具有非光滑核的积分-微分算子的Cauchy问题和鞅问题
The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels
摘要：
我们考虑由\[ Lu(x) =\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(α) 1_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x)) k(x,y) \sd y . \]定义的线性积分-微分算子$L$。这里，对于小的$y$，核$k(x,y)$的行为类似于$|y|^{-d-α}$、$α\in (0,2)$，并且在第一个变量中是Hölder连续的，下面给出精确的定义。这项工作的目的是双重的。一方面，我们研究了与$L$相对应的Cauchy问题的唯一可解性。另一方面，我们研究了$L$的鞅问题。对确定性抛物型方程的解析结果保证了鞅问题的适定性。我们的战略遵循经典的Stroock-Varadhan路径。这些假设考虑到了迄今为止尚未处理的案件。
英文摘要：
We consider the linear integro-differential operator $L$ defined by \[ Lu(x) =\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(α) 1_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x)) k(x,y) \sd y . \] Here the kernel $k(x,y)$ behaves like $|y|^{-d-α}$, $α\in (0,2)$, for small $y$ and is Hölder-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
