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MASSIMILIANO GUBINELLI AND SAMY TINDEL

56 pages
ROUGH EVOLUTION EQUATIONS MASSIMILIANO GUBINELLI AND SAMY TINDEL Abstract. We show how to generalize Lyons' rough paths theory in order to give a pathwise meaning to some non-linear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we apply the theory to a class of 1d SPDEs driven by a space-time fractional Brownian motion. Contents 1. Introduction 1 2. Algebraic integration in one dimension 4 2.1. Increments 4 2.2. Computations in C? 8 2.3. Dissection of an integral 9 3. Algebraic integration associated to a semigroup 11 3.1. Analytical semigroups 11 3.2. Convolutional increments 11 3.3. Computations in C? 13 3.4. Fractional heat equation setting 16 4. Young theory 18 4.1. Young integration 18 4.2. Young SPDEs 19 4.3. Application: the fractional heat equation 22 5. Rough evolution equations: the linear case 28 5.1. Strategy 28 5.2. Integration of weakly controlled paths 29 5.3. Linear evolution problem 32 5.4. Application: stochastic heat equation 33 5.5. The algebra of a rough path 37 6. Polynomial non-linearities 38 6.1. Formal expansions and trees 38 6.2. Algebraic computations 40 6.3. A space of integrable paths 42 6.4. The Brownian case 45 6.5. Diagrammatica 46 6.6. More complex graphs 50 References 54 Date: March 4, 2008. 2000 Mathematics Subject Classification.

  • young integral

  • fractional brownian

  • holder exponent

  • setting

  • linear evolution

  • rough path

  • young theory

  • rough evolution

  • hts ?

  • equation


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ROUGHEVOLUTIONEQUATIONS
MASSIMILIANOGUBINELLIANDSAMYTINDEL
Abstract.
WeshowhowtogeneralizeLyons’roughpathstheoryinordertogiveapathwise
meaningtosomenon-linearinfinite-dimensionalevolutionequationassociatedtoananalytic
semigroupanddrivenbyanirregularnoise.Asanillustration,weapplythetheorytoaclass
of1dSPDEsdrivenbyaspace-timefractionalBrownianmotion.

Contents
1.Introduction
2.Algebraicintegrationinonedimension
2.1.Increments
2.2.Computationsin
C

2.3.Dissectionofanintegral
3.Algebraicintegrationassociatedtoasemigroup
3.1.Analyticalsemigroups
3.2.Convolutionalincrements
3.3.Computationsin
C
ˆ

3.4.Fractionalheatequationsetting
4.Youngtheory
4.1.Youngintegration
4.2.YoungSPDEs
4.3.Application:thefractionalheatequation
5.Roughevolutionequations:thelinearcase
5.1.Strategy
5.2.Integrationofweaklycontrolledpaths
5.3.Linearevolutionproblem
5.4.Application:stochasticheatequation
5.5.Thealgebraofaroughpath
6.Polynomialnon-linearities
6.1.Formalexpansionsandtrees
6.2.Algebraiccomputations
6.3.Aspaceofintegrablepaths
6.4.TheBrowniancase
6.5.Diagrammatica
6.6.Morecomplexgraphs
References

Date
:March4,2008.
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2

MASSIMILIANOGUBINELLIANDSAMYTINDEL

1.
Introduction
Thispapercanbeseenaspartofanongoingprojectwhoseaimistogiveapathwisedefinition
tostochasticPDEs.Indeed,theroughpaththeory[8,13,17,18]anditsvariants[9,7]have
nowreachedacertainlevelofmaturity,leadingtoaproperdefinitionofdifferentialequations
drivenbyirregularsignalsandinparticularbyafractionalBrownianmotion[2].Startingfrom
thisobservation,wehavetriedin[12]todefineandsolvethefollowinggeneralproblem:let
B
beaseparableBanachspace,and
A
:
D
(
A
)
→B
theinfinitesimalgeneratorofananalytical
semigroup
{
S
t
;
t

0
}
on
B
,inducingthefamily
{B
α
;
α

R
}
with
B
α
=
D
((

A
)
α
).Letalso
f
beafunctionfrom
B
to
L
(
B

α
,
B

α
)foragiven
α>
0and
x
anoisyinput,consideredasa
functionfrom
R
+
to
B

α
.Then,for
T>
0,considertheequation
dy
t
=
Ay
t
dt
+
f
(
y
t
)
dx
t
,t

[0
,T
]
,
(1)
withaninitialcondition
y
0
∈B
.Themainexamplewehaveinmindisthecaseofthe1-
dimensionalheatequationin[0
,
1],namely
B
=
L
2
([0
,
1]),
A
=ΔwithDirichletboundary
conditions,theusualSobolevspaces
B
α
=
H
α
=
W
2
α,
2
,and
x
afractionalBrownianmotion
withHurstparameter
H
takingvaluesin
B

