论文标题
在边界处的dirichlet形式变性的热内核估计值
Heat kernel estimates for Dirichlet forms degenerate at the boundary
论文作者
论文摘要
本文的目的是对两种类型的纯粹的双向估计来建立两种类型的纯粹不连续的对称马尔可夫进程的热核，在$ \ mathbb r^d $的上半空间中具有跳跃内核在边界处变性。跳跃内核是$ j（x，y）= \ Mathcal b（x，y）| x-y |^{ - α-d} $，$α\ in（0,2）$，其中功能$ \ mathcal b $取决于四个参数，并且可能在边界上消失。我们的结果是边界上跳跃核的非本地运算符的热核的第一个尖锐的两侧估计值。第一种类型的过程是$ \ overline {\ mathbb r}^d _+$的保守马尔可夫进程，带有跳跃内核$ j（x，y）$。根据参数所属的区域，热核估计值具有三种不同的形式，其中两种与所有以前已知的热核估计值不同。第二种过程是在临界电位或撞到半空间边界时杀死的上述过程。我们确定他们的热核估计值具有近似分解特性，其生存概率衰减是距离边界距离的力量，在这种距离的距离中，功率取决于临界电位中的常数。
The goal of this paper is to establish sharp two-sided estimates on the heat kernels of two types of purely discontinuous symmetric Markov processes in the upper half-space of $\mathbb R^d$ with jump kernels degenerate at the boundary. The jump kernels are of the form $J(x,y)=\mathcal B(x,y)|x-y|^{-α-d}$, $α\in (0,2)$, where the function $\mathcal B$ depends on four parameters and may vanish at the boundary. Our results are the first sharp two-sided estimates for the heat kernels of non-local operators with jump kernels degenerate at the boundary. The first type of processes are conservative Markov processes on $\overline{\mathbb R}^d_+$ with jump kernel $J(x,y)$. Depending on the regions where the parameters belong, the heat kernels estimates have three different forms, two of them are qualitatively different from all previously known heat kernel estimates. The second type of processes are the processes above killed either by a critical potential or upon hitting the boundary of the half-space. We establish that their heat kernel estimates have the approximate factorization property with survival probabilities decaying as a power of the distance to the boundary, where the power depends on the constant in the critical potential.
