Page 1
FractionalFokker?PlanckEquationforNonlinear
StochasticDi?erentialEquationsDrivenby
Non?GaussianLevyStableNoises
D? Schertzer? M?Larc hev?eque
Lab oratoiredeMod?elisationenM?ecanique?Tour ???Boite???
Universit?ePierre etMarie Curie?
?PlaceJussieu? F??????ParisCedex???France?
J?Duan
Department ofMathematical Sciences
ClemsonUniversity
Clemson? SC??????USA?
V?V?Y anovsky
T urbulenceResearch?InstituteforSingleCrystals
NationalAcad?Sci?Ukraine?
Leninave? ???Kharkov???????Ukraine
S?Lovejoy
PhysicsDepartment? McGillUniv ersity
????UniversityStreetMontreal? H?A?T??Quebec?Canada?
submittedto J?Math?Phys?? Oct??????
Abstract
TheF okker?Planckequation hasbeenveryusefulforstudying dynamic
behavior ofstochasticdi?erentialequationsdrivenbyGaussiannoises?In
thispaper?wederiveaFractionalFokker?Planckequationfortheprob?
abilitydistributionofparticles whosemotionisgovernedbyanonlinear
Langevin?typeequation?whichisdrivenbyanon?GaussianLevy?stable
noise?WeobtaininfactamoregeneralresultforMarkovianprocesses
generatedbystochasticdi?eren tialequations?
PACSnumbers???????j???????w? ?????Cb??????a?
CorrespondenceshouldbeaddressedtoD?Schertzer?fax?????
????????? e?mail?schertze?ccr?jussieu?fr??
?
Page 2
?Introductionandstatementofthe prob?
lem
TheFokker?Planckequationisoneofthemost celebratedequationsin
Physics?sinceithasbeenveryusefulforstudyingdynamicbeha viorof
stochasticdi?erentialequationsdriv enbyGaussiannoises?There has
recen tlybeenamushroomingin terest?????????????inthefactthat
theprobabilitydensityofa?linear?Levy motionsatis?esageneralized
Fokker?Planckequationinvolvingfractional ordersof di?erentiation?This
essentiallycorrespondstoare?interpretation ofthecharacteristicfunction
ofaLevymotionandsigni?cant applicationsseemtorequireitsnonlinear
generalization?
Wethereforeconsider thefollowingnonlinearLangevin?likeequation
for astochastic ?real?quantityX?t??
dX?t??m?X?t??t? dt???X?t??t?dL???
where thedrivingsourceisaLevystable motionL?i?e?amotion?e?g?
???? whoseincrements?Larestationaryandindependentforanytime
lag?tandcorrespondtoindependent?identicallydistributedLevystable
variables???????????????? LetusrecallthataLevystable motion
isde?ned?asare itsincrements?byfourparameters?itsLevystability
index?????????itsskewness ??????????its center??t
anditsscaleparameterD?t?D????Brownian motioncorresponds
to thelimitcase????whichalso implies?? ??and tothe?normal?
di?usion la w?ThevarianceVar?X?t??X?t
?
??of thedistance traveledby
a brownian particle?ist wiceits scaleparameter andtherefore yieldsthe
classical Einsteinrelation?V ar?X?t??X?t
?
????D?t?t
?
??
The linearcase?which istheuniquecase studied un tilnow?corre?
sp ondsto?
m?X?t??
?
t??m?
?
C onst??
?
??X?t??t????
?
C onst?
?
???
X?t??X?t
?
? is alsoa Levymotion which hasthe same Levystability
index??butwithap ossibledi?erentcen terortrend?whenm?? ?? and
scaleor amplitude?when??????
Inthe nonlinearcases???X?t??t??? andm?X?t??t?arenonlinear
functions ofX?t?andt?which satisfy certainregularit yconstraintstobe
discussedlater?Weclaimthat?
Prop osition?Thetransitionpr ob abilitydensity?
