Bismut's gradient formula for vector bundles

0.1 MB·

by  Elton P. Hsu

BISMUT’S GRADIENT FORMULA FOR VECTOR BUNDLES
6 ELTONP.HSUANDZHENANWANG
1
0
2 ABSTRACT. We prove a general Bismut’s formulafor the gradientof a
classofsmoothWienerfunctionalsovervectorbundlesofacompactRie-
n
mannianmanifold. Thisgeneralformulacanbeusedrepeatedlyforob-
a
tainingprobabilisticrepresentationofhigherordercovariantderivatives
J
of solutionsofthe heat equationsimilarto the classicalBismut’srepre-
1
sentationforthecovariantgradientoftheheatkernel.
1

]
R
(preliminaryversion)
P
.
h
CONTENTS
t
a
m 1. Introduction 1
2. Notationsandthestatementoftheproblem 3
[
3. Commutationofthegradientandtheheatsemigroup 5
1
4. Integrationbyparts 7
v
5 5. Secondcommutationandde´nouement 10
8 References 11
2
2
0
.
1
1. INTRODUCTION
0
6 Let M be a compact Riemannian manifold of dimension n and P the
x
1
law ofBrownianmotionon M startingfrom x. Thenowclassical Bismut’s
:
v
formula ([1]) is a probabilistic representation of the gradient of the heat
i
X semigroup
ar Ptf(x) = Exf(Xt) = p(t,x,y)f(y)dy,
ZM
where p(t,x,y) is the transition density function of the Brownian X on M
(the heat kernel). Several interesting objects are involved in this formula.
On the orthonormal frame bundle O(M) of M the scalarized Ricci cur-
vature tensor is realized as an O(n)-invariant, End(Rn)-valued function
Ric : u 7→ Ric ∈ End(Rn). Let {U } be a horizontal lift of the Brownian
u t
1991MathematicsSubjectClassification. Primary-60D58;secondary-28D05.
Keywordsandphrases. Bismut’sformula,Brownianmotion,Riemannianmanifold,vec-
torbundle.
TheresearchofthefirstauthorwassupportedinpartbyaSimonsFoundationCollabo-
rationGrantforMathematiciansandaresearchgrantadministeredbytheChineseUniver-
sityofScienceandTechnology.
1
2 ELTONP.HSUANDZHENANWANG
motion X = {X } to theframe bundle O(M). The firstobject ofinterestis
t
themultiplicative Feynman-Kacfunctional M = {M }definedby
t
1
dM + Ric M = 0, M = I.
t 2 Ut t 0
The second object of interest is the so-called stochastic anti-development
W = {W }oftheBrownianmotion M. ItistheeuclideanBrownianmotion
t
which drivesthestochasticdifferentialequationfor thehorizontalBrown-
ianmotionU,namely,
n
dU = ∑ H(U )◦dWi,
t i t t
i=1
where{H }arethecanonicalhorizontalvectorfieldsonO(M). Theclassi-
i
calBismut’sformulaisgivenby
T
T∇E f(X ) = E f(X ) M dW .
x T x T s s
(cid:20) Z0 (cid:21)
Since u(t,x) = E f(X ) is the solution of the heat equation on M with
x t
the initial function u(0,x) = f(x), the above identity gives a probabilistic
representationofthecovariantgradientofthesolution. Thisrepresentation
isequivalenttothefollowingprobabilisticrepresentationofthegradientof
theheatkernel:
t
T∇ logp(t,x,y) = E M dW ,
x x,y;T s s
(cid:20)Z0 (cid:21)
E P
where istheexpectationwithrespecttothelaw ofaBrownian
x,y;T x,y;T
bridgefrom x toyintime T. Itcanberegardedasaformofintegrationby
parts formula because ofthe absence of the gradientoperator on the right
side;seeHsu[2]foradetailedexpositionforthisformula.
