MASSIMILIANO GUBINELLI AND SAMY TINDEL

Description

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

Sujets

• Setting
• Équation
• multiplication

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English

Extrait

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