α
.Noticeinparticularthatwewishtoconsidera
noise
x
whichisirregularinbothtimeandspace.Then,in[12],wegavealocalexistenceand
uniquenessresultforequation(1),byconsideringitinitsmildform
Zty
t
=
S
t
y
0
+
S
ts
f
(
y
s
)
dx
s
,
(2)
0wherewelet
S
ts
=
S
t

s
andinterpretingtheintegralinthismildformulationasaYoung
integral.Oncetheequationissetundertheform(2),themainproblemoneisfacedwithisto
quantifytheregularizationofthesemi-group
S
ts
ontheterm
f
(
y
s
)
dx
s
,andthentoelaborate
therightfixedpointargumentinordertosolvetheequation.Thegeneralresultsof[12]could
beappliedinthecaseofthestochasticheatequationdrivenbyafractionalBrownianmotion
withHurstparameter
H>
1
/
2.Theyshouldbecomparedwiththereference[19],wherea
non-linearfractionalSPDEissolvedthankstosomefractionalcalculusmethods,butwhere
x
isasmoothnoiseinspace.
Inthecurrentarticle,wewouldliketogoonestepfurtherwithrespectto[12],andsetthe
basisofarealroughpathexpansioninordertodefineandsolveequation(2),whichwould
allowtoconsider,inthecaseoftheheatequationin[0
,
1],afractionalBrownianmotionwith
Hurstparameter
H

1
/
2.Thistaskisquitelongandinvolved,butletussummarizeatthis
pointsomeoftheideaswehavefollowed:
(1)
Wewillrecastequation(2)inasuitablewayforexpansionsaccordingtothefollowing
simpleobservation:wehavetriedtosolveourevolutionequationbymeansofitsinfinite
dimensionalsetting,sinceitallowstoconsider
x
and
y
asfunctionsofauniqueparameter
t

[0
,T
],whichmakesitsroughpathtypeanalysiseasier(see[11]and[23]foramultiparametric
setting).However,whenwecometotheapplicationstotheheatequation,wewillconsider
theevolutionequationin[0
,T
]
×
[0
,
1]undertheform
Z
1
Z
t
Z
1
y
(
t,ξ
)=
G
t
(
ξ,η
)
y
0
(
η
)

+
G
t

s
(
ξ,η
)
σ
(
y
s
(
η
))
x
(
ds,dη
)
,
(3)
000where
G
standsforthefundamentalsolutiontotheheatequation,
σ
:
R

R
isaregular
function,and
x
(
ds,dη
)isunderstoodasthedistributionalderivativeofareal-valuedcontinuous
processon[0
,T
]
×
[0
,
1].Thisdefinitionofourequationisofcourseequivalentto(2)when
f
isconsideredasthepointwisenon-linearoperator[
f
(
y
t
)](
ξ
)

σ
(
y
t
(
ξ
)).Now,whenwritten

ROUGHEVOLUTIONEQUATIONS

3

underitsmultiparametricform(3),theequationisalsoequivalentto
Z
1
Z
t
Z
1
y
(
t,ξ
)=
G
t
(
ξ,η
)
y
0
(
η
)

+
G
t

s
(
ξ,η
)
x
(
ds,dη
)
σ
(
y
s
(
η
))
,
000andithappensthatthissimplereformulationismuchmoreconvenientforourfutureexpansions
thantheoriginalone.Whenwegobacktotheoriginalinfinitedimensionalsetting,wecan
recast(2)into
Z
ty
t
=
S
t
y
0
+
S
ts
dx
s
f
(
y
s
)
,
(4)
0where
f
isnowasmoothfunctionfrom
B
to
B
,and
x
willbeunderstoodasaHo¨lder-continuous
processtakingvaluesinaspaceof
deregularizing
operatorsfrom
B
toadistributionalspace
B

ζ
foracertain
ζ>
0.Theproduct
dx
s
f
(
y
s
)willthenberegularizedagainbytheaction
of
S
ts
,inawaywhichwillbequantifiedlateron.Noticethattheform(4)ofourevolution
equationisalittleunusualintheSPDEtheory,butmakessenseinourcontext.
(2)
InsteadofconsideringRiemannsumslikein[12]orlikeintheoriginalLyons’theory
[17],ouranalysiswillbebasedonthetheoryof
generalizeddifferentials
,called
k
-increments,
containedin[9].Roughlyspeaking,thistheoryisbasedonthefactthatanelementaryoperator,
Rtcalled
δ
,cantransformanintegral
s
dg
u
[
h
u

h
s
],seenasafunctionofthevariables
s
and
t
,
intoafinitedifferenceproduct(
g
t

g
s
)(
h
t

h
s
).Furthermore,undersomeadditionalregularity
propertieson
g
and
h
,theoperator
δ
canbeinverted,anditsinverseΛ,called
sewingmap
(from[7]),willbethebuildingstoneofourextensionofthenotionofintegral.Noticethat,
whenever
g
and
h
areHo¨lder-continuouswithHo¨lderexponent
>
1
/
2,thisextensioncoincides
RtwiththeusualYoungintegral.Whenweconsideranintegraloftheform
s
dg
u
φ
(
g
u
)fora
Ho¨lder-continuousfunction
g
withHo¨lderexponentin(1
/
3
,
1
/
2]admittingaLevyarea,our
definitionofintegralalsocoincideswithLyons’one,asshownin[9].Infact,iftheusualrough
paththeorygivesaricherpointofviewonthealgebraicstructureofthepath
x
,itisworth
mentioningthatourapproachhasatleasttwoadvantages:
(1)Onceourunusualsettingisassimilated,itbecomesquiteeasytofigureouthowagiven
expansionintermsof
x
canbeleaded.Andindeed,itwillbecomeclearthroughoutthe
paper,thatthe
k
-incrementstheoryprovidesatoolallowingsomenaturalcomputations
forourgeneralizedintegrals.
(2)TheonlystepwhereadiscretizationprocedureisneededistheconstructionoftheΛ
mapalludedtoabove,andthisavoidssomeofthecumbersomecalculationswhichare
oneofthemainingredientsoftheroughpaththeory.
Wehopethatthispaperwilladvocatefortheuseofthe
k
-incrementstheory,whichobviously
doesnotexcludetheotherapproaches[17,7].
(3)
Thefactthatwearedealingwithanevolutionproblemwillforceustochangesomeofthe
algebraicstructurewewillrelyon,especiallyifonewantstotakeadvantageoftheregularizing
effectof
S
t
.Thiswillleadustointroduceanoperator
a
ts
=
S
ts

