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Nearly Hyperharmonic Functions are Infima of Excessive Functions
Journal of Theoretical Probability ( IF 0.8 ) Pub Date : 2019-06-20 , DOI: 10.1007/s10959-019-00927-8 Wolfhard Hansen , Ivan Netuka
Journal of Theoretical Probability ( IF 0.8 ) Pub Date : 2019-06-20 , DOI: 10.1007/s10959-019-00927-8 Wolfhard Hansen , Ivan Netuka
Let \(\mathfrak {X}\) be a Hunt process on a locally compact space X such that the set \(\mathcal {E}_{\mathfrak {X}}\) of its Borel measurable excessive functions separates points, every function in \(\mathcal {E}_{\mathfrak {X}}\) is the supremum of its continuous minorants in \({\mathcal {E}}_{{\mathfrak {X}}}\), and there are strictly positive continuous functions \(v,w\in {\mathcal {E}}_{{\mathfrak {X}}}\) such that v / w vanishes at infinity. A numerical function \(u\ge 0\) on X is said to be nearly hyperharmonic, if \(\int ^*u\circ X_{\tau _V}\,\text {d}P^x\le u(x)\) for every \(x\in X\) and every relatively compact open neighborhood V of x, where \(\tau _V\) denotes the exit time of V. For every such function u, its lower semicontinuous regularization \(\hat{u}\) is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that \( u=\inf \{w\in {\mathcal {E}}_{{\mathfrak {X}}}:w\ge u\}\) for every Borel measurable nearly hyperharmonic function on X. Principal novelties of our approach are the following: 1. A quick reduction to the special case, where starting at \(x\in X\) with \(u(x)<\infty \) the expected number of times the process \({\mathfrak {X}}\) visits the set of points \(y\in X\), where \(\hat{u}(y):=\liminf _{z\rightarrow y} u(z)
中文翻译：
近高谐波函数是过度函数的内层
令\(\mathfrak {X}\) 是局部紧空间X 上的Hunt 过程，使得其Borel 可测过剩函数的集合\(\mathcal {E}_{\mathfrak {X}}\) 分离点， \(\mathcal {E}_{\mathfrak {X}}\) 中的每个函数都是它在 \({\mathcal {E}}}_{{\mathfrak {X}}}\) 中的连续次幂的上界，并且存在严格正连续函数 \(v,w\in {\mathcal {E}}_{{\mathfrak {X}}}\) 使得 v / w 在无穷远处消失。X 上的数值函数 \(u\ge 0\) 被称为接近超谐波，如果 \(\int ^*u\circ X_{\tau _V}\,\text {d}P^x\le u( x)\) 对于每个 \(x\in X\) 和 x 的每个相对紧凑的开邻域 V，其中 \(\tau _V\) 表示 V 的退出时间。对于每个这样的函数 u，其下半连续正则化 \ (\hat{u}\) 是多余的。这篇论文的主要目的是给出一个简短的，
更新日期：2019-06-20
中文翻译：
近高谐波函数是过度函数的内层
令\(\mathfrak {X}\) 是局部紧空间X 上的Hunt 过程，使得其Borel 可测过剩函数的集合\(\mathcal {E}_{\mathfrak {X}}\) 分离点， \(\mathcal {E}_{\mathfrak {X}}\) 中的每个函数都是它在 \({\mathcal {E}}}_{{\mathfrak {X}}}\) 中的连续次幂的上界，并且存在严格正连续函数 \(v,w\in {\mathcal {E}}_{{\mathfrak {X}}}\) 使得 v / w 在无穷远处消失。X 上的数值函数 \(u\ge 0\) 被称为接近超谐波，如果 \(\int ^*u\circ X_{\tau _V}\,\text {d}P^x\le u( x)\) 对于每个 \(x\in X\) 和 x 的每个相对紧凑的开邻域 V，其中 \(\tau _V\) 表示 V 的退出时间。对于每个这样的函数 u，其下半连续正则化 \ (\hat{u}\) 是多余的。这篇论文的主要目的是给出一个简短的，
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