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8
MohamedHmissi,HassenMejri
paperisdevotedtotherepresentationbyPβ-exitlaws,wherePβisthesub-
ordinatedsemigroupofPbymeansof;ji.e
(1.2)
P
tf:=/
β
0
∞
Psf;t(ds)j
f∈L2(m)jt>0.
Moreprecisely,wesupposethat;admitsacompleteBernsteinfunctionfof
theform
f(r)=/
0
∞
(1−e1sr)!(s)ds
forsomefunction!on]0j∞[(cf.Section3.2).Thisfamilyofsubordinators
containsmanyinterestingexamplesanditisconsideredin[5].
Inthiscontext,weprovethefollowingintegralrepresentation:Lethbea
Pβ-potential,i.eh≥0,Pβ
th≤hjlim
t→0
P
th=handP
β
th∈D(Aβ)thedomain
β
ofthegeneratorAβofPβ.ThenthereexistsauniquePβ-exitlaww=(wt)t>0
suchthat
(1.3)
h=/
0
∞
wsds.
Iffisboundedthenwtisexplicitlygivenby
wt=/
0
∞
(P
th−PsP
β
th)!(s)ds.
β
Byconsideringtheone-sidedstablesubordinatorsoforderO∈]0j1[,wededuce
arepresentationfortheoriginalsemigroup.Namely,ifuisaP-potential,then
thereexistauniqueP-exitlawϕsuchthat
(1.4)
u=/
0
∞
ϕsds.
Asimilarproblemisstudiedin[12]byconsideringC1-subordinatorsinstead
ofsubordinatorswithcompleteBernsteinfunctions.
Theintegralrepresentation(1.4)isalreadyobtainedinmanypapers,but
underanadditionalhypothesisonP.Forexample,Pissupposedtobe(al-
most)symmetricin[6–8],absolutelycontinuousin[9–10]andlatticein[11,13].