?t?t?p?x?tjx?t??Pr?X?t??xjX?t??x????
c orresp onding tothe nonlinearsto chasticdi?er entialequation
?Eq???? with????or???? is thesolutionofthefollowing
FractionalFokker?Planckequation??
?
Page 3
?
?t
p?x?tjx
?
?t
?
???
?
?x
????x?t??m?x?t??p?x?tjx
?
?t
?
?
?D ?????
???
??????
?
?x
????
???????
???x?t?
?
p?x?tjx
?
?t
?
????
where????isde?nedby?
??????????tan
??
?
???
andwherethefractionalpowersoftheLaplacian?willbediscussedin
Sect???Proposition?andEq?? willbeestablishedforscalarprocesses
?i?e???
?
?
?x
?
? anditsextension tovector processeswillbediscussedand
presentedinSect???
ThisF ractionalFokk er?Planc kequationwillbeestablished withthe
helpofthemuchmore generalproposition?
Proposition? TheinverseFourier tr ansformof thesec ond
char acteristicfunctionorcumulant generatingfunction ofthe
increments ofa Markovproc essX?t?gener atesbyc onvolution
theF okker?Plancke quation ofevolution ofitstransition prob?
abilityp?x?tjx
?
?t
?
??
We willdemonstrate thispropositionina straigh tforward?y etrigorous
way? Moreprecisely?wewill establishthe following?
?p
?t
?x?tjx
?
?t
?
??
Z
dy
?
e
K
?t
?x?yjy?t?p?y?tjx
?
?t
?
? ???
where
e
K isthe inverseF ouriertransformof thecum ulant generatingfunc?
tion of theincrements?Its argumen tswillbecomeexplicitin Sect???
This notonly holdsforprocesseswith stationary andindependent
increments? as inthe linearcase?Eq??? butforanyMarkov process?in?
cluding thosede?nedby the non?linear Langevin?likeequation?Eq??with
m??Const?????Const??? Asaconsequenceof Eq???wewilldemonstrate
thefollowing?
Proposition?The Kr amers?Moyalcoe?cientsA
n
of theFokker?
PlanckequationofaMarkov processX?t??
?p
?t
?x?tjx
?
?t
?
??
X
n?J
?
n
?x
n
?A
n
?x?t?p?x?tjx
?
?t
?
?????
aredir ectlyrelated to the cumulantsC
n
oftheincrements?
A
n
?x?t??
????
n
n?
C
n
?x?t????
wherethesetJ of theindicesnisf???g inthemostclassical case?e?g?
????whichisaparticularcaseofJ?Nwhich correspondstoananalytic
expansion ofcumulan ts?
?
Page 4
Wewill demonstratethis property?Prop????whichatbestisonly
mentionedin fewstandardtextbooksonthe?classical?Fokker?Plank
equation?aswellasitsgeneralizationfornonanalyticcumulantexpan?
sions?i?e?thereare non integersindicesn?J?Thislatterproperty?
discussed inSect???willbeexploited inSect??inorder toderive Prop??
withJ?f???g???????
?The cumulantgeneratingfunctionof
the increments
The?rst andsecond?conditional?c haracteristicfunctionsarerespectively
themomentgeneratingfunctionZ
X
?k?t?t
?
jx
?
?t
?
? andthecum ulant
generating functionK
X
?k?t?t
?
jx
?
?t
?
??asso ciatedwith the transition
probabilityp?x?tjx
?
?t
?
? ofa processX?t??They arede?nedby theF ourier
transformofthelatter?withkbeing the conjugatevariable ofx?x
?
?
F?p?x?tjx
?
?t
?
???Z
X
?k?t?t
?
jx
?
?t
?
????
?exp?K
X
?k?t?t
?
jx
?
?t
?
?? ????
?E?exp?ik?X?t??X
?
??jX?t
?
???X
?
?????
whereE??j?? denotethe conditionalmathematical expectation?F andF
??
respectivelytheF ourier?transform anditsinverse?
