Universal properties of harmonic functions on trees

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• 作者：Evgeny Abakumov^a ; ^{evgueni.abakoumov@u-pem.fr" class="auth_mail" title="E-mail the corresponding author} ; Vassili Nestoridis^b ; ^{vnestor@math.uoa.gr" class="auth_mail" title="E-mail the corresponding author} ; Massimo A. Picardello^c ; ^{picard@mat.uniroma2.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Infinite trees ; Transition coefficients ; Harmonic functions ; Boundary of a tree ; Universal harmonic functions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：445
• 期：2
• 页码：1181-1187
• 全文大小：307 K

文摘

We consider an infinite locally finite tree T equipped with nearest neighbor transition coefficients, giving rise to a space of harmonic functions. We show that, except for trivial cases, the generic harmonic function on T   has dense range in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X1630018X&_mathId=si1.gif&_user=111111111&_pii=S0022247X1630018X&_rdoc=1&_issn=0022247X&md5=c7e053f7de8a8be5769db98b892406ab" title="Click to view the MathML source">Cclass="mathContainer hidden">class="mathCode">$\mathbb{C}$. By looking at forward-only transition coefficients, we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of T. We also study algebraic genericity, spaceability and frequent universality of these phenomena.