The Levy Laplacian

160 Pages·2005·1.905 MB·English

by  M. N. Feller

CAMBRIDGETRACTSINMATHEMATICS
GeneralEditors
b. bollobas,w. fulton,a. katok,f. kirwan,
p. sarnak,b. simon,b. totaro
166TheLe´vyLaplacian
TheLe´vyLaplacianisaninfinite-dimensionalgeneralizationofthewell-known
classicalLaplacian. Itstheoryhasbeenincreasinglywell-developedinrecentyears
andthisbookisthefirstsystematictreatmentofit.
Thebookdescribestheinfinite-dimensionalanaloguesoffinite-dimensionalresults,
andmoreespeciallythosefeaturesthatappearonlyinthegeneralizedcontext. It
developsatheoryofoperatorsgeneratedbytheLe´vyLaplacianandthesymmetrized
Le´vyLaplacian,aswellasatheoryoflinearandnonlinearequationsinvolvingit.
TherearemanyproblemsleadingtoequationswithLe´vyLaplaciansandto
Le´vy–Laplaceoperators,forexamplesuperconductivitytheory,thetheoryofcontrol
systems,theGaussrandomfieldtheory,andtheYang–Millsequation.
Thebookiscomplementedbyexhaustivebibliographicnotesandreferences. The
resultisaworkthatwillbevaluedbythoseworkinginfunctionalanalysis,partial
differentialequationsandprobabilitytheory.
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The Le´vy Laplacian
M. N. FELLER
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Contents
Introduction page1
1 TheLe´vyLaplacian 05
1.1 Definitionoftheinfinite-dimensionalLaplacian 05
1.2 ExamplesofLaplaciansforfunctionsoninfinite-
dimensionalspaces 09
1.3 Gaussianmeasures 13
2 Le´vy–Laplaceoperators 22
2.1 Infinite-dimensionalorthogonalpolynomials 23
2.2 Thesecond-orderdifferentialoperatorsgenerated
bytheLe´vyLaplacian 30
2.3 Differentialoperatorsofarbitraryordergenerated
bytheLe´vyLaplacian 33
3 SymmetricLe´vy–Laplaceoperator 40
3.1 ThesymmetrizedLe´vyLaplacianonfunctionsfromthedomain
ofdefinitionoftheLe´vy–Laplaceoperator 40
3.2 TheLe´vyLaplacianonfunctionsfromthedomainofdefinition
ofthesymmetrizedLe´vy–Laplaceoperator 44
3.3 Self-adjointnessofthenon-symmetrized
Le´vy–Laplaceoperator 48
4 Harmonicfunctionsofinfinitelymanyvariables 53
4.1 Arbitrarysecond-orderderivatives 54
4.2 Orthogonalandstochasticallyindependentsecond-order
derivatives 59
4.3 Translationallynon-positivecase 64
v
vi Contents
5 LinearellipticandparabolicequationswithLe´vyLaplacians 68
5.1 TheDirichletproblemfortheLe´vy–Laplace
andLe´vy–Poissonequations 68
5.2 TheDirichletproblemfortheLe´vy–Schro¨dinger
stationaryequation 84
5.3 TheRiquierproblemfortheequationwithiterated
Le´vyLaplacians 86
5.4 TheCauchyproblemfortheheatequation 88
6 Quasilinearandnonlinearellipticequations
withLe´vyLaplacians 92
6.1 TheDirichletproblemfortheequation(cid:1) U(x)= f(U(x)) 92
L
6.2 TheDirichletproblemfortheequation
f(U(x),(cid:1) U(x))= F(x)94
L
6.3 TheRiquierproblemfortheequation
(cid:1)2U(x)= f(U(x)) 96
L
6.4 TheRiquierproblemfortheequation
f(U(x),(cid:1)2U(x))=(cid:1) U(x)99
L L
6.5 TheRiquierproblemfortheequation
f(U(x),(cid:1) U(x),(cid:1)2U(x))=0 103
L L
7 NonlinearparabolicequationswithLe´vyLaplacians 108
7.1 TheCauchyproblemfortheequations∂U(t,x)/∂t =
f((cid:1) U(t,x))and∂U(t,x)/∂t = f(t,(cid:1) U(t,x)) 108
L L
7.2 TheCauchyproblemfortheequation∂U(t,x)/∂t =
f(U(t,x),(cid:1) U(t,x)) 115
L
7.3 TheCauchyproblemfortheequationϕ(t,∂U(t,x)/∂t)=
f(F(x),(cid:1) U(t,x)) 121
L
7.4 TheCauchyproblemfortheequation f(U(t,x),
∂U(t,x)/∂t,(cid:1) U(t,x))=0 126
L
AAppendix.Le´vy–Dirichletformsandassociated
Markovprocesses 133
A.1 TheDirichletformsassociated
withtheLe´vy–Laplaceoperator 133
A.2 Thestochasticprocessesassociated
withtheLe´vy–Dirichletforms 137
Bibliographicnotes 142
References 144
Index 152
Introduction
The Laplacian acting on functions of finitely many variables appeared in the
worksofPierreLaplace(1749–1827)in1782.Afternearlyacenturyandahalf,
theinfinite-dimensionalLaplacianwasdefined.In1922PaulLe´vy(1886–1971)
introducedtheLaplacianforfunctionsdefinedoninfinite-dimensionalspaces.
