非均匀集与小集上的调和测度
Unavoidable sets and harmonic measures living on small sets
摘要：
给定$ R^d$中的一个连通开集$U\ne\emptyset$，$d\ge 2$，$U$中的一个相对闭集$A$称为\emph{unavoidable in $U$}，如果布朗运动，从$x\in U\setminus A$开始并在离开$U$时被杀死，几乎肯定击中$A$，或者等价地，如果$x$关于$U\setminus A$的调和测度在$A$上有质量$1$。首先证明了一个新的不可避免集的判定准则，该准则有助于在$U$中构造越来越小的不可避免集。从$U$的任意香槟子域开始（这是得到省略了一个局部有限的联合成对不相交的封闭球$\overline B(z, r_z)$，$z\in Z$，满足$\sup_{z\in Z} r_z/\mbox{dist}(z,U^c)<1$），一个组合的准则和存在的小的非极性紧集的康托型产生一个集$A$上的调和测度为$U\setminus A$是活的，并具有Hausdorff维数$d-2$，如果$d=2$，对数Hausdorff维数$1$这也可以对欧几里得空间上的Riesz势（各向同性$α$ -稳定过程）和$C^{1,1}$开子集上的删失稳定过程进行。最后，在平衡空间$(X,\mathcal W)$的非常一般的设置中，函数$1$是调和的（它不仅涵盖了大量的二阶偏微分方程，而且还涵盖了非局部情况，例如，由Riesz势给出的，迷向单峰Lévy过程或截尾稳定过程）给出了$X$的香槟子集$X\setminus A$的一个构造，它具有小的不可避免集$A$，它推广（并部分改进）了经典情况下的最近构造。
英文摘要：
Given a connected open set $U\ne\emptyset$ in $ R^d$, $d\ge 2$, a relatively closed set $A$ in $U$ is called \emph{unavoidable in $U$}, if Brownian motion, starting in $x\in U\setminus A$ and killed when leaving $U$, hits $A$ almost surely or, equivalently, if the harmonic measure for $x$ with respect to $U\setminus A$ has mass $1$ on $A$. First a new criterion for unavoidable sets is proven which facilitates the construction of smaller and smaller unavoidable sets in $U$. Starting with an arbitrary champagne subdomain of $U$ (which is obtained omitting a locally finite union of pairwise disjoint closed balls $\overline B(z, r_z)$, $z\in Z$, satisfying $\sup_{z\in Z} r_z/\mbox{dist}(z,U^c)<1$), a combination of the criterion and the existence of small nonpolar compact sets of Cantor type yields a set $A$ on which harmonic measures for $U\setminus A$ are living and which has Hausdorff dimension $d-2$ and, if $d=2$, logarithmic Hausdorff dimension $1$. This can be done as well for Riesz potentials (isotropic $α$-stable processes) on Euclidean space and for censored stable processes on $C^{1,1}$ open subsets. Finally, in the very general setting of a balayage space $(X,\mathcal W)$ on which the function $1$ is harmonic (which covers not only large classes of second order partial differential equations, but also non-local situations as, for example, given by Riesz potentials, isotropic unimodal Lévy processes or censored stable processes) a construction of champagne subsets $X\setminus A$ of $X$ with small unavoidable sets $A$ is given which generalizes (and partially improves) recent constructions in the classical case.
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