Non‐geometric rough paths on manifolds 2022 John Armstrong.pdf

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Received:28May2021Revised:24November2021Accepted:29November2021
DOI:
JournaloftheLondon
RESEARCHARTICLEMathematicalSociety
Non-geometricroughpathsonmanifolds
JohnArmstrong1DamianoBrigo2ThomasCass2
EmilioRossiFerrucci2
1DepartmentofMathematics,King’s
CollegeLondon,London,UKAbstract
2DepartmentofMathematics,ImperialWeprovideatheoryofmanifold-valuedroughpaths
CollegeLondon,London,UKofbounded3>-variation,whichwedonotassume
,
EmilioRossiFerrucci,Departmentofrelyingonthevectorspace-valuedtheoryofFrizand
Mathematics,ImperialCollegeLondon,Hairer(Acourseonroughpaths,2014),andcoordinate-
180Queen’sGate,SouthKensington,
LondonSW72AZ,(butconnection-dependent)definitionsofthe
Email:roughintegralofcotangentbundle-valuedcontrolled
-******@,andofroughdifferentialequationsdrivenby
Fundinginformationaroughpathvaluedinanothermanifold,aregiven.
EPSRC,Grant/AwardNumber:Whenthepathistherealisationofsemimartingale,
EP/S026347/1;CentreforDoctoral
TraininginFinancialComputing&werecoverthetheoryofItôintegrationandstochastic
Analytics,Grant/AwardNumber:differentialequationsonmanifolds(Émery,Stochastic
EP/L015129/1calculusinmanifolds,1989).Weproceedtopresentthe
extrinsiccounterpartstoourlocalformulae,andshow
howtheseextendtheworkinCassetal.(.
.(3)111(2015)1471–1518)tothesettingof
non-geometricroughpathsandcontrolledintegrands
moregeneralthan1-,we
turntoparalleltransportandCartandevelopment:the
lackofgeometricityleadsustomakethechoiceofa
connectiononthetangentbundleofthemanifold,
whichfiguresinanItôcorrectiontermintheparal-
lelismroughdifferentialequation;suchconnection,
whichisnotneededinthegeometric/Stratonovich
setting,isrequiredtosatisfypropertieswhichguarantee
©©
underthetermsoftheCreativeCommonsAttributionLicense,whichpermitsuse,distributionandreproductioninanymedium,provided
theoriginalworkisproperlycited.
.Soc.(2)2022;106:756–:.
NON-GEOMETRICROUGHPATHSONMANIFOLDS757
well-definedness,linearity,andoptionallyisometric-

fewexamplesthatexploretheadditionalsubtleties
introducedbyourchangeinperspective.
MSC2020
53C05,60L20(primary)
Contents
INTRODUCTION......................................757
..........................759
-valuedroughpaths...............................759
......................762
............................769
...............................771
,ROUGHINTEGRATION,ANDRDEsONMANIFOLDS.......772
.............................781
..............792
ACKNOWLEDGEMENTS..................................816
REFERENCES........................................816
INTRODUCTION
Thetheoryofroughpaths,firstintroducedin[25],hasasitsprimarygoalthatofprovidinga
rigorousmathematicalframeworkforthestudyofdifferentialequationsdrivenbyhighlyirreg-