Ever since its appearance 30 years ago, this explicit representation has
found applications in many areas. Besides its obvious significance in sto-
chastic analysis on manifolds, it has also played important roles in cer-
tain problems in financial mathematics involving computations of option
greeks. One naturally considers possible generalizations of this beautiful
formulainseveraldirections. Twopossibledirectionscometomindimme-
diately: higher order derivatives and functions in more general spaces. It
is an important observation due to Norris [5] that these two possible gen-
eralizations can be combinedinto oneframework,in which thefunction f
isreplacedbyasmoothmaponavectorbundleandtheprocessX islifted
to the vector bundle by introducing, in addition to a compatible connec-
tion onthe vectorbundle, also a vertical motion driven by X itself. In this
way,theformulaforhigherderivativecan simplybeobtainedbyfromthe
general formula by a judicious choice of the bundle map and the vertical
motion.
Bismut[1]derivedhis formulaby aperturbationmethodin pathspace,
whichwasverymuchinlinewiththetechniquesavailableatthetimewhen
BISMUT’SFORMULA 3
Malliavin calculus was studied intensively. This method was followed by
andlargebyNorris[5]inhisgeneralizationofBismut’sformulaintheset-
ting we have just mentioned. The technical difficulties involved in this
methodistremendous. Infact, uptonowNorris’paperhasnotbeenfully
digestedbytheprobabilisticcommunity. Inviewofthenewideasandtech-
nical tools introduced during the intervening decades (mainly gradient-
heat semigroup commutation relations and Itoˆ calculus for diffusion pro-
cessesonmanifolds)andimportanceoftheseresults,itishighlyappropri-
atethatwerevisittheseresultswiththegoaloffindingaproofmoreinline
with the current state of art of stochastic analysis on manifolds. In addi-
tion,in ourrecentinvestigationoffunctionalinequalities in pathandloop
spacesofaRiemannianmanifold,wefeeltheneedtoextendtheframework
adopted in Norris [5] by considering maps between two different vector
bundles instead a self-map on a single vector bundle. Thus the purpose
ofthiscurrentworkistouseItoˆ calculus toproveaprobabilistic represen-
tation of the gradient of the solution of a class of heat equations between
twovectorbundles. Thebasicideaofthecurrentapproach isexplainedin
Hsu[3]forthesimplestcaseofatrivialvectorbundle.
ACKNOWLEDGMENT. Theoriginofthecurrentworkcan betracedback
totheearlycollaborationofthefirstauthorwithProfessorsMarcArnaudon
(Bordeaux),ZhongminQian(Oxford)andAntonThalmaier(Luxembourg)
forthepurposeoffindingan alternative approachtoNorris[5]. Although
this collaboration did not result in a joint publication, the discussion with
these individuals helped clarify many confusions and misunderstandings
concerning the problem. Their early contribution to the current project is
hereby gratefully acknowledged. The authors would also like to express
their gratitude to the School of Mathematics of the University of Science
andTechnologyofChina.TheSchoolprovidedfinancialsupportfortheau-
thors’ visit in the summer of 2014, during which a portion of the present
workwascarriedout.
2. NOTATIONS AND THE STATEMENT OF THE PROBLEM
WeworkonaRiemannianmanifold Mofdimensionm. Letτ : E → M
1 1
andπ : E → MbetwoRiemannianvectorbundlesover Mofdimensions
2 2
m and m , respectively. Let x = {x , t ≥ 0} be a Riemannian Brownian
1 2 t
motion on M and W = {W , t ≥ 0} its anti-development, which is a stan-
t
dardBrownianmotionin Rm adaptedtothefiltrationof x. Thehorizontal
liftofx,i.e. thehorizontalBrownianmotionU = {U , t ≥ 0},isdefinedby
t
∂U = H(U )◦∂Wi,
t i t t
where{H } are canonical horizontal vectorfieldson F(M). The(stochas-
i
tic)paralleltransportalongthepath x[0,t] isτ = U U−1.