F?f??
?
f?k??
Z
?
??
dx exp?ikx?f?x? ????
F
??
?
?
f??f?x??
Z
?
??
dk
??
exp??ikx?
?
f?k?????
Thecorresponding quantitiesforincrements?X??t??X?t??t??X?t??
correspondingtoagiv en time lag?t???arede?nedina similarwa y?
F?p?x??x?t??tjx?t????Z
X
?k??tjx?t?????
?exp??K
X
?k??tjx?t??????
?E?exp?ik?X?t??t??X??jX?t??X? ????
wherek istheconjugatevariableof?x? Thecum ulants oftheincremen ts
C
n
are theco e?cien ts of theTaylorexpansionof?K
X
?
?K
X
?k??tjx?t???t
X
n?J
?ik?
n
n?
C
n
?x?t??o??t? ????
Asalreadymentioned?theclassicalcase corresponds to ananalytic
expansionof?K
X
?i?e?J?N? whereaswe willbe interestedbya non?
analyticexpansionJ?f???g?
?
Page 5
? Processeswithstationaryandindepen?
dentincrements
Letus ?rstconsiderthe simplesub?case ofaprocesswith stationaryand
independentincremen ts?It correspondstoC
n
?x?t??C
n
?Const?in
Eqs??? ?? and asalreadydiscussedinSect???itincludesthe linear case
?Eq???of theLangevin?likeequation?Eq????
Howeverweb elievethat thefollowingderivation isnotonlysomewhat
p edagologicalonthe role ofthecharacteristicfunctions forthe nonlinear
case?butterserthanderivations previously presentedforthe linearcase?
Thestationarity oftheincremen tsimpliesthatthetransition proba?
bility dependsonlyonthe time andspace lags?i?e??
p?x?tjx
?
?t
?
??p?x?x
?
?t?t
?
?????
andsimilarly? thec haracteristicfunctions ofthe incremen ts arenolonger
conditioned?forinstance?
Z
X
?k?t?t
?
jx
?
?t
?
??Z
X
?k?t?t
?
?????
K
X
?k?t?t
?
jx
?
?t
?
??K
X
?k?t?t
?
? ????
On theother hand?the independence oftheincremen tsimpliesthat
thetransition probabilitiessatisfyaconv olution?ov eranypossible inter?
mediatepositiony? forany given timelag?t?
??t???p?x?x
?
?t??t?t
?
??
Z
dyp?x?y??t?p?y?x
?
?t?t
?
?????
andthe correspondingcharacteristic functionsmerelyfactor?resp?add??
Therefore?weha ve?
Z
X
?k?t??t?t
?
??Z
X
?k?t?t
?
??Z
X
?k?t?t
?
???Z
X
?k??t???? ????
Thisinturn leadsto?
Z
X
?k?t??t?t
?
??Z
X
?k?t?t
?
??Z
X
?k?t?t
?
??K
X
?k??t??o??t?????
ItsinverseFouriertransform yields?
p?x?t??tjx
?
?t
?
??p?x?tjx
?
?t
?
??
Z
dyF
??
??K
X
?k??t??p?y?x
?
?t?t
?
??o??t?
????
Thisdemonstrates?in thelimit?t? ??Prop??andEq???aswell as
Prop??? sinceEq???corresponds? withthe help ofEq????to?
?
Page 6
p?x?t??tjx
?
?t
?
??p?x?tjx
?
?t
?
???t
X
n
?C
n
????
n
n?
Z
dy?
?n?
x?y
p?y?tjx
?
?t
?
???o??t?
????
where?
n
x
denotes then
th
derivativeofthe Diracfunction?Therefore?we
obtain?
?
?t
p?x?tjx
?
?t
?
??
X
n?J
A
n
?
n
?x
n
p?x?tjx
?
?t
?
?????
which correspondsto thelinear caseofEq???
