Gel'fand三元组与无限网络的边界
Gel'fand triples and boundaries of infinite networks
摘要：
本文从随机积分的角度研究了连通赋权图$G$的边界理论。对于$G$上Dirichlet有限函数的Hilbert空间\HE，我们构造了一个Gel'fand三元组$S \ci {\mathcal H}_{\mathcal E} \ci S'$ .这就产生了$S'$上的概率测度$\mathbb{P}$和${\mathcal H}_{\mathcal E}$到$L^2(S',\mathbb{P})$中的等距嵌入，从而给出了边界作为$S'$中某类“分布”的具体表示。在以前的论文中，我们证明了一个离散高斯-格林恒等式的无限网络产生的边界表示的谐波函数的有限能量，给出了一定的限制。在本文中，我们使用的技术，从随机积分，使边界$\operatorname{bd}G$精确的测度空间，并获得一个边界整表示为$S'$上的积分。
英文摘要：
We study the boundary theory of a connected weighted graph $G$ from the viewpoint of stochastic integration. For the Hilbert space \HE of Dirichlet-finite functions on $G$, we construct a Gel'fand triple $S \ci {\mathcal H}_{\mathcal E} \ci S'$. This yields a probability measure $\mathbb{P}$ on $S'$ and an isometric embedding of ${\mathcal H}_{\mathcal E}$ into $L^2(S',\mathbb{P})$, and hence gives a concrete representation of the boundary as a certain class of "distributions" in $S'$. In a previous paper, we proved a discrete Gauss-Green identity for infinite networks which produces a boundary representation for harmonic functions of finite energy, given as a certain limit. In this paper, we use techniques from stochastic integration to make the boundary $\operatorname{bd}G$ precise as a measure space, and obtain a boundary integral representation as an integral over $S'$.
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。