Idfor
t

s
,andamodified
δ
operator,called
δ
ˆ,definedby
δ
ˆ=
δ

a
.Thewholeincrementtheorywillhavetobebuild
againbasedonthismodifiedoperator,andwewillseethatitisreallysuitablefortheevolution
settinginducedby(4).Inparticular,wewillbeabletodefineanalogsoftheLevyareaand
ofthehigherorderiteratedintegrals,whichareofcoursehardertoexpressthaninthefinite
dimensionalcase,butcanbewritten,inthebilinearcase(thatis
σ
(
r
)=
r
in(3)),as
Z
t
Z
u
Z
t
232X
ts
=
S
tu
dx
u
S
uv
dx
v
S
vs
,X
ts
=
S
tu
dx
u
X
us
,
etc.(5)
sssObviously,aconvenientdefinitionofiteratedintegralsisthekeytoreachthecaseofaHo¨lder
continuousnoiseoforder

1
/
2.

4

MASSIMILIANOGUBINELLIANDSAMYTINDEL

(4)
Thewholeintegrationtheorycanbeexpressedinanabstractway,byjustsupposinga
certainsetofassumptionsonsomeincrementaloperatorslike
X
2
and
X
3
.However,wewill
trytochecktheseassumptionsinsomeinterestingcases,liketheinfinite-dimensionalfractional
BrownianmotionforourYoungtypeintegration,ortheinfinite-dimensionalBrownianmotion
forourstep3expansion,basedon
X
2
and
X
3
.Noticethattheroughexpansionsforthe
fractionalBrownianmotionshouldbeinvestigatedindetailstoo,butoneisfacedwithan
additionalprobleminthissituation:ononehand,aStratonovichtypeintegrationrequiresa
lotofregularityinspaceforthenoise,duetothewell-knownpresenceofsometraceterms.
Ontheotherhand,theSkorokhodintegraldoesn’tfulfillthealgebraicrequirementsweask
forourintegralextension.Adiscussionoftheseproblemsandsomeideastosolvethemwill
beincludedattheendofthepaper,butforsakeofconciseness,wewillpostponeacomplete
developmentofthisparttoasubsequentpaper,andstickheretotheBrowniancase.
Thispaperisstructuredasfollows.InSect.2werecallthebasicsetupof[9]whichallows
toembedthetheoryofroughpathsinatheoryofintegrationof
generalizeddifferentials
,
calledhere
k
-
increments
.Wewroteitwiththeaimofhavingaself-containedandpedagogical
introductiontothetopic.Howeverwegivealsoanewandveryelementaryproofofthe
existenceofthebasicintegrationmapΛof[9].InSect.3weintroduceandstudyamodified
coboundaryinducedbytheoperator
δ
ˆonthecomplexofincrements,usingtheadditional
dataprovidedbyananalyticsemigroup
S
,insuchawaythatthenewcomplexcanbeshown
toactsimplyonconvolutionintegralsoftheformappearingineq.(4)andontheiriterated
versions.Thisnewcomplexmaintainmanyofthepropertiesoftheoriginalcomplex(e.g.
itscohomologyistrivial)anditisshownthatwhenequippedwithHo¨lder-likenormswhich
measures“smallness”oftheincrements,itadmitsamap,calledΛˆhere,whichisthemaintool
forbuildinganintegration(orbetter,convolution)theoryoverthose1-incrementswhichare
goodenough(again,inasuitablesense,tobespecifiedinduetime).Akeyfeatureofthis
perturbedcomplexisthat,duetotheconvolutionwiththesemigroup
S
,“space”and“time”
regularityofincrementsdependsoneachother:wecangainspaceregularitybyloosingsome
timeregularityandvice-versa.Thispropertywillbeessentialforthesolutionoftheevolution
problembyfixed-pointarguments.InSect.4.2weusethetheoryoutlinedinSect.3.2to
definetheconvolutionintegralintheYoungsenseandsolveaclassofnon-linearevolution
problems,reobtainingsomeresultsofthework[12].Noticethatwewillalsoimprovesome
ofourpreviousresultscontainedin[12],inthesensethatwewillbeabletoconstructglobal
solutionstoourevolutionequationsintheYoungcontext.InSect.5westudythe(bi)-linear
evolutionproblem
Zty
t
=
S
t
y
0
+
S
ts
dx
s
y
s
.
(6)
0Wewillalsointroduceanotionofrough-pathsuitablefornoisesdrivingevolutionequations.
Byexploitingthispath-wisetechniqueweareabletoobtainautomaticallytheflowsemigroup
oftheequationandwewillshowhowtoexpressthissemigroupasaconvergentseriesof
iterated-integralswhicharetheliftofthestep-3roughpathusedintheconstructionofthe
solution.TheninSect.6weturntoanon-linearcaseofevolutionsystem,namelythecaseof
thequadratictypeequation
Zty
t
=
S
t
y
0
+
S
ts
dx
s
B
(
y
s