andintegrationinapplicable,andmotivatesthedefinitionofroughpath,apathaccompanied
byfunctions,satisfyingcertainalgebraicandanalyticconstraints,whichpostulatethevaluesof
its(otherwiseundefined)-
tionagainsttheroughpathandofroughdifferentialequation(RDE)drivenby,whichbear
theimportantfeatureofbeingcontinuousinthesignal,accordingtoappropriatelydefined-
,includingtothecase
inwhichisgivenbytherealisationofastochasticprocess,forwhichitconstitutesapathwise
approachtostochasticintegration,extendingtheclassicalstochasticanalysisofsemimartingales.
Animportantfeaturethataroughpathcansatisfyisthatofbeinggeometric:thiscanbeinter-
pretedasthestatementthatitobeystheintegrationbypartsandchangeofvariablelawsoffirst-
ordercalculus,
moststudied[20],andappliestosemimartingalesthroughtheuseoftheStratonovichintegral.
Othernotionsofstochasticintegration,however,cannotbemodelledbygeometricroughpaths,
theItôintegralbeingtheprime(butnottheonly[18])example.
Sincesmoothmanifoldsaremeanttoprovideageneralsettingforordinarydifferentialcalculus
tobecarriedout,itisnaturaltoaskhow‘rougher’:.
758ARMSTRONGetal.
thecontextofstochasticcalculus,thisquestionhasledtoarichliteratureonBrownianmotion
,ithasbeenraisedanumberoftimeswithregardtoroughpaths[5,8,
11,15]and[4]inthesettingofBanachmanifoldsandarbitrary-,
however,onlythecaseofgeometricroughpathshasbeendiscussed.
Themaingoalofthispaperistoconstructatheoryofmanifold-valuedroughpathsofbounded
-variation,with<3,
ensuresthatwemaydrawonthefamiliarsettingof[19]forvectorspace-valuedroughpaths;drop-
pingthisrequirementwouldrequirethemorecomplexalgebraictoolsof[22].Ourtheoryincludes
definingroughintegrationanddifferentialequations,bothfromtheintrinsicandextrinsicpoints
ofview,andshowinghowtheclassicalnotionsofparalleltransportandCartandevelopmentcan
beextendedtothecaseofnon-geometricroughpaths.
AlthoughthedefinitionoftheItôintegralonmanifoldshasbeenknownfordecades,
Stratonovichcalculushasbeenpreferredinthevastmajorityoftheliteratureonstochasticdif-
,therearephenomenathatarebestcapturedbyItôcalculus,
,threeoftheauthors
recentlyshowedhowaconcreteprobleminvolvingtheapproximationofstochasticdifferential
equations(SDEs)withonesdefinedonsubmanifoldsnecessitatestheuseofItônotation,and
thattheresultnaturallyprovidedbyprojectingtheStratonovichcoefficientsissuboptimalingen-
eral[1,2].ThereasonthatStratonovichintegrationandgeometricroughpathsarepreferredin
differentialgeometryisthattheyadmitasimplecoordinate-freedescription,asisalsoremarked
on[25,].Animportantpoint,however,thatwewishtomakeinthispaperisthefollowing:
aninvarianttheoryofintegrationagainstnon-geometricroughpathsmayalsobegiven,albeit
onethatdependsonthechoiceofalinearconnectiononthetangentbundleofthemanifold.
Althoughgeometricroughpaththeorystillretainstheimportantpropertyofbeingconnection-
invariant,allroughpathsmaybetreatedinacoordinate-freemanner,since,whilemanifoldsmay
notadmitglobalcoordinatesystems,
principleleadstothecommonmisconceptionthatItôcalculus/non-geometricroughintegration
cannotbecarriedoutonmanifolds,evenincaseswhereaconnectionisalreadyindependently
andcanonicallyspecified,forexample,
differentialgeometrythefocusisnotonthestochasticintegralperse,whichisviewedasatoolto
investigatelawsofprocessesdefinedonRiemannianWienerspace:inthiscontextitiscertainly
,however,isonpath-
wiseintegrationitself:forthisreason,webelieveittobeofvaluetobuildupthetheoryinaway
thatisfaithfultothechoiceofthecalculus,asspecifiedthroughtheroughpath.
Thispaperisorganisedasfollows:inSection1wereviewthetheoryofvectorspace-valued
roughpathsofbounded3>-variation,controlledroughintegrationsandRDEs,relying(witha
fewmodificationsandadditions)on[19].
InSection2,wedevelopthetheoryattheheartofthepaper:thisentailsdefiningroughpathson
manifoldsandtheircontrolledintegrandsinacoordinate-freemannerbyusingpushforwardsand
pullbacksthroughcharts,showinghowthechoiceofalinearconnectiongivesrisetoadefinition
ofroughintegral,‘transferprinciple’philoso-
phy[17]ofreplacingallinstancesofEuclideanspaceswithsmoothmanifolds,whichmeansthat
boththedrivingroughpathandthesolutionarevaluedin(possiblydifferent)
werestrictourtheorytosemimartingaleswerecovertheknownframeworkforItôintegration
andSDEsonmanifolds[16].
InSection3,weswitchfromthelocaltotheextrinsicframework,andshowhowourtheory
extendsthatof[8]tonon-geometricintegratorsandcontrolledintegrandsmoregeneralthan:.
NON-GEOMETRICROUGHPATHSONMANIFOLDS759
1-
onthepath,
areconfiningourselvestotheRiemanniancase(withthemetricbeinginducedbyanembedding),
whileintherestofthepaperweallowforgeneralconnections.
Finally,inSection4wereturntoourlocalcoordinateframeworktocarryouttheconstructions
ofparalleltransportalongroughpathsandtheresultingnotionofCartandevelopment,or‘rolling
withoutslipping’,acornerstoneofstochasticdifferentialgeometrywhichyieldsaconvenientway

