射杀稳定过程
Shot-down stable processes
AI总结：
本文研究了欧几里得空间中分数拉普拉斯算子的射杀稳定过程，探讨了其转移密度、狄利克雷形式及格林函数的性质。
摘要：
射杀过程是一种强马尔可夫过程，当跳跃到给定向量空间的开子集的补集时会被湮灭或射杀。由于射杀时间的特殊性质，此类过程提出了非局部微分方程的新类型边界条件。在本文中，我们构造了欧几里得空间中分数拉普拉斯算子的射杀过程。对于光滑有界集$D$，我们研究了其转移密度并刻画了狄利克雷形式。我们证明了对应的格林函数与在$D$上具有狄利克雷条件的分数拉普拉斯算子的格林函数是可比的。然而，对于非凸$D$，射杀稳定过程的转移密度与$D$上的分数拉普拉斯算子的狄利克雷热核是不可比的。
英文摘要：
The shot-down process is a strong Markov process which is annihilated, or shot down, when jumping over or to the complement of a given open subset of a vector space. Due to specific features of the shot-down time, such processes suggest new type of boundary conditions for nonlocal differential equations. In this work we construct the shot-down process for the fractional Laplacian in Euclidean space. For smooth bounded sets $D$, we study its transition density and characterize Dirichlet form. We show that the corresponding Green function is comparable to that of the fractional Laplacian with Dirichlet conditions on $D$. However, for nonconvex $D$, the transition density of the shot-down stable process is incomparable with the Dirichlet heat kernel of the fractional Laplacian for $D$.
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