具有各向同性单峰格林函数过程的非均匀球集
Unavoidable collections of balls for processes with isotropic unimodal Green function
摘要：
假设在$R^d$上有一个右连续Markov半群$d\ge 1$，使得它的势核由一个函数$G_0=g(|\cdot|)$的卷积给出，其中$g$是递减的，在零点处有一个较弱的衰减性质，在无穷远处有一个很弱的衰减性质.这不仅涵盖了布朗半群（经典势理论）和各向同性$α$ -稳定半群（Riesz势），还涵盖了更一般的各向同性Lévy过程，其中特征函数具有一定的低标度性质，以及各种几何稳定过程。总是存在相应的寻线过程。$R^d$的一个子集$A$被称为不可避免的，如果进程以概率$1$命中$A$，无论它从哪里开始。我们知道，对于任何两两不相交球的局部有限并，$B(z,r_z)$，$z\in Z$，这是不可避免的，$\sum_{z\in Z} g(|z|)/g(r_z)=\infty$。另外，假设对于某些$\varepsilon>0$，$|z-z'|\ge \varepsilon |z| (g(|z|)/g(r_z))^{1/d}$，无论何时$z,z'\in Z$，$z\ne z'$，证明了相反的情况。如果球的位置是规则的，也就是说，如果它们的中心保持一定的最小相互距离，那么一定大小的每个球都与$Z$和$r_z=g(ϕ(|z|))$相交，其中$ϕ$是一个递减函数。所得结果推广了A. Mimica和Z.冯德拉切克
英文摘要：
Let us suppose that we have a right continuous Markov semigroup on $R^d$, $d\ge 1$, such that its potential kernel is given by convolution with a function $G_0=g(|\cdot|)$, where $g$ is decreasing, has a mild lower decay property at zero, and a very weak decay property at infinity. This captures not only the Brownian semigroup (classical potential theory) and isotropic $α$-stable semigroups (Riesz potentials), but also more general isotropic Lévy processes, where the characteristic function has a certain lower scaling property, and various geometric stable processes. There always exists a corresponding Hunt process. A subset $A$ of $R^d$ is called unavoidable, if the process hits $A$ with probability $1$, wherever it starts. It is known that, for any locally finite union of pairwise disjoint balls $B(z,r_z)$, $z\in Z$, which is unavoidable, $\sum_{z\in Z} g(|z|)/g(r_z)=\infty$. The converse is proven assuming, in addition, that, for some $\varepsilon>0$, $|z-z'|\ge \varepsilon |z| (g(|z|)/g(r_z))^{1/d}$, whenever $z,z'\in Z$, $z\ne z'$. It also holds, if the balls are regularly located, that is, if their centers keep some minimal mutual distance, each ball of a certain size intersects $Z$, and $r_z=g(ϕ(|z|))$, where $ϕ$ is a decreasing function. The results generalize and, exploiting a zero-one law, simplify recent work by A. Mimica and Z. Vondracek.
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