?More generalMarkov pro cesses
In thecase ofaMarkov process which do esnot havestationary andin?
dependent increments?there isnolongerasimple conv olution equation
?Eq???? ofthetransitionprobabilities? nora simplefactorization ofchar?
acteristicfunctions ?Eq?????How ever? theformer satis?esageneralized
conv olutionequation which correspondsto theChapman?Kolmorogorov
iden tity ????v alidfor any Markov pro cessX?t??
??t???p?x?t??tjx
?
?t
?
???
Z
dyp?x?t??tjy?t?p?y?tjx
?
?t
?
? ????
which indeedreduces toamere convolution?Eq???? in thecase of pro?
cesseswith stationaryand indep endentincremen ts?This identity canbe
written under the equivalentform?
p?x?t??tjx
?
?t
?
??
Z
dy
Z
dk
??
e
?iky??K
X
?k??tjy?t?
p?y?tjx
?
?t
?
?????
Notingthatwehave?
p?x?tjx
?
?t
?
??
Z
dyp?y?tjx
?
?t
?
?
Z
dk
??
e
?iky
????
we obtain?
p?x?t??tjx
?
?t
?
??p?x?tjx
?
?t
?
???t
Z
dyF
??
??K
X
?k??tjy?t??p?y?tjx
?
?t
?
??o??t?
????
Inthelimit?t???thiscorresponds toProp?? andEq? ??When
J?N?it yieldswith thehelp of Eq????
?p?x?tjx
?
?t
?
???t
X
n?N
Z
dy?
?n?
x?y
?
????
n
n?
C
n
?y?t?p?y?tjx
?
?t
?
???o??t?????
Thelimit?t?? correspondstoEq?? anddemonstratesProp??for
any ?classical?Markow process?
?
Page 7
?Extensiontofractionalorders
Inthetwoprevious sections?Sects?????? thefactthat theindicesn?J
shouldbeintegers interveneatb estonlyinthecorrespondencebetween
?integer order?di?eren tiation
?
n
?x
n
?inEq???andpow ersof theconjugate
v ariablek
n
?in Eq????? How ever?bythev ery de?nition offractional
di?erentiation?e?g???? ???this corresp ondenceholds alsofornon integer
orders? Howev er? there isnota uniquede?nition offractionaldi?eren tia?
tionandtherefore?asdiscussed in somedetailsin?????we cannot expect
tohavea uniqueexpression oftheFractionalFokker?Planckequation?
Since itwillbe su?cientfor thefollowing to consideranexpansion
ofthecharacteristicfunction inv olving fractionalpow ersof onlythewave
n umb erjkj? it isinteresting to considerRiesz?sde?nition ofa fractional
di?erentiation? Indeed?the lattercorrespondstoconsiderfractionalpow?
ers ofthe Laplacian?
?????
???
f?x??F
??
?jkj
?
?
f?k??????
whichhas furthermore theadv an tageofb eingvalid for thev ectorcases?
How ev er?wewill see in Sect?? thatingeneralitdoes notapply ina
straightforw ardmannerford?dimensional stableL?evymotions? Indeed
thelatter in troduces rather ?one?dimensional?directionalLaplacians?i?e?
?one?dimensional?Laplacians
?
along
??
a giv en
?
directionu
?ju
?
j? ??
?
?
????
u
?
???
f?x??F
??
?j?k?u?j
?
?
f?k ??????
where????? denotesthescalar product?On the otherhand?itwillbeuseful
toconsider thefractionalpowerofthecontractionofthe Laplaciantensor
??
?
i?j
?
?
?x
i
?
?x
j
????
bya tensor?
?withthefollowing de?nition?
????
?? ??
?
?
?
?F?j?k
?? ??
?k
j
?
?
??F
??
?j?
?k
j? ????
? Levy case
Thesecond characteristicfunction oftheincremen ts?Lofthe?scalar?
Levy forcingisthefollowing?