y
s
)
,
0where
B
standsforthepointwisemultiplicationoffunctions.Thisrequirestheadditional
carefulintroductionofacollectionofaprioriincrementsindexedbyplanartrees,andan
associatenotionofcontrolledpath.Finally,allourresultswillbeappliedintheconcretecase
ofthestochasticheatequationonthecircle,inasettingrecalledatSect.3.4.Thecaseofa

ROUGHEVOLUTIONEQUATIONS

5

fractionalBrowniancaseishandledinthespecialsituationoftheYoungtheory,whilewestick
totheexampleofaninfinite-dimensionalBrownianmotionintheroughersituation.Webuild
theroughpathassociatedtothislatternoiseandprovideconcreteconditionswherethetheory
outlinedintheprevioussectionscanbefruitfullyapplied.Asystematicstudyoftheregularity
propertiesoftheincrementaloperatorsdefinedas
X
2
or
X
3
in(5)willalsobeprovidedat
Sections6.5and6.6,thankstosomeFeynmandiagramstechniques.
2.
Algebraicintegrationinonedimension
Theintegrationtheoryintroducedin[9]isbasedonanalgebraicstructure,whichturns
outtobeusefulforcomputationalpurposes,buthasalsoitsowninterest.Sincethissetting
isquitenon-standard,comparedwiththeonedevelopedin[17],andsincewewillelaborate
onitthroughoutthepaper,wewillrecallbrieflyhereitsmainfeatures.Wealsoprovidean
elementaryproofoftheexistenceoftheΛmap.
2.1.
Increments.
Asmentionedintheintroduction,theextendedintegralwedealwithis
basedonthenotionofincrement,togetherwithanelementaryoperator
δ
actingonthem.
However,thissimplestructuregivesraisetoanicetopologicalstructurethatwewilldescribe
brieflyhere:firstofall,foranarbitraryrealnumber
T>
0,avectorspace
V
,andaninteger
k

1,wedenoteby
C
k
(
V
)thesetoffunctions
g
:[0
,T
]
k

V
suchthat
g
t
1
∙∙∙
t
k
=0whenever
t
i
=
t
i
+1
forsome
i

k

1.Suchafunctionwillbecalleda(
k

1)
-increment
,andwewillset
C

(
V
)=

k

1
C
k
(
V
).Theoperator
δ
alludedtoabovecanbeseenasacoboundaryoperator
actingon
k
-increments,inducingacochaincomplex(
C


),andisdefinedasfollowson
C
k
(
V
):
1+kXδ
:
C
k
(
V
)
→C
k
+1
(
V
)(
δg
)
t
1
∙∙∙
t
k
+1
=(

1)
i
g
t
1
∙∙∙
t
ˆ
i
∙∙∙
t
k
+1
,
(7)
1=iwhere
t
ˆ
i
meansthatthisparticularargumentisomitted.Thenafundamentalpropertyof
δ
,
whichiseasilyverified,isthat
δδ
=0,where
δδ
isconsideredasanoperatorfrom
C
k
(
V
)to
C
k
+2
(
V
).Wewilldenote
ZC
k
(
V
)=
C
k
(
V
)

Ker
δ
and
BC
k
(
V
):=
C
k
(
V
)

Im
δ
,respectively
thespacesof
k
-cocycles
andof
k
-coboundaries
,followingstandardconventionsofhomological
algebra.
Somesimpleexamplesofactionsof
δ
,whichwillbetheoneswewillreallyusethroughout
thepaper,areobtainedbyletting
g
∈C
1
and
h
∈C
2
.Then,forany
t,u,s

[0
,T
],wehave
(
δg
)
ts
=
g
t

g
s
,
and(
δh
)
tus
=
h
ts

h
tu

h
us
.
(8)
Furthermore,itisreadilycheckedthatthecomplex(
C


)is
acyclic
,i.e.
ZC
k
+1
(
V
)=
BC
k
(
V
)
forany
k

1,orotherwisestated,thesequence
0

R
→C
1
(
V
)

δ
→C
2
(
V
)

δ
→C
3
(
V
)

δ
→C
4
(
V
)
→∙∙∙
(9)
isexact.Inparticular,thefollowingbasicproperty,whichwelabelforfurtheruse,holdstrue:
Lemma2.1.
Let
k

1
and
h
∈ZC
k
+1
(
V
)
.Thenthereexistsa(nonunique)
f
∈C
k
(
V
)
such
that
h
=
δf
Proof.
Thiselementaryproofisincludedin[9],seealsoProp.3.1below.Letusjustmention
that
f
t
1
...t
k
=
h
t
1
...t
k
0
isapossiblechoice.
¤

Remark
2.2
.
ObservethatLemma2.1impliesthatalltheelements
h
∈C
2
(
V
)suchthat
δh
=0canbewrittenas
h
=
δf
forsome(nonunique)
f
∈C
1
(
V
).Thuswegetaheuristic
interpretationof
δ
|
C
2
(
V
)
:itmeasureshowmuchagiven1-incrementisfarfrombeingan
exact
incrementofafunction(i.e.afinitedifference).