t 0 t
4 ELTONP.HSUANDZHENANWANG
FollowingNorris[5],weintroduceadiffusionprocessonE drivenbyx
1
(orequivalently,byW)inthefollowingmanner,
∂y = v ∂t+v ◦∂Wi,
t 0 1,i t
where v ,v are vector fields on E . This process is a lift of the Brownian
0 1 1
motionon M,i.e., π y = x . Thepreciseinterpretationoftheequationfor
1 t t
yis
∂τ−1y = τ−1tv ∂t+τ−1v ◦∂W .
t t 0 t 1 t
Consider a smooth map between the two vector bundles f : E → E
1 2
whichrespectsthefibres. Thenτ−1f(x ,y )isanE -valuedrandomvari-
T T T 2,x
ableandtheexpectationE τ−1f(x ,y )isanelementinE whichvaries
x,y T T T 2,x
smoothly as a function of the initial point (x ,y ) = (x,y). The focus of
0 0
thispaperistogiveaprobabilisticrepresentationofthecovariantgradient
∇E τ−1f(x ,y )(withrespecttothexvariable).
x,y T T T
For a section of a vector bundle, it will be much more convenient for
ourexpositionifwe dealwith its scalarization, which is a euclidean space
valued function on the frame bundle of that vector bundle. We introduce
theframebundleF(M) = ∪ U ,where
x∈M x
U : Rm → T M
0 x
U = (U0,U1,U2) = U1 : Rn1 → E1,x .
U2 : Rn2 → E2,x
Each Ui is an isometry between its respective domain and target spaces.
Thescalarizationofthebundlefunction f : E → E isdefinedby
1 2
F : F(M)×Rn1 → Rn2
F(U,Y) := U−1f(x,U Y).
2 1
Let Y = {Y, t ≥ 0} be the scalarization of the process y. Then it is the
t
solutionofthestochasticdifferentialequation
∂Y = V (U ,Y)∂t+∑V (U ,Y)◦∂Wi,
t 0 t t 1,i t t t
i
whereV (U ,Y) := U−1v (x ,U Y) andV (U ,Y) := U−1v (x ,U Y)
0 t t 1,t 0 t 1,t t 1,i t t 1,t 1,i t 1,t t
arethescalarization ofv (x ,y )andv (x ,y ),respectively.
0 t t 1,i t t
Foreaseofnotation,wecombinethetwoprocessesUandYintoasingle
processZ = (U ,Y). Itisclearthat
t t t
∂Z = (0,V )∂t+∑(H,V )◦∂Wi.
t 0 i 1,i t
i
Tofurthersimplifythenotation,wewillwrite(H,0),(0,V ),and(0,V )
i 0 1,i
simly as H, V , and V , respectively. whenever there is not possibility of
i 0 1,i
confusion. ThenthegeneratorofthediffusionprocessZ is
t
1
L = ∑(H +V )2+V .
i 1,i 0
2
i
BISMUT’SFORMULA 5
And F(z,t) = E {F(Z )}isthesolutiontoequation
z t
∂F
= LF.
∂t
Thegoalofthispaperistofindaprobabilisticrepresentationofthegradient
of the solution to this heat equation over the vector bundles. Our main
object of interest ∇E τ−1f(x ,y ), becomes ∇HF(z,T). Here ∇H is the
x,y T T T
horizontalgradientdefinedby
∇HF = ∑(H F)e ,
j j
j
where{ej}nj=11 isthecanonicalorthonormalbasisinRn1.
3. COMMUTATION OF THE GRADIENT AND THE HEAT SEMIGROUP
Thefirststeptowardsestablishingtheprobabilisticrepresentationofthe
gradient of a solution to the heat equation is to commute the gradient op-
eratorwiththeheatsemigroup. Inthiswaywepassthegradientoperator
undertheexpectationonthebundlemap. Thiswillbefollowedbyaninte-
grationbypartsargumenttoremovethegradientoperatorfromthebundle
map.
