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AnnalesMathematicaeSilesianae22(2008),7-26
PraceNaukoweUniwersytetuŚląskiegonr2703,Katowice
ONREPRESENTATIONBYEXITLAWSFORSOME
BOCHNERSUBORDINATEDSEMIGROUPS
MohamedHmissi,HassenMejri
Abstract.Thispaperisdevotedtotheintegralrepresentationofpotentials
byexitlawsintheframeworkofsub-Markoviansemigroupsofkernelsacting
onL2(m).WemainlyinvestigatesubordinatedsemigroupsinBochnersense
bymeansofsubordinatorswithcompleteBernsteinfunctions.Asapplication,
wegivearepresentationfortheoriginalsemigroup.
1.Introduction
LetP=(Pt)t≥0beasub-MarkoviansemigroupofkernelsonL2(m).A
P-exitlawisafamilyϕ=(ϕt)t>0ofL2
+(m)satisfyingthefunctionalequation
(1.1)
Psϕt=ϕs+tj
sjt>0.
ThisnotionisfirstintroducedbyDynkin[5]intheframeworkofpotential
theorywithoutreferencemeasure.Then,theintegralrepresentationofpoten-
tialsbyexitlawswasinvestigatedinmanypapers(cf.[1]and[6–14]).Asitis
known,thisallowsexplicitformulasfortheenergyandthecapacity(cf.[6–8]
andtherelatedreferences).
Now,let;=(;t)t>0beaBochnersubordinator,thatisavaguelycontinu-
ousconvolutionsemigroupofsub-probabilitymeasureson[0j+∞[.Thepresent
Received:14.12.2008.Revised:25.02.2009.
(2000)MathematicsSubjectClassification:47D03,31C15,39B42,60J99.
Keywordsandphrases:sub-Markoviansemigroup,potential,exitlaw,subordinator,
completeBernsteinfunction,one-sidedstablesubordinator.
