Derivative and divergence formulae for diffusion semigroups

0.21 MB·

by  Anton Thalmaier

DERIVATIVE AND DIVERGENCE FORMULAE
FOR DIFFUSION SEMIGROUPS
7
1
0
ANTONTHALMAIERANDJAMESTHOMPSON
2

n MathematicsResearchUnit,FSTC,UniversityofLuxembourg
a
6,rueRichardCoudenhove-Kalergi,1359Luxembourg,GrandDuchyofLuxembourg
J

3
1 Abstract. ForasemigroupPtgeneratedbyanellipticoperatoronasmoothmanifoldM,
weusestraightforwardmartingaleargumentstoderiveprobabilisticformulaeforPt(V(f)),
] notinvolvingderivativesof f,whereVisavectorfieldonM.Fornon-symmetricgenera-
R
tors,suchformulaecorrespondtothederivativeoftheheatkernelintheforwardvariable.
P Asanapplication,theseformulaecanbeusedtoderivevariousshift-Harnackinequalities.
.
h
t
a
m
Introduction
[
ForaBanachspaceE,e EandaMarkovoperatorPon (E),itisknownthatcertain
b
1 ∈ B
estimatesonP( f)areequivalenttocorrespondingshift-Harnackinequalities. Thiswas
e
v ∇
provedbyF.-Y.Wangin[18]. Forexample,forδ (0,1)andβ C((δ , ) E;[0, )),
5 e∈ e∈ e ∞ × ∞
2 heprovedthatthederivative-entropyestimate
6
3 P( f) δ P(flogf) (Pf)logPf +β (δ, )Pf
e e
0 ∇ ≤ − ·
1. holdsforanyδ≥δe(cid:12)(cid:12)(cid:12)andpos(cid:12)(cid:12)(cid:12)itive(cid:0)f ∈Cb1(E)ifandonlyif(cid:1)theinequality
0 1 pr p 1
17 (Pf)p≤ P(fp(re+·)) exp Z0 1+(p 1)sβe r+r(−p 1)s,·+sre!ds!
(cid:0) (cid:1) − −
: holdsfor any p 1/(1 rδ ), r (0,1/δ ) and positive f (E). Furthermore, he also
v ≥ − e ∈ e ∈Bb
i provedthatifC 0isaconstantthentheL2-derivativeinequality
X ≥
r P( f)2 CPf2
a ∇e ≤
(cid:12) (cid:12)
holdsforanynon-negative f ∈Cb1(E(cid:12)(cid:12))ifand(cid:12)(cid:12)onlyiftheinequality
Pf P f(αe+ ) + α CPf2
≤ · | |q
holds for any α R and non-negativ(cid:0)e f ((cid:1)E). The objective of this article is to find
b
∈ ∈B
probabilistic formulae for P (V(f)) from which such estimates can be derived, for the
T
caseinwhichP istheMarkovoperatorassociatedtoanon-degeneratediffusionX ona
T t
smooth,finite-dimensionalmanifoldM,andV avectorfield.
E-mail address:[email protected], [email protected].
Date:January16,2017.
2010MathematicsSubjectClassification. 58J65,60J60,53C21.
Key words and phrases. Diffusion semigroup, Heat kernel, Gradient estimate, Harnack inequality, Ricci
curvature.
1
2 DERIVATIVEANDDIVERGENCEFORMULAE
InSection1wesupposethatMisaRiemannianmanifoldandthatthegeneratorofX is
t
∆+Z,forsomesmoothvectorfieldZ. Anynon-degeneratediffusiononasmoothmanifold
inducesaRiemannianmetricwithrespecttowhichitsgeneratortakesthisform.Thebasic
strategyisthentousetherelationV(f)=div(fV) fdivVtoreducetheproblemtofinding
−
a suitable formulafor P (div(fV)). Such a formulawas givenin [3] for the case Z =0,
T
whichweextendtothegeneralcasewithTheorem1.16. Indoingso,wedonotmakeany
assumptionsonthederivativesofthecurvaturetensor,asoccurredin[2]. Foranadapted
processh withpathsintheCameron-MartinspaceL1,2([0,T];R),withh =0andh =1
t 0 T
andundercertainadditionalconditions,weobtaintheformula
P (V(f))(x)
T
= E f(X (x))(divV)(X (x))
T T
−
1(cid:2) (cid:3) T
E f(X (x)) V(X (x)),// Θ (divZ)(X(x))h h˙ Θ 1dB
−2 " T D T T TZ0 (cid:16) t t− t(cid:17) −t tE#
whereΘistheAut(T M)-valuedprocessdefinedbythepathwisedifferentialequation
x
d
dtΘt=−//t−1 Ric♯+(∇.Z)∗−divZ //tΘt
(cid:16) (cid:17)
with Θ =id . Here // denotes the stochastic parallel transport associated to X(x),
0 TxM t t
whose antidevelopmentto T M has martingale part B. In particular, B is a diffusion on