Considertheequationfor∇HF
∂∇HF
= ∇HLF
∂t
= L∇HF+[∇H,L]F
Wehave
1 1 1
L = ∑H2+ ∑V2 + ∑(HV +V H )+V .
2 i 2 1,i 2 i 1,i 1,i i 0
i i i
Byfurtherexpandthecommutatorbracket,weobtain
∂∇HF 1
= L∇HF− Ric∇HF+(1)+(2)+(3)+(4)+(5)
∂t 2
where
(I) (1) = ∑ ∑−Ω(2)(H F)+∑(DH F)(Ω(1)Y) e
ji i i ji j
!
j i i
1
+ ∑ ∑−(HΩ(2))F+∑(DF)((HΩ(1))Y) e
2 i ji i ji j
!
j i i
6 ELTONP.HSUANDZHENANWANG
(II) (2) = ∑ ∑(D2F)(H V )(V ) e
j 1,i 1,i j
!
j i
1
+ ∑DF ∑(DH V )(V )+(DV )(H V ) e
j 1,i 1,i 1,i j 1,i j
2
!
j i
1
(III) (3) = ∑ ∑(DH F)(H V ) e + ∑DF ∑H H V e
i j 1,i j i j 1,i j
2
! !
j i j i
(IV) (4) = ∑ ∑(D2F)(V )(Ω(1)Y) e
1,i ji j
!
j i
+∑ ∑−Ω(2)(DF)(V ) e
ji 1,i j
!
j i
1
+ ∑DF ∑(DV )(Ω(1)Y) e
2 1,i ji j
!
j i
(V) (5) = ∑(DF)(H V )e
j 0 j
j
In the above expansion, −1Ric∇HF +(1) comes from [∇H,∑H2], (2)
2 i
comes from [∇H,∑V2], then (3)+(4) comes from [∇H,∑HV +V H ]
i i 1,i 1,i i
and (5) comes from [∇H,V ]. We also group (3)+(4) so that all terms in
0
their(correspondingcommutatorwithcurvaturegoto(4)andtheremain-
ingtermsgoto(3). Thereasonfordoingsoistoemphasizetheeffectofthe
curvatureofthebundleE and E .
1 2
Forclarification, wesayafewwordsaboutthenotationsusedherecon-
cerning the curvature tensors (see Kobayashi and Nomizu [4]). Let H be
i
thecanonicalhorizontalvectorfieldsoftheproductbundleTM⊕E ⊕E .
1 2
Then the commutator Ω = [H ,H] is a vertical vector field on the same
ij j i
productbundlewiththreecomponents,i.e.,
Ω = (Ω(0),Ω(1),Ω(2)) ∈ o(m)×o(n )×o(n ),
ij ji ji ji 1 2
whereΩ(l) arethe(scalarized)curvaturetensorsforl = 0(onthemanifold
ji
M),1(forthevectorbundleE ),and2(forthevectorbundleE ).
1 2
In the above computation, we have also adoptedthe following conven-
tions. If Q is a vector in Rn1, the notation DQ denotes the row vector
(DiQ)ni=11;ifQhasalsoothercoordinates,sayitisavectorinRm×Rn1,then
DQ will just be 0in Rm and (DiQ)ni=11 in Rn1. Weuse (D2Q)(α)(β), where
QisamapfromRn1 andα,βarevectorsinRn1,todenote∑i,j(DiDjQ)αiβj.
BISMUT’SFORMULA 7
Wenowwanttoapply Feynman-KactechniquetodealwithRic∇HF as
inHsu[2]. LetM bethesolutionofthematrix-valueddifferentialequation
t
1
∂M + M Ric ∂t = 0,M = I.
t 2 t Ut 0
UsingItoˆ’sformula,itisstraightforwardtoverifythat
∂M ∇HF(Z ,T−t)+M Φ(Z ,T−t)∂t
t t t t
isamartingale,where
Φ = (1)+(2)+(3)+(4)+(5).