The infinite-dimensional analysis inspired by the book of Le´vy Lec¸ons
d’analyse fonctionnelle [93] attracted the attention of many mathematicians.
This attention was stimulated by the very interesting properties of the Le´vy
Laplacian(whichoftendonothavefinite-dimensionalanalogues)anditsvari-
ousapplications.
Inawork[68](publishedposthumouslyin1919)Gaˆteauxgavethedefinition
ofthemeanvalueofthefunctionaloveraHilbertsphere,obtainedtheformula
forcomputationofthemeanvaluefortheintegralfunctionalsandformulated
andsolved(withoutexplicitdefinitionoftheLaplacian)theDirichletproblem
for a sphere in a Hilbert space of functions. In this work he called harmonic
thosefunctionalswhichcoincidewiththeirmeanvalues.
In a note written in 1919 [92], which complements the work of Gaˆteaux,
Le´vy gave the explicit definition of the Laplacian and described some of its
characteristicpropertiesforthefunctionsdefinedonaHilbertfunctionspace.
In 1922, in his book [93] and in another publication [94] Le´vy gave the
definitionoftheLaplacianforfunctionsdefinedoninfinite-dimensionalspaces
anddescribeditsspecificfeatures.Moreoverhedevelopedthetheoryofmean
valuesandusingthemeanvalueovertheHilbertsphere,solvedtheDirichlet
problemforLaplaceandPoissonequationsfordomainsinaspaceofsequences
andinaspaceoffunctions,obtainedthegeneralsolutionofaquasilinearequa-
tion.Wehavementionedhereonlyafewofagreatnumberofresultsgivenin
Le´vy’sbookwhichistheclassicalworkoninfinite-dimensionalanalysis.
Thesecondhalfofthetwentiethcenturyandthebeginningoftwenty-first
century follows a period of development of a number of trends originated
1
2 Introduction
in [93], and the infinite-dimensional Laplacian has become an object of
systematic study. This was promoted by the appearance of its second edition
Proble`mesconcretsd’analysefonctionnelle[95]in1951andtheappearance,
largelyduetotheinitiativeofPolishchuk,ofitsRussiantranslation(editedby
Shilov) in 1967. During this period, there were published, among others, the
worksof:Le´vy[96],Polishchuk[111–125],Feller[36–66],Shilov[132–135],
Nemirovsky and Shilov [102], Nemirovsky [100, 101], Dorfman [28–33],
Sikiryavyi [137–145], Averbukh, Smolyanov and Fomin [10], Kalinin [82],
Sokolovsky [146–151], Bogdansky [13–22], Bogdansky and Dalecky [23],
Naroditsky [99], Hida [75–78], Hida and Saito [79], Hida, Kuo, Potthoff
andStreit[80],Yadrenko[158],Hasegawa[72–74],KuboandTakenaka[85],
GromovandMilman[69],Milman[97,98],Kuo[86–88],Kuo,ObataandSaito
[89, 90], Saito [126–129], Saito and Tsoi [130], Obata [103–106], Accardi,
GibiliscoandVolovich[4],Accardi,RoselliandSmolyanov[5],Accardiand
Smolyanov[6],AccardiandBogachev[1–3],Zhang[159],Koshkin[83,84],
Scarlatti[131],Arnaudon,BelopolskayaandPaycha[9],Chung,JiandSaito
[26],Le´andreandVolovich[91],Albeverio,BelopolskayaandFeller[8].
