Hunt’s hypothesis (H) and triangle property of the Green function

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• 作者：Wolfhard Hansen^a ; ^{hansen@math.uni-bielefeld.de" class="auth_mail" title="E-mail the corresponding author} ; Ivan Netuka^b ; ^{netuka@karlin.mff.cuni.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 31D05 ; secondary ; 60J45 ; 60J60 ; 60J75
• 刊名：Expositiones Mathematicae
• 出版年：2016
• 出版时间：2016
• 年：2016
• 卷：34
• 期：1
• 页码：95-100
• 全文大小：194 K

文摘

Let pan id="mmlsi100" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si100.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=c1b8a540867c0c0ff0a9b158e1630798" title="Click to view the MathML source">Xpan>pan class="mathContainer hidden">pan class="mathCode">$X$pan>pan>pan> be a locally compact abelian group with countable base and let pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si10.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=b8ad77a8cede94a7285142c3091b90f1" title="Click to view the MathML source">Wpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{W}$pan>pan>pan> be a convex cone of positive numerical functions on pan id="mmlsi100" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si100.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=c1b8a540867c0c0ff0a9b158e1630798" title="Click to view the MathML source">Xpan>pan class="mathContainer hidden">pan class="mathCode">$X$pan>pan>pan> which is invariant under the group action and such that pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si4.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=ccfe2a62912a8cfc141bd9750a573a60" title="Click to view the MathML source">(X,W)pan>pan class="mathContainer hidden">pan class="mathCode">$(X,\mathcal{W})$pan>pan>pan> is a balayage space or (equivalently, if pan id="mmlsi318" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si318.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=8685e308bf95feb1261a46b43686a904" title="Click to view the MathML source">1∈Wpan>pan class="mathContainer hidden">pan class="mathCode">$1\in \mathcal{W}$pan>pan>pan>) such that pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si10.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=b8ad77a8cede94a7285142c3091b90f1" title="Click to view the MathML source">Wpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{W}$pan>pan>pan> is the set of excessive functions of a Hunt process on pan id="mmlsi100" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si100.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=c1b8a540867c0c0ff0a9b158e1630798" title="Click to view the MathML source">Xpan>pan class="mathContainer hidden">pan class="mathCode">$X$pan>pan>pan>, pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si10.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=b8ad77a8cede94a7285142c3091b90f1" title="Click to view the MathML source">Wpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{W}$pan>pan>pan> separates points, every function in pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si10.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=b8ad77a8cede94a7285142c3091b90f1" title="Click to view the MathML source">Wpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{W}$pan>pan>pan> is the supremum of its continuous minorants in pan id="mmlsi10" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si10.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=b8ad77a8cede94a7285142c3091b90f1" title="Click to view the MathML source">Wpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{W}$pan>pan>pan>, and there exist strictly positive continuous pan id="mmlsi11" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si11.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=b2c1598d4a3fa52d46e4ed8cf3f1d566" title="Click to view the MathML source">u,v∈Wpan>pan class="mathContainer hidden">pan class="mathCode">$u,v\in \mathcal{W}$pan>pan>pan> such that pan id="mmlsi12" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si12.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=7b0e4689a6d871995eee5122c1315168" title="Click to view the MathML source">u/v→0pan>pan class="mathContainer hidden">pan class="mathCode">$u/v\to 0$pan>pan>pan> at infinity.

Assuming that there is a Green function pan id="mmlsi13" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si13.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=cde4c28c77e98506335514e59027bcc6" title="Click to view the MathML source">G>0pan>pan class="mathContainer hidden">pan class="mathCode">$G>0$pan>pan>pan> for pan id="mmlsi100" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si100.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=c1b8a540867c0c0ff0a9b158e1630798" title="Click to view the MathML source">Xpan>pan class="mathContainer hidden">pan class="mathCode">$X$pan>pan>pan> which locally satisfies the triangle inequality pan id="mmlsi15" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0723086914000784&_mathId=si15.gif&_user=111111111&_pii=S0723086914000784&_rdoc=1&_issn=07230869&md5=9f7765ded9859de8f889b717487ddbaf" title="Click to view the MathML source">G(x,z)∧G(y,z)≤CG(x,y)pan>pan class="mathContainer hidden">pan class="mathCode">$G(x,z)\wedge G(y,z)\le CG(x,y)$pan>pan>pan> (true for many Lévy processes), it is shown that Hunt’s hypothesis (H) holds, that is, every semipolar set is polar.