E
Takingtheexpectationwithrespectto ,wehave
z
∇HE F(Z ) =E M ∇HF(Z ,T−t)
z T z t t
t
+E M Φ(Z ,T−s)∂s.
z s s
Z0
Integratingover tfrom0to T and usingintegrationbypartsonthesec-
ondtermontherightside,wehave
T
(1) T∇HE F(Z ) = E M ∇HF(Z ,T−t)∂t
z T z t t
Z0
T
+E N Φ(Z ,T−t)∂t.
z t t
Z0
Here N = (T −t)M . With this last identity, we have accomplished the
t t
taskwehavesetourselvesatthebeginningofthissection,namely,wehave
passedthe gradient operatorthroughthe expectationto act on the bundle
map f,hererepresentedbyitsscalarizationF. Aswepointedoutabove,the
nextstepistoremovetheexpectationbyanintegrationbypartsargument
inthepathspace.
4. INTEGRATION BY PARTS
The first term on the right side of (1) contains the gradient on F. From
theheatequation ∂F = LFwehavebyItoˆ’sformula,
∂t
∂F(Z ,T−t) = ∑(H +V )F∂Wi.
t i 1,i t
i
8 ELTONP.HSUANDZHENANWANG
Foranyv ∈ Rm andr ∈ {1,2,...,n },wehave
2
T
E hM ∇HF,vi∂t
z t r
Z0
T
= E h∇HF,M†vi∂t
z r t
Z0
T T
= E h∇HF,∂W i hM†v,∂W i
z r t t t
Z0 Z0
T T
= E F (Z )−E F(Z )− ∑V F ∂Wi hv,M ∂W i
z r T z r T 1,i r t t t
Z0 i !Z0
T T
= E F (Z ) hv,M ∂W i −E h∑V F e,M†vi∂t
z r T t t z 1,i r i t
(cid:18) Z0 (cid:19) Z0 i
T T
= E F (Z ) hv,M ∂W i −E hM ∑V F e,vi∂t
z r T t t z t 1,i r i
(cid:18) Z0 (cid:19) Z0 i
Thiscanbewrittenmorecompactlyas
T
(2) E M ∇HF∂t
z t
Z0
T T
= E F(Z ) M ∂W −E M ∑V Fe ∂t.
z T t t z t 1,j j
(cid:18) Z0 (cid:19) Z0 j
Theleftsideaboveisjustthefirsttermontherightsideof(1). Notethatwe
haveremovedthegradientoperatorfrom F. Theremainingdifferentiation
on F areinthefibredirections.
Thedealingswiththeremainingtermsin(1)andthesecondtermonthe
rightsideof(2)
T
E N Φ(Z ,T)−M ∑(DF)(V )e ∂t
z t t t t 1,j j
Z0 j
areadmittedlytechnical,butintheprocesswewillsingleoutthetermsin-
volvingsomeverticalderivativesonwhichwecanperformasecondcom-
mutationoperation.
We will adopt the following notations: N¯j stands for the horizontal lift
t
of N e andKj(l) forthe2ndand3rdcoordinatesof[N¯j,H ]. Notethat N is
t j t,i t i t
symmetric,thereforeinexplicitcomponentswehave
N e = ∑Nk e , N¯j = ∑Nj H , and Kj(l) = ∑Ω(l)Nj .
t j t,j k t t,k k t,i ki t,k
k k k
WehRecallthatΦ = (1)+(2)+(3)+(4)+(5). Wehave
N Φ(Z ,T−t)−M ∑(DF)(V )e = −A + A + A
t t t 1,j j 1 2 3
j
BISMUT’SFORMULA 9
where
1
A = ∑∑ Kj(2)(H F)+ (HKj(2))F+Kj(2)(DF)(V ) e
1 t,i i 2 i t,i t,i 1,i j
j i (cid:18) (cid:19)
A = ∑∑ DH F+(D2F)(V ) N¯jV +Kj(1)Y e
2 i 1,i t 1,i t,i t j
j i
(cid:16) (cid:16) (cid:17)(cid:17)
(cid:0) (cid:1)
A = ∑DF B +(N¯jV )−∑V Mj e
3 t,j t 0 1,i t,i j
!
j i
1
B = ∑ (HKj(1))Y +(DN¯jV )(V )
t,j 2 i t,i t t 1,i 1,i
i
(cid:16)
+(DV )(N¯jV )+(DV )(Kj(1)Y)+H N¯jV .
