Hunt过程的Liouville性质、Wiener检验和不可避免集
Liouville property, Wiener's test and unavoidable sets for Hunt processes
摘要：
设$(X,\mathcal W)$是平衡空间$1\in \mathcal W$，或等价地，设$\mathcal W$是局部紧空间$X$上Hunt过程的过剩函数的集合，$\mathcal W$具有可数基，使得$\mathcal W$分离点，$\mathcal W$中的每个函数是其连续子变量的上确界，并且存在严格正连续的$u,v\in \mathcal W$，使得$u/v\to 0$在无穷远处。我们假设存在一个关于$X$的格林函数$G>0$，一个关于$X$的度量$ρ$和一个具有加倍性质的递减函数$g\colon[0,\infty)\to (0,\infty]$使得$G\approx g\circρ$ .假设常数函数$1$是调和的，球是相对紧凑的，它表明，每一个正调和函数是常数（刘维性质）和维纳的测试在无穷大显示，如果一个给定的$A$在$X$是不可避免的，也就是说，如果该过程击中$A$的概率为1，无论它开始。应用结果表明，对于$X$上的适当测度$λ$，具有一定分离性质的两两不相交球$B(z,r_z)$，$z\in Z$的局部有限并集是不可避免的，当且仅当对于某点$x_0\in X$，级数$\sum_{z\in Z} g(ρ(x_0,z))/g(r_z) $发散.结果推广，并利用0 - 1定律的命中概率，简化最近的工作S。加德纳和M. Ghergu，A. Mimica和Z.冯德拉\v切克，和作者。
英文摘要：
Let $(X,\mathcal W)$ be a balayage space, $1\in \mathcal W$, or - equivalently - let $\mathcal W$ be the set of excessive functions of a Hunt process on a locally compact space $X$ with countable base such that $\mathcal W$ separates points, every function in $\mathcal W$ is the supremum of its continuous minorants and there exist strictly positive continuous $u,v\in \mathcal W$ such that $u/v\to 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $ρ$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property such that $G\approx g\circρ$. Assuming that the constant function $1$ is harmonic and balls are relatively compact, is is shown that every positive harmonic function is constant (Liouville property) and that Wiener's test at infinity shows, if a given set $A$ in $X$ is unavoidable, that is, if the process hits $A$ with probability one, wherever it starts. An application yields that locally finite unions of pairwise disjoint balls $B(z,r_z)$, $z\in Z$, which have a certain separation property with respect to a suitable measure $λ$ on $X$ are unavoidable if and only if, for some/any point $x_0\in X$, the series $\sum_{z\in Z} g(ρ(x_0,z))/g(r_z) $ diverges. The results generalize and, exploiting a zero-one law for hitting probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and Z. Vondra\v cek, and the author.
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。
