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OnrepresentationbyexitlawsforsomeBochnersubordinatedsemigroups
2.Preliminaries
9
Let(EjE)beameasurablespaceandletmbeaσ-finitepositivemeasure
on(EjE).WedenotebyL2(m)theBanachspaceofsquareintegrable(classes
of)functionsdefinedonE,by"."2theassociatednormandbyL2
+(m)the
positiveelementsofL2(m).Moreover,inthesequel,equalityandinequality
holdsalwaysm-a.e.(i.e.almosteverywherewithrespecttom).
Inthissectionwesummarizesomeknownresults(cf.[2],[3]and[15–18]).
2.1.Sub-Markoviansemigroup
AkernelonEisamappingN:E×E→[0j∞[suchthat
1.x→N(xjA)ismeasurableforeachA∈E.
2.A→N(xjA)isameasureon(EjE)foreachx∈E.
LetNbeakernelonE.Forf∈L2(m),wedefine
Nf(x):=/
E
f(g)N(xjdg)j
x∈E.
IfN(L2(m))⊂L2(m),wesaythatNisakernelonL2(m).IfN1≤1jNis
saidtobesub-Markovian.
Asub-MarkoviansemigrouponEisafamilyP:=(Pt)t≥0ofsub-Markovian
kernelsonL2(m)suchthatP0=I,
1.PsPt=Ps+tforallsjt>0,
2.lim
t→0
"Ptf−f"2=0foreveryf∈L2(m),
3."Ptf"2≤"f"2foreacht>0andf∈L2(m).
LetPbeasub-MarkoviansemigrouponE.TheassociatedL2(m)-generator
Aisdefinedby
Af:=lim
t→0
1
t
(Ptf−f)
onitsdomainD(A)whichisthesetofallfunctionsf∈L2(m)forwhichthis
limitexistsinL2(m).Itisknownthat:
1.D(A)isdenseinL2(m)andAisclosed.
2.Ifu∈D(A)thenPtu∈D(A)andA(Ptu)=PtAujforeacht>0.