1,i t 1,i 1,i t,i t i t 1,i
(cid:17)
Here as will be in the sequel, we have omitted the ubiquitous (Z ,T−t)
t
fortheeaseofnotation. Theruleforregroupingthetermsheremayappear
to be mysterious. Basically everything related to the curvature on E is
2
included in A , and the remaining terms follow the rule that the second
1
ordertermsof Fgoto A ,andthefirstordertermsgoto A .
2 3
Weconsidertheterm A . From
1
∂F = ∑(H F+(DF)(V ))∂Wi
i 1,i t
i
wehave
A ∂t = ∑ ∑Kj(2)(H F+(DF)(V )) ∂te
1 t,i i 1,i j
!
j i
+ 1 ∑ ∑(H Kj(2))F ∂te
2 i t,i j
!
j i
= ∑ ∑Kj(2)∂Wi ·(∂F)e
t,i t j
!
j i
1
+ ∑ ∑(∂Kj(2))·(∂Wi) Fe
2 t,i t j
!
j i
= ∑ ∂Gj(2)·∂F e +∑(∂Gj(2))Fe +∂S
t j t j t
j j
(cid:16) (cid:17)
= ∑∂(Gj(2)F)e +∂S′
t j t
j
10 ELTONP.HSUANDZHENANWANG
j(l)
HereG arematrixvaluedprocessdefinedbyby
t
t
Gj(l) = ∑Kj(l)◦∂Wi,
t t,i t
Z0 i
and S and S′ are martingales. Integrating and taking the expectation we
t t
have
T
E A ∂t = E ∑ Gj(2)F(Z ) e .
z 1 z T T j
Z0 j
(cid:16) (cid:17)
Remark 4.1. As we mentioned before this paragraph of calculation, the
curvature on E gives rise toa separatetermin the formula, and this term
2
turnsouttoonlyrelyonthe0thordertermoftheterminalvalueof F.
5. SECOND COMMUTATION AND DE´NOUEMENT
Before starting the calculations of A ,A , it is perhaps helpful to point
2 3
out that if the vector bundle E flat, then A vanishes, but A and A re-
2 1 2 3
main. We therefore anticipate terms involving the derivative DF. Every
term in A and A includes the 0th or 1st order term of DF. Following
2 3
Hsu [3], we carry out a second commutation computation. From the heat
equationfor F wehave
∂DF
= LDF+[D,L]F
∂t
= LDF+A +A
4 5
where
A = ∑ (D2F)(V )+DH F (DV ),
4 1,i i 1,i
i
(cid:0) (cid:1)
1
A = DF∑ (D2V )(V )+(DV )(DV )+DHV +DV .
5 1,i 1,i 1,i 1,i i 1,i 0
2
i (cid:18) (cid:19)
(cid:0) (cid:1)
Here (D2P)Q stands for the row vector (∑ P Q ). As a consequence we
i ij i
have
∂(DF)(Z ,T−t) = ∑ (D2F)(V )+DH F ∂Wi− A ∂t−A ∂t
t 1,i i t 4 5
i
(cid:0) (cid:1)
Wenowuseamethodsimilar totheoneusedindealingwith A toabsorb
1
A and A .
4 5
for this purpose we introduce the processesRn1-valued processesYj =
0
0 called the derived processes in Norris [5]. They are determined by the
stochasticdifferentialequation
∂Yj = Cj ∂Wi+Cj∂t+Cj ∂Wi+Cj∂t,
t 2,i t 3 4,i t 5
where
Cj = N¯jV +Kj(1)Y,
2,i t 1,i t,i t