緊李群上隨機(jī)游動的弱收斂.pdf

1、摘要本文主要考慮緊李群上的隨機(jī)游動的弱收斂間題該同題等價(jià)干緊李群上卷積測度的弱收斂.應(yīng)用PeterWeyl定理(即緊李群上的傅立葉變換)，我們給出緊李群上隨機(jī)游動弱收斂的一些條件.我們分共扼不變的隨機(jī)游動和一般的隨機(jī)游動兩種情況給出相似但不相同的刻畫.作為前者的一個(gè)簡單特例，我們討論了緊李群SU(2)上共扼不變的隨機(jī)游動.并由此啟發(fā)我們用這篇文章里得到的結(jié)果，推出緊李群上所有弱收斂的隨機(jī)游動最后必然弱收斂到一個(gè)李子群上的Haar測度.關(guān)

2、祖詞:弱收斂，隨機(jī)游動，緊李群，卷積，PeterWeyl定理，緊李群上的傅立葉變換.CHAPTER1INTRODUCTIONWestudytheweakconvergenceofrandomwalksorequivalentlythatofconvolutionproductsofprobabilitymeasuresincompactLieGroups.Diaconis[21(1988)andRosenthal[71(1994)hav

3、eusedFouriermethodstostudythedistributionalconvergenceofrandomwalkstoHaarmeasuresonfinitegroupsandinsomespecialcasesLiegroups.WeherewilstudytheconvergenceofrandomwalksincompactLiegroupstoHaarmeasuresusingFouriermethodsba

4、sedonPeterWeyltheorem.ThismethodhasbeenappliedinLiao[51andLiao[6]recentlytostudytheFourierexpansionsofthedistributiondensitiesofLevyprocessesincompactLiegroupsandtoobtaintheexponentialconvergenceofthedistributionstoHaarm

5、easures.Theconvergenceconsideredintheseworksareconvergenceunderthetotalvariationnorm.Inthepresent.paperonlyweakconvergencewillbeconsideredTheItoKawadaTheoremprovidesacompletecharacterizationfortheweakconvergenceofconvolu

6、tionproductsofprobabilitymeasuresoncompacttopologicalgroupsHoweverthepresentapproachbasedonthePeterWeylTheoreminthecaseofcompactLiegroupsissimplerandprovidesanalternativecharacterization.ALiegroupGisagroupandamanifoldsuc

7、hthatboththeproductmapGxG〕(9h)ghEGandtheinversemapG〕9、91〔Garesmooth.InthesequelwewillalwaysuseGtodenoteacompactLiegroupandassumethatanymeasureonGinourdiscussionisdefinedon13(G)theBorel二一algebraonG.RandomwalksinGandconvol

8、utionsoftwomeasuresinGaredefinedasfollows.Definition1.1.ArandomwalkonaLieGroupGis玖=XXYn.whereXlX.K3areasequenceofindependentandidenticallydistributedGvaluedrandomvariables.Definition1.2.LetpandvbetwomeasuresonGthentheirc