6

MASSIMILIANOGUBINELLIANDSAMYTINDEL

Noticethatourfuturediscussionswillmainlyrelyon
k
-incrementswith
k

2,forwhich
wewillusesomeanalyticalassumptions.Namely,wemeasurethesizeoftheseincrementsby
Ho¨ldernormsdefinedinthefollowingway:for
f
∈C
2
(
V
)let
|
f
ts
|
µ
k
f
k
µ

sup
µ
,
and
C
1
(
V
)=
{
f
∈C
2
(
V
);
k
f
k
µ
<
∞}
.
s,t

[0
,T
]
|
t

s
|
Inthesameway,for
h
∈C
3
(
V
),set
|h|k
h
k
γ,ρ
=sup
tus
(10)
s,u,t

[0
,T
]
|
u

s
|
γ
|
t

u
|
ρ
)(XXk
h
k
µ

inf
k
h
i
k
ρ
i


ρ
i
;
h
=
h
i
,
0

i
<µ,
iiPwherethelastinfimumistakenoverallsequences
{
h
i
∈C
3
(
V
)
}
suchthat
h
=
i
h
i
andfor
allchoicesofthenumbers
ρ
i

(0
,z
).Then
k∙k
µ
iseasilyseentobeanormon
C
3
(
V
),andwe
tesµC
3
(
V
):=
{
h
∈C
3
(
V
);
k
h
k
µ
<
∞}
.
µ+1Eventually,let
C
3
(
V
)=

µ>
1
C
3
(
V
),andremarkthatthesamekindofnormscanbeconsid-
µeredonthespaces
ZC
3
(
V
),leadingtothedefinitionofsomespaces
ZC
3
(
V
)and
ZC
31+
(
V
).
Withthesenotationsinmind,thefollowingpropositionisabasicresultwhichisatthecore
ofourapproachtopath-wiseintegration:
Proposition2.3
(ThesewingmapΛ)
.
Thereexistsauniquelinearmap
Λ:
ZC
31+
(
V
)

C
21+
(
V
)
(the
sewing
map)suchthat
δ
Λ=
Id
ZC
3
(
V
)
.
µµFurthermore,forany
µ>
1
,thismapiscontinuousfrom
ZC
3
(
V
)
to
C
2
(
V
)
andwehave
1+1k
Λ
h
k
µ

µ
k
h
k
µ
,h
∈ZC
3
(
V
)
.
(11)
2−2Proof.
Forsakeofcompleteness,weincludeaproofofthisresulthere,whichismoreelementary
thantheoneprovidedin[9],andwhichwillbegeneralizedatTheorem3.5.Fornotational
sake,wewillomitthedependencein
V
inourfunctionalspaces,andwriteforinstance
C
3
insteadof
C
3
(
V
).Letthen
h
beanelementof
ZC
3
µ
⊂ZC
31+
forsome
µ>
1.
µStep1:
Letusfirstprovetheuniquenessofthe1-increment
M
∈C
2
suchthat
δM
=
h
.Indeed,
let
M,M
ˆbetwoelementsof
C
2
µ
satisfying
δM
=
δM
ˆ=
H
andset
Q
=
M

M
ˆ.Then
δQ
=0
and
Q
∈C
2
µ
.InvokingLemma2.1,thereexistsanelement
q
∈C
1
suchthat
Q
=
δq
,butsince
µ>
1,
q
isafunctionon[0
,T
]withzeroderivative,i.e.aconstantandthen
Q
=0.
µStep2:
Letusconstructnowaprocess
M
∈C
2
,with
µ>
1,satisfying
δM
=
h
.Since
δh
=0,
invokingagainLemma2.1,weknowthatthereexistsa
B
∈C
2
suchthat
δB
=
h
.Pick
s,t

[0
,T
],suchthat
s<t
inordertofixideas,andfor
n

0,considerthedyadicpartition
{
r
in
;
i

2
n
}
oftheinterval[
s,t
],where
(
t

s
)
i
r
in
=
s
+
n
,
for
i

2
n
.
(12)
2Then,for
n

0set
n1−2XnM
ts
=
B
ts

B
r
in
+1
,r
in
.
(13)
0=i

ROUGHEVOLUTIONEQUATIONS

7

Thenitisreadilycheckedthat
M
t
0
s
=0.Furthermorewehave
2
n
X

1
³´
1+nnM
ts

M
ts
=
B
r
2
ni
++12
,r
2
ni
+1

B
r
2
ni
++11
,r
2
ni
+1

B
r
2
ni
++12
,r
2
ni
++11
0=i2
n
X

12
n
X

1
=(
δB
)
r
2
ni
++12
,r
2
ni
++11
,r
2
ni
+1
=
h
r
2
ni
++12
,r
2
ni
++11
,r
2
ni
+1
,
i
=0
i
=0
andsince
h
∈C
3
µ
with
µ>
1,weobtain
¯¯
nn
+1
¯¯
k
h
k
µ
(
t

s
)
µ
M
ts

M
ts

2
n
(
µ

1)
,
whichyieldsthat
M
ts

lim
n
→∞
M
tns
exists,andsatisfiesinequality(11).
nnnStep3:
Letusconsidernowageneralsequence
{
π
n
;
n

