Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Cauchy problem and the martingale problem for integro-differential
  operators with non-smooth kernels   We consider the linear integro-differential operator $L$ defined by \[ Lu(x)
=\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(\alpha) 1_{\{|y|\leq 2\}}(y)y \cdot \nabla
u(x)) k(x,y) \sd y . \] Here the kernel $k(x,y)$ behaves like
$|y|^{-d-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is H\"older-continuous
in the first variable, precise definitions are given below. The aim of this
work is twofold. On one hand, we study the unique solvability of the Cauchy
problem corresponding to $L$. On the other hand, we study the martingale
problem for $L$. The analytic results obtained for the deterministic parabolic
equation guarantee that the martingale problem is well-posed. Our strategy
follows the classical path of Stroock-Varadhan. The assumptions allow for cases
that have not been dealt with so far.

8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th