MASSIMILIANO GUBINELLI AND SAMY TINDEL

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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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
)
dη
+
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
)
dη
+
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
-incrementstheoryprovidesatoolallowingsomenaturalcompu