1
}
ofpartitions
{
r
0
,r
1
,...,r
k
n
,
r
kn
n
+1
}
of[
s,t
],with
s
=
r
0
n
<r
1
n
<...<r
kn
n
<r
kn
n
+1
=
t
.Weassumethat
π
n

π
n
+1
,and
lim
n
→∞
k
n
=

.Set
knXπM
ts
n
=
B
ts

B
r
ln
+1
,r
ln
.
(14)
0=lItiseasilyseenthatthereexists1

l

k
n
suchthat
2
|
t

s
|
|
r
ln
+1

r
ln

1
|≤
(15)
knPicknowsuchanindex
l
,andletustransform
π
n
into
π
ˆ,where
©
nnnnnn
ª
π
ˆ=
r
0
,r
1
,...,r
l

1
,r
l
+1
,...,r
k
n
,r
k
n
+1
.
Then,asinthepreviousstep,
π
ˆ
π
n
π
n
M
ts
=
M
ts

(
δB
)
r
ln
+1
,r
ln
,r
ln

1
=
M
st

h
r
ln
+1
,r
ln
,r
ln

1
,
usingthedefinitionofthespace
C
3
µ
andthebound(15)wehave
¶µ¯¯¯¯
t

s
µ
¯
M

ˆ
s

M
tπs
n
¯

2
µ
k
h
k
µ
.
knRepeatingnowthisoperationuntilweendupwiththetrivialpartition
π
ˆ
0
≡{
s,t
}
,forwhich
M

ˆ
t
0
=0,weobtain:
∞knXX|
M
tπs
n
|≤
2
µ
k
h
k
µ
|
t

s
|
µ
j

µ

2
µ
k
h
k
µ
|
t

s
|
µ
j

µ

c
µ,h
|
t

s
|
µ
.
j
=1
j
=1
πHence,thereexistsasubsequence
{
π
m
;
m

1
}
of
{
π
n
;
n

1
}
suchthat
M
ts
m
convergestoan
element
M
ts
,satisfying
M
ts

c
µ,h
|
t

s
|
µ
.Withthesameconsiderationsasin[13],itcanalso
becheckedthatthatthelimit
M
doesnotdependontheparticularsequenceofpartitionswe
havechosen,andthuscoincideswiththeoneconstructedatStep2.
Step4:
Itremainstoshowthat
δM
=
h
.Considerthen0

s<u<t

T
,andtwosequences
ofpartitions
π
uns
and
π
tnu
of[
s,u
]and[
u,t
]respectively,whosemeshestendto0as
n
→∞
.
nnnSetalso
π
ts
=
π
tu

π
us
.Fromthepreviousstep,onecanconstructeasilysomesubsequences
mmmmmm
π
tu

us

ts
,with
π
ts
=
π
tu

π
us
,suchthat
mmmππlim
M
tu
tu
=
M
tu
,
lim
M
uπs
us
=
M
us
,
lim
M
ts
ts
=
M
ts
.
m
→∞
m
→∞
m
→∞

8

MASSIMILIANOGUBINELLIANDSAMYTINDEL

Callnow
k
tms
(resp.
k
tmu
,k
ums
)thenumberofpointsofthepartition
π
tms
(resp.
π
tmu

ums
).Then,
adirectcomputation,usingdefinition(14),showsthatforany0

i

2
n
wehave:
π
m
π
m
π
m
M
ts
ts

M
tu
su

M
us
ut
k
tms
+
X
k
ums
+1
k
X
tmu
k
tmu
+
X
k
ums
+1
=(
δB
)
tus


B
r
lm
+1
r
lm

B
r
lm
+1
r
lm

B
r
lm
r
lm
+1

=(
δB
)
tus
=
h
tus
.
l
=0
l
=0
l
=
k
tmu
+1
Takingthelimit
m
→∞
inthelatterrelation,weget(
δM
)
tus
=
h
tus
,whichendstheproof.
¤Wecannowgiveanalgorithmforacanonicaldecompositionofthepreimageofthespace
ZC
31+
(
V
),orinotherwords,ofafunction
g
∈C
2
(
V
)whoseincrement
δg
issmoothenough:
µCorollary2.4.
Takeanelement
g
∈C
2
(
V
)
,suchthat
δg
∈C
3
(
V
)
for
µ>
1
.Then
g
canbe
decomposedinauniquewayas
g
=
δf

δg,
where
f
∈C
1
(
V
)
.
Proof.
Elementary,see[9].
¤
Atthispointtheconnectionofthestructureweintroducedwiththeproblemofintegration
ofirregularfunctionscanbestillquiteobscuretothenon-initiatedreader.Howeversomething
interestingisalreadygoingonandthepreviouscorollaryhasaveryniceconsequencewhich
isthesubjectofthefollowingproperty.
Corollary2.5
(Integrationofsmallincrements)
.
Forany1-increment
g
∈C
2
(
V
)
,suchthat
δg
∈C
31+
,set
δf
=(
Id

