpha$-stable process $Y$. Moreover, we prove that the
above SDE has a unique weak solution.

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The categories of this paper are: acc-phys
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con