?K
L
?k??t???t?ik??Djkj
?
??? i?
k
jkj
??k?????o??t?????
where??k??? is de?nedby?
???????k?????????tan
??
?
???????k????
?
?
l ogjkj????
?
Page 8
ConsideringanIto?likeforw ard integrationofEq??? theincrements?L
generatesthe following ??rst?c haracteristicfunctionfor theincremen ts
?X ofthe motionX?t??
?Z
X
?k??tjx?? x?t??e
ikm?X ?t?
?Z
?L
?k??tjx?t???o??t?????
whichyields thefollo wingelemen tarycum ulantgeneratingfunction?K
X
?
?K
X
?k??tjx?t???t?ikm?x?t?? ik???x?t?????
?Djkj
?
???i?
k
jkj
??k??????x?t?
?
??o??t? ????
andwhichis ofthe sametypeas Eq????withJ?f???g? Therefore?as
discussed inSect??? we havefractionaldi?erentiations in thecorresponding
Eq???which willpreciselycorrespond toEq??? and therefore establishes
Prop? ??
Letus discussbrie?ythe regularit yconstrain tsthat shouldbe satis?ed
by the nonlinear function? ??X?t??t???andm??X?t??t??Obviously?they
shouldbe measurable?Onthe otherhand?theuniquenessofthe solution
shouldrequire? as for theclassicalnonlinearFokker?Planckequation?e?g?
??????aLipschitzconditionforboth? ??X?t??t? andm??X?t??t??
?Extension tov ectorprocesses
With but oneimp ortant exception?the extensionofthepreviousresults
tohigherdimensionsisrather straightforward?The startingp ointofthis
extensionisthefollowingnonlinear stochasticequation?X
?t??R
d
??
dX
?t??m?X?t??t?dt???X?t??t??dL????
wherem
and?are thenaturalv ector?respectivelytensor?extensions
ofthedeterministic?liketrend? respectiv elymodulation of therandom
driving force?L
isad?dimensionalLevy stablemotion and?asdiscussed
below?theexpressionofitscharacteristicfunction correspondstothe
sourceofthe di?cultyinextendingthe scalar resultstohigh dimensions?
Onthecontrary?itisstraightforwardtocheckthat Props? ??? arev alid
inthed?dimensional case?with thefollowingextensions?x
?R
d
?forEq?
??
?p
?t
?x?tjx
?
?t
?
??
Z
dy
?
e
K
?t
?x?yjy?t?p?y?tjx
?
?t
?
?????
andforEq???n
?J?N
d
?jnj?
P
d
i??
n
i
??
?p
?t
?x?
tjx
?
?t
?
??
X
n
?J
?
jnj
?x
n
?
?
?x
n
?
?
???x
n
d
d
?A
n
?x?t?p?x?tjx
?
?t
?
??????
therelation to thecumulan tsC
n
oftheincremen ts isnow?
?
Page 9
A
n
?x?t??
????
jnj
?n
?
???n
?
?????n
d
??
C
n
?x?t?????
Onthe otherhand? Eq???yields thefollo wingextensionto Eq????
?Z
X
?k??tjx?t??e
ik
?m?x ?t?
?Z
?
?L
?k??tjx?t?????
andthereforewehav e?
?K
X
?k??tjx?t??ik ?m?x?t???K
L
??
?
?k
??tjx?t??o??t? ????
Let us recall thata stableL? evyvector intheclassicalsense ??? ??? ???
?see???? foradiscussion andageneralization?corresp ondstothe limit of
asum ofjumps? withapow er?lawdistribution? along randomdirections
u
??B
?
?B
?
b eingthe unit ball?distributedaccording toa ?positiv e?
measured??u
?? Thelatter? whichgeneralizes thescaleparameterD ofthe
scalar case?isthesourceof thedi?culty sinceingeneraltheprobability
distributionofastableL?evyvector depends on thismeasure?and therefore
isa nonparametricdistribution? How ev er?asdiscussed belo w?there is at
leasta trivialexception?thecase ofisotropicstableL? evyvectors?