Λ
δ
)
g
.Then
nX(
δf
)
ts
=lim
g
t
i
+1
t
i
,
|
Π
ts
|→
0
i
=0
wherethelimitisoveranypartition
Π
ts
=
{
t
0
=
t,...,t
n
=
s
}
of
[
t,s
]
whosemeshtendsto
zero.The1-increment
δf
istheindefiniteintegralofthe1-increment
g
.
Proof.
Justconsidertheequation
g
=
δf

δg
andwrite
X
n
X
n
X
n
S
Π
=
g
t
i
+1
t
i
=(
δf
)
t
i
+1
t
i
+(Λ
δg
)
t
i
+1
t
i
i
=0
i
=0
i
=0
nX=(
δf
)
ts
+(Λ
δg
)
t
i
+1
t
i
.
0=iThenobservethat,duetothefactthatΛ
δg
∈C
31+
(
V
),thelastsumconvergestozero.
¤
2.2.
Computationsin
C

.
Forsakeofsimplicity,letusassume,untilSection3,that
V
=
R
,
andset
C
k
(
R
)=
C
k
.Thenthecomplex(
C


)isan(associative,non-commutative)graded
algebraonceendowedwiththefollowingproduct:for
g
∈C
n
and
h
∈C
m
let
gh
∈C
n
+
m
the
elementdefinedby
(
gh
)
t
1
,...,t
m
+
n

1
=
g
t
1
,...,t
n
h
t
n
,...,t
m
+
n

1
,t
1
,...,t
m
+
n
+1

[0
,T
]
.
(16)
Inthiscontext,thecoboundary
δ
actasagradedderivationwithrespecttothealgebra
structure.Inparticularwehavethefollowingusefulproperties.
Proposition2.6.
Thefollowingdifferentiationrulesholdtrue:

ROUGHEVOLUTIONEQUATIONS

9

(1)
Let
g,h
betwoelementsof
C
1
.Then
δ
(
gh
)=
δgh
+
gδh.
(17)
(2)
Let
g
∈C
1
and
h
∈C
2
.Then
δ
(
gh
)=
δgh
+
gδh,δ
(
hg
)=
δhg

hδg.
Proof.
Wewilljustprove(17),theotherrelationsbeingequallytrivial:if
g,h
∈C
1
,then
[
δ
(
gh
)]
ts
=
g
t
h
t

g
s
h
s
=
g
t
(
h
t

h
s
)+(
g
t

g
s
)
h
s
=
g
t
(
δh
)
ts
+(
δg
)
ts
h
s
,
whichprovesourclaim.
¤Theiteratedintegralsofsmoothfunctionson[0
,T
]areobviouslyparticularcasesofelements
of
C
whichwillbeofinterestforus,andletusrecallsomebasicrulesfortheseobjects:co
R
nsider
f,g
∈C
1

,where
C
1

isthesetofsmoothfunctionsfrom[0
,T
]to
R
.Thentheintegral
dgf
,
whichwillbedenotedby
J
(
dgf
),canbeconsideredasanelementof
C
2

.Thatis,for
s,t

[0
,T
],weset
µZ¶Z
tJ
ts
(
dgf
)=
dgf
=
dg
u
f
u
.
sstThemultipleintegralscanalsobedefinedinthefollowingway:givenasmoothelement
h
∈C
2

and
s,t

[0
,T
],weset
µZ¶Z
tJ
ts
(
dgh
)

dgh
=
dg
u
h
us
.
sstInparticular,thedoubleintegral
J
ts
(
df
3
df
2
f
1
)isdefined,for
f
1
,f
2
,f
3
∈C
1

,as
µZ¶Z
t
¡¢
J
ts
(
df
3
df
2
f
1
)=
df
3
df
2
f
1
=
df
u
3
J
us
df
2
f
1
.
sstandif
f
1
,...,f
n
+1
∈C
0

,weset
Zt¢¡J
ts
(
df
n
+1
df
n
∙∙∙
df
2
f
1
)=
df
un
+1
J
us
df
n
∙∙∙
df
2
f
1
,
(18)
swhichdefinestheiteratedintegralsofsmoothfunctionsrecursively.
Thefollowingrelationsbetweenmultipleintegralsandtheoperator
δ
willalsobeusefulin
theremainderofthepaper:
Proposition2.7.
Let
f,g
betwoelementsof
C
1

.Then,recallingtheconvention(16),itholds
tahtδf
=
J
(
df
)

(
J
(
fdg
))=0

(
J
(
dgdf
))=(
δg
)(
δf
)=
J
(
dg
)
J
(
df
)
,
and,ingeneral,
¡¢
n
X

1
¡¢¡¢
δ
J
(
df
n
∙∙∙
df
1
)=
J
df
n
∙∙∙
df
i
+1
J
df
i
∙∙∙
df
1
.
1=iProof.
Hereagain,theproofiselementary,andwewilljustshowthethirdoftheserelations:
wehave,for
s,t

[0
,T
]
,
ZZttJ
ts
(
dgdf
)=
dg
u
(
f
u

f
s
)=
dg
u
f
u

K
ts
,
sswith
K
ts
=(
g
t

g
s
)
f
s
.Thefirsttermoftherighthandsideiseasilyseentobein
ZC
2
.Thus
δ
(
J
(
dgdf
))
tus
=

(
δK
)
tus
=[
g
t

g
u
][
f
u

f
s
]
,
whichgivestheannouncedresult.