transportasa-valuedRDEdrivenbyan-valuedroughpath,thelackofgeometricityleads
ustorequirethechoiceofaconnectionnotjustonthetangentbundleofbutalsoofoneonthe
,andwe
identifycriteria(formulatedintermsoftheformerconnection)thatguaranteewell-definedness,
linearity,and,ifisRiemannian,
connectiongiverisetodifferentdefinitionsofparalleltransportandCartandevelopment,which
areonlydetectableatasecond-orderlevel,andallcollapsetothesameRDEwhentheroughpath
,threeexamplesfor
howaconnectiononmaybeliftedtooneonaredrawnfromtheliterature;acasenot
analyseduntilnowconcernstheLevi-CivitaconnectionoftheSasakimetric,whichresultsinpar-

toexploreafewadditionaltopicsinstochasticanalysisonmanifolds,suchasCartandevelopment
inthepresenceoftorsion,withapathwiseemphasis.
WewouldliketothankMartinHairerandZhongminQianfortheirhelpfulcomments,which
resultedinanimprovementofoneoftheresultsinSection3.
1BACKGROUNDONROUGHPATHS
Inthissection,wereviewthecoretheoryoffinite-dimensionalvectorspace-valued(controlled)
roughpaths,[19],
withthecaveatthatweareworkinginthesettingofarbitrarycontrolfunctions,asopposedto
Hö-invariantframework,
andofallowingustoconsideralargerclassofpaths(forexample,allsemimartingales,andnotjust
Brownianmotion).Otherauthorshavealreadybeentreatingcontrolledroughpathsinthesetting
ofbounded-variation[9,].Whenaresultinthisfirstsectionisstatedwithout
proof,itisunderstoodthattheproofcanbefoundin[19,Chapters1–10],possiblywithtrivial

quantitativeaspectsofroughpathsareleftout,astheywillnotberelevantforthetranspositionof
-dimensional,andsincewewillrelyon
arbitrarychartstomakethemanifold-valuedtheorycoordinate-free,wewillusefixedcoordinates
toexpressallofourformulae.
-valuedroughpaths
Throughoutthisdocument,willbearealnumber∈[1,3);wewillnotexcludethecaseof∈
[1,2)inwhichthetheoryreducestoYoungintegration,andremainsvalidwithtrivialadjustments.
Acontrolon[0,]isacontinuousfunctiondefinedonthesubdiagonalΔ∶={(,)∈[0,]2∣
:.
760ARMSTRONGetal.
⩽},suchthat(,)=0for0⩽⩽and(,)+(,)⩽(,)for0⩽⩽⩽⩽.The
functi