ManyproblemsofmodernscienceleadtoequationswithLe´vyLaplacians
andLe´vy–Laplacetypeoperators.Theyappear,forexample,insuperconduc-
tivitytheory[24,71,152,155],thetheoryofcontrolsystems[121,122],Gauss
randomfieldtheory[158]andthetheoryofgaugefields(theYang–Millsequa-
tion)[4],[91].
Le´vy introduced the infinite-dimensional Laplacian acting on a function
U(x) bytheformula
(cid:1) U(x )=2lim M(x0,(cid:4))U(x)−U(x0)
L 0 (cid:4)→0 (cid:4)2
(the Le´vy Laplacian), where M(x0,(cid:4))U(x) is the mean value of the function
U(x) overtheHilbertsphereofradius(cid:4)withcentreatthepoint x .
0
Given a function defined on the space of a countable number of variables
wehave
1 (cid:1)n ∂2U
(cid:1) U(x ,...,x ,...)= lim ,
L 1 n n→∞n ∂x2
k=1 k
whileforfunctionsdefinedonafunctionalspacewehave
(cid:2)b
1 δ2U(x)
(cid:1) U(x(t))= ds,
L b−a δx(s)2
a
whereδ2U(x)/δx(s)2isthesecond-ordervariationalderivativeof U(x(t)).
Introduction 3
Butalready,in1914,Volterra[154]haduseddifferentsecond-orderdiffer-
entialexpressionssuchas
(cid:2)b
δ2V(x)
(cid:1) V(x(t))= ds
0 δx(s)δx(s)
a
(theVolterraLaplacian),whereδ2V(x)/δx(s)δx(τ)isthesecondmixedvaria-
tionalderivativeof V(x(t)).In1966Gross[70]andDalecky[27]independently
definedtheinfinite-dimensionalellipticoperatorofthesecondorderwhichin-
cludestheLaplaceoperator
(cid:1) V(x(t))=TrV(cid:4)(cid:4)(x),
0
whereV(cid:4)(cid:4)(x) istheHessianofthefunction V(x) atthepoint x.Forafunction
V definedonafunctionalspace,(cid:1) V(x(t))istheVolterraLaplacian,andfor
0
functionsdefinedonthespaceofacountablenumberofvariables,wehave
(cid:1)∞ ∂2V
(cid:1) V(x ,...,x ,...) = .
0 1 n ∂x2
k=1 k
Thereexistsanumberofotherexamplesofsecond-orderinfinite-dimensional
differentialexpressionswhichconsiderablydifferfromthedifferentialexpres-
sionsofLe´vytype.Thecorrespondingreferencescanbefoundinthebibliog-
raphytothemonographsofBerezanskyandKondratiev[12]andDaleckyand
Fomin[27].
The present book deals with the problems of the theory of equations with
the Le´vy Laplacians and Le´vy–Laplace operators. It is based on the author’s
papers[36–38,40,50–66]andthepaper[8].
InChapter1wegivethedefinitionoftheLe´vyLaplaciananddescribesome
ofitsproperties.
Intheforewordtohisbook[95],Le´vywrote:‘Inthetheorieswhichwemen-
tioned,weessentiallyfacethelawsofgreatnumberssimilartothelawsofthe
theoryofprobabilities...’.TheprobabilistictreatmentoftheLe´vyLaplacian
in the second, third, and fourth chapters allows us to enlarge on a number of
its interesting properties. Let us mention some of them. The Le´vy Laplacian
givesrisetooperatorsofarbitraryorderdependingonthechoiceofthedomain
of definition of the operator. There is a huge number of harmonic functions
of infinitely many variables connected with the Le´vy Laplacian. The natural
domainofdefinitionoftheLe´vyLaplacianandthatofthesymmetrizedLe´vy
Laplaciandonotintersect.Startingfromthenon-symmetrizedLe´vyLaplacian,
onecanconstructasymmetricandevenaself-adjointoperator.