Corresp
The
onding
scalar
to
?Eq????
ourprevious
corresp
remarks?
onds
a ?classical?stable L?evyvector
hasthefollo wing?Fourier? cumulant generatingfunction?
K
L
?k???t?i?k????
Z
u??B
?
?ik
?u?
?
d??u???o??t?????
which yieldswiththehelpoftheEq????
?
?t
e
K
X
?k???div?m?? ????F
??
?
Z
u??B
?
?i?
?
?x
?t??k?u?
?
d??u??????
caseto?
??p??????p???d??u??Dcos?
??
?
??p?
?u???
? ???p??
?u???
?????
F orany dimensiond? thesecondterm on therighthandside ofEq???
corresponds toa fractional di?erentiationoperatoroforder??This op?
eratorcan besligh tlyre?arranged?With thehelp of theo ddd?
?
?u? and
evend?
?
?u?parts of themeasured??u??
?d?
?
?u
??d??u??d???u???d?
?
?u??d??u??d???u?????
andtheiden tity??being theHea visidefunction??
?ik?
?
?jkj
?
???k?e
i
??
?
????k?e
?i
??
?
?????
onecanwritetheextension ofEq??under thefollowingform?
?
Page 10
?
?t
p?x
?tjx
?
?t
?
???div?m?x?t????x?t?????p?x?tjx
?
?t
?
??
??????
?? ??
?
?
?
?
?
?
?
???r??
?
?????
?? ??
?
?
???
?
?
?
?
?p?x?
tjx
?
?t
?
? ????
wherethesymmetricandan tisymmetricoperatorsare de?ned?similarly
toEq???? inthefollo wing manner?
?????
?? ??
?
?
?
?
?
?
?
?
Z
u??B
?
d?
?
?u?F
??
?j????x?t??k?u?j
?
?????
???r??
?
?????
?? ??
?
?
???
?
?
?
?
?
Z
u??B
?
d?
?
?u?F
??
???i?
?
?x
?t??k?u?j??
?
?x
?t??k?u?j
???
? ????
Ingeneral? eachoperator correspondstoarathercomplex integra?
tion ?whichisindicatedby thesymb ol???
?
?ofdirectionalfractional
Laplacians?Eq????? How ever? thesymmetric operator becomes simpler
asso on as theev enpartd?
?
ofthe measured? isisotropic? Indeed?the
integrationoverdirections yieldsonlyaprefactorD?
????
????
?
?
?
?
?
?
?
?D???????
?
?
?
?
D?
Z
u??B
?
d?
?
?u?j?u
?
?u
?j
?
????
and for???thiscorrespondstothe classical term??
?? ??
?
? ofthe
standardd?dimensionalF okker?Planckequation? Ifd? itself isrotation
inv ariant?then theasymmetric op eratorvanishes? sinced?
?
??? If
furthermore??
isrotation invarian t?i?e???? ??thenoneobtainsthe
followingFractionalFokker?Planckequation?
?
?t
p?x
?tjx
?
?t
?
???div ??
???x?t??m?x?t??p?x?tjx
?
?t
?
?????
?D ?????
???
???x?t?
?
p?x
?tjx
?
?t
?
?????
Therefore?as onemightexpectit?due totherotation symmetries?this
corresponds toa rathertrivialextensionofthestandardgaussiancase?a
fractionalpow er?ofthed?dimensionalLaplacian?as inthepure scalar
case?Eq????Ob viously?the integrationperformed inEq???isalsogreatly
simpli?edassoonasd??u?is discrete?i?e? itssupp ortcorrespondstoa
discreteset ofdirectionsu
i
?
