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武汉大学数学与统计学院

发布时间:2019-10-07 编辑 :本站 / 66次点击
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武汉大学数学与统计学院

Inthistalk,Iwillpresentsomeresultsfromongoingprojectsonthepoten$D\subset\mathbbR^d$definedviaDirichletformswithjumpkernelsoftheform$J(x,y)=j(|x-y|)B(x,y)$andcriticalkillingfunctions$\kappa(x)$.Here$j(|x-y|)$istheL\evydensityofanisotropicstableprocess(ormoregenerally,asubordinateBrownianmotion)in$\R^d$.Themainnoveltyisthattheterm$B(x,y)$tendsto0when$x$or$y$approachtheboundaryof$D$.Undersomegeneralassumptionson$B(x,y)$,weconstructthecorrespondingprocessandprovethatnon-negativeharmonicfunctionsoftheprocesssatisfytheHarnackinequalityandCarleson,$\beta_1,\beta_2,\beta_3$,roughlygoverningthedecayoftheboundarytermneartheboundaryof$D$.Inthesecondpart,wespecializetothecaseofthehalf-space$D=\mathbbR_+^d=\{x=(\wt{x},x_d):\,x_d0\}$,the$\alpha$-stablekernel$j(|x-y|)=|x-y|^{-d-\alpha}$andthekillingfunction$\kappa(x)=cx_d^{-\alpha}$,$\alpha\in(0,2)$,where$c$dependson$p$and$B$.OurmainresultinthispartisaboundaryHarnackprinciplewhichsaysthat,forany$p(\alpha-1)_+$,therearevaluesoftheparameters$\beta_1,\beta_2,\beta_3$and$c$suchthatnon-negativeharmonicfunctions$f(x)$oftheprocessmustdecayattherate$x_d^p$$\beta_1,\beta_2,\beta_3$and$c$forwhichtheboundaryHarnackprinciplefailsdespitethefactthatCarlesonsestimateisvalid.。