01

MASSIMILIANOGUBINELLIANDSAMYTINDEL

¤2.3.
Dissectionofanintegral.
Thepurposeofthissectionisnottoprovideanaccountonall
thecomputationscontainedin[9].However,wewillgointosomesemi-heuristicconsiderations
that,hopefully,willshedsomelightonthewaywewillsolveroughPDEslateron:withthe
notation
R
sofSection2.2inmind,wewilltrytogive,intuitivelyspeaking,ameaningtothe
integral
ϕ
(
x
)
dx
=
J
(
ϕ
(
x
)
dx
)foranon-smoothfunction
x
∈C
1
.Noticethat,inthesequel,
x
shouldbeconsideredasavectorvaluedfunction,sincethewholetheorycanbehandledvia
theDoss-Soussmanmethodologyintherealcase.However,wewillpresentthemainideas
ofthealgorithmbelowasif
x
wererealvalued,thegeneralizationfrom
R
to
R
n
beingjusta
matterof(cumbersome)notations.
2.3.1.
TheYoungcase.
Thefirstideaonecanhaveinmindinordertodefine
J
(
dxϕ
(
x
))isto
performanexpansionaroundtheincrement
dx
:indeed,inthesmoothcase,wehave
J
(
dxϕ
(
x
))=
δxϕ
(
x
)+
J
(
dxdϕ
(
x
))
.
(19)
Ifwewishtoextendtherighthandsideof(19)toanon-smoothcase,weseethatthefirst
termisharmless,sinceitisdefinedindependentlyoftheregularityof
x
,by
[
δxϕ
(
x
)]
ts
=[
x
t

x
s
]
ϕ
(
x
s
)=[
δxϕ
(
x
)]
ts
,
for
s,t

[0
,T
]
.
Thelasttermof(19)ismoreproblematicandweproceedtoits
dissection
bytheapplication
of
δ
:invokingProposition2.7,weget,inthesmoothcase,that
δ
(
J
(
dxdϕ
(
x
)))=
δxδ
(
ϕ
(
x
))
,
i.e.[
δ
(
J
(
dxdϕ
(
x
)))]
tus
=[
δx
]
tu
[
δ
(
ϕ
(
x
))]
us
.
(20)
Nowther.h.s.of(20)iswelldefinedindependentlyoftheregularityof
x
.Thus,if
δxδ
(
ϕ
(
x
))

C
31+
,whichhappenswhen
x
∈C
1
α
with
α>
21
and
ϕ

C
1
(
R
),thenProposition2.3canbe
applied,andΛ[
δxδ
(
ϕ
(
x
))]isdefinedunambiguously.Hence,owingto(20),weset
J
(
dxdϕ
(
x
))=Λ(
δxδ
(
ϕ
(
x
)))
dnaJ
(
dxϕ
(
x
))=
δxϕ
(
x
)+Λ(
δxδ
(
ϕ
(
x
)))=(Id

Λ
δ
)[
δxϕ
(
x
)]
,
(21)
wherethelastequalityisduetoProposition2.6andtothefactthat
δδx
=0.Noticeonceagain
thatthisconstructionisvalidwhenever
x
∈C
1
α
with
α>
21
and
ϕ

C
1
(
R
),anditiseasily
shown,alongthesamelinesasintheproofofProposition2.3thattheintegral
J
(
dxϕ
(
x
))
definedby(21)correspondstotheusualYoungintegral.
2.3.2.
Caseofa
α
-Ho¨lderpathwith
31
<α<
21
.
Theconstruction(21)doesn’tworkif
x
6∈
+2/1C
1
.However,if
x
∈C
1
α
with
α>
31
,wecanproceedfurtherintheexpansionofequation
(19)byobservingthat,stillinthesmoothcase,wehave,for
s,t

[0
,T
],
Z
t
Z
t
Z
t
Z
u
[

(
x
)]
u
=
dx
u
ϕ
0
(
x
u
)=[
x
t

x
s
]
ϕ
0
(
x
s
)+
dx
u
dx
v
ϕ
00
(
x
v
)
,
ssssoraccordingtothenotationsofSection2.2,
¢¡¢¡δϕ
(
x
)=
J
(

(
x
))=
J
dxϕ
0
(
x
)=
δxϕ
0
(
x
)+
J
dxdϕ
0
(
x
)
.
(22)
Injectingthisequalityinequation(19),thanksto(18),weobtain
¢¡J
(
dxϕ
(
x
))=
δxϕ
(
x
)+
J
(
dxdx
)
ϕ
0
(
x
)+
J
dxdxdϕ
0
(
x
)
.
(23)
Letusassumenowthatwearegivenaprocess
J
(
dxdx
)
∈C
2
,usually(andsomewhatimprop-
erly)calledtheLevyareaof
x
,suchthat
δ
(
J
(
dxdx
))=
δxδx
and
J
(
dxdx
)
∈C
22
α
.
(24)