Onthe otherhand?let usnotethattheframew orkofgeneralized stable
L?evyvectors??? ??allo ws onetointroduceamuchstrongeranisotropythan
thethemeasured?doesiffor classical stableL? evyv ectors?This there?
fore diminishes theimportanceoftheasymmetryof thelatter? Indeed?
thecomponen tsofa generalizedstableL? evyvector do nothave neces?
sarily thesameL? evystabilityindex?the latterbeinggeneralizedintoa
??
Page 11
secondrank tensor?Similarly? thedi?erentialoperatorsinvolvedinthe
correspondingFractionalFokker?Planckequation havenolongera unique
order of di?erentiation? Thisisrathereasy tocheck in case ofadiscrete
measure d??u? andwe willexploreelsewhere thegeneralcase?
?Conclusion
We havederivedaFractionalF okk er?Planckequation? i?e?a kineticequa?
tion which inv olves fractionalderiv ativ es?for the evolutionof theproba?
bilitydistribution ofnonlinearstoc hastic di?erentialequations drivenby
non?GaussianLevystablenoises?We?rstestablished thisequation in
thescalarcase?whereithasarathercompactexpressionwith thehelpof
fractionalpow ersoftheLaplacian? andthendiscussed itsextension to the
v ector case?ThisFractionalF okker?Planckequationgeneralizes broadly
previous resultsobtainedfora linearLangevin?like equationwithaL? evy
forcing?asw ellas thestandardF okker?Planck equationforanonlinear
Langevin equationwithaGaussianforcing?
? Ac knowledgmen ts
Wew ould liketothankDr?JamesBrannanforhelpfuldiscussions?P art
of thisworkwasperformedwhile DanielSchertzerw asvisiting Clemson
Univ ersity?
References
???
???
F
F
ogedb
ristedt?
y?H?C??
B??
Phys?R
L?
ev?E?
A
???
dern
?? ??????????
Approac
??????
to
???Chec hkin? A?V??D? Schertzer?A?V?T ur?V?V?Y anovsky?Ukr?J
Phys??
Gardiner?
?????
C?W??
???????
Handb
???????
ok
??? Compte?A Phys?Rev?E???? ??????????????????
???Marsan?D??D? Schertzer? S?Lovejoy? J? Geophys?Res????D???????
????????????
??? Chaves?A?S?? Ph ys?LettersA? ????????? ??????
???Yanovsky?V?V?? A?V?Chechkin? D? Sc hertzerandA? V?T ur?sub?
mittedto PhysicaA??????
Gray?MohProbabilit yTheory ?
Birkhauser? Boston???????
??? ofStaoc hasticMethods forPhysics?Chem?
istryandtheNaturalSciences??Springer?V erlag?Berlin ???????
???L?evyP??Theoriedel?AdditiondesVariablesAl?eatoires?Gauthier?
Villars?Paris???????
????KhintchineA?Y?and L?evy P??C? R?Acad?Sci??Paris????????
???????
??
Page 12
???? Gnedenko B?V?andKolmogorovA?N??LimitDistributions forSums
ofIndependent RandomVariables? AddisonWesley?Reading? MA?
???????
????FellerW?? AnIntro duction toProbabilityTheory anditsApplica?
tions?John?Wiley? Sons? New?York???????
????ZolotarevV?M?? One?dimensionalStable Distributions?MM?V ol? ???
AMS???????
????Miller? K?S??An Intro ductionto theFractionalCalculus andFrac?
tional Di?erential Equations?J?Wiley?sons? NewYork???????
????PaulauskasV?J????????Some remarksonMultivariate StableDistri?
butions? J?MultivariateAnal??? ???????
????NikiasC?L? and M?Shao ???????Signal Processingwith alpha?stable
distributions andapplic ations?John WileyandSons?New?York?
????Schertzer?D??M?Larc hev?eque? J? Duan? S? Lov ejoy?Submitted toJ?
MultivariateAnal??July ???????
????Oksendal?B?Stoc hasticDi?erentialEquations? Springer?Verlag??th
ed????? pp????????
??
Download full-text