x
Rn generated by the Laplacian; it is a standard Brownian motion sped up by 2, so that
dBidBj=2δ dt. Choosingh explicitlyyieldsa formulafromwhich estimatesthen can
t t ij t
bededuced,asdescribedinSubsection1.5.
The problem of finding a suitable formula for P (V(f)) is dual to that of finding an
T
analogousoneforV(P f). AformulaforthelatteriscalledtheBismutformula[1]orthe
T
Bismut-Elworthy-Liformula,onaccountof[6]. WeprovideabriefproofofitinSubsec-
tion1.3,sincewewouldliketocompareittoourformulaforP (V(f)). Ourapproachto
T
theseformulaeisbasedonmartingalearguments;integrationbypartsisdoneatthelevelof
localmartingales. Underconditionswhichassurethatthelocalmartingalesaretruemar-
tingales,thewantedformulaearethenobtainedbytakingexpectations.Theyallowforthe
choiceofafiniteenergyprocess.Dependingontheintendedtype,conditionsareimposed
eitherontherightendpoint,asintheformulaforP (V(f)),ortheleftendpoint,asinthe
T
formulaforV(P f). Theformulafor P (V(f))requiresnon-explosivity;theformulafor
T T
V(P f) does not. From the latter can be deduced Bismut’s formula for the logarithmic
T
derivativeinthebackwardvariablexoftheheatkernel p (x,y)determinedby
T
(P f)(x)= f(y)p (x,y)vol(dy), f C (M).
T T b
Z ∈
M
FromourformulaforP (V(f))canbededucedthefollowingformulaforthederivativein
T
theforwardvariabley:
1 T
( logp (x, )) = E // Θ (divZ)(X(x))h h˙ Θ 1dB X (x)=y .
∇ T · y −2 " T TZ0 (cid:16) t t− t(cid:17) −t t(cid:12) T #
(cid:12)
In Section 2 we considerthe generalcase in which M is a smooth(cid:12)manifold and X a
t
non-degeneratediffusionsolvingaStratonovichequationoftheform
dX =A (X)dt+A(X) dB.
t 0 t t t
◦
We denote by TX the derivative (in probability) of the solution flow. Using a similar
t
approachtothatofSection1,andavarietyofgeometricobjectsnaturallyassociatedtothe
DERIVATIVEANDDIVERGENCEFORMULAE 3
equation,weobtain,undercertainconditions,theformula
P (V(f))
T
m
= E f(X )A V,A (X )
T i i T
− h i
Xi=1 (cid:2) (cid:3)
1 T
E f(X ) V(X ),Ξ Ξ 1 (traceˆA )(X)h h˙ A(X)dB +2hAAdt
−2 " T (cid:28) T TZ0 −t (cid:16)(cid:16) ∇ 0 t t− t(cid:17) t t t 0 (cid:17)(cid:29)#
with
t
Ξ = TX TX TX 1 (˘A ) + ˘A +traceˆA (Ξ ) ds,
t t− tZ0 s− (cid:16)(cid:16) ∇ 0 ∗ ∇ 0 ∇ 0(cid:17) s (cid:17)
m
AA= (˘A ) + ˘A T˘( ,A) (A) + A ,T˘( ,A) (A) ,
0 ∇ 0 ∗ ∇ 0 · i ∗ i 0 · i ∗ i
Xi=1(cid:16) (cid:17)(cid:16) (cid:17) (cid:2) (cid:3)
wheretheoperators ˆA , ˘A andT˘( ,A)aregivenateachx Mandv T Mby
0 0 i x
∇ ∇ · ∈ ∈
ˆ A = A(x) d(A ( )A ( )) (v) (dA ) (v,A ) ,
v 0 ∗ 0 x ∗ x 0
∇ · · −
˘ A = A(x)d(cid:0) A( ) A ( ) (v), (cid:1)
∇v 0 · ∗ 0 · x
T˘(v,A) = A(x)(d(cid:0)A ) (v,A).(cid:1)
i x ∗ x i
This formula has the advantage of involving neither parallel transport nor Riemannian
curvature,bothtypicallydifficulttocalculateintermsofA.
1. IntrinsicFormulae
1.1. Preliminaries. LetMbeacompleteandconnectedn-dimensionalRiemannianman-
ifold, theLevi-Civitaconnectionon M andπ: O(M) M theorthonormalframebun-
∇ →
dleover M. Let E M be anassociatedvectorbundlewith fibreV andstructuregroup
→
G=O(n).Theinducedcovariantderivative
: Γ(E) Γ(T M E)
∗
∇ → ⊗
determinestheso-calledconnectionLaplacian(orroughLaplacian)(cid:3)onΓ(E),
(cid:3)a=trace 2a.
∇
Notethat 2a Γ(T M T M E)and((cid:3)a) = 2a(v,v) E wherev runsthrough
∗ ∗ x i i i x i
∇ ∈ ⊗ ⊗ ∇ ∈
anorthonormalbasisofTxM. Fora,b Γ(E)ofcPompactsupportitisimmediatetocheck
∈
that
(cid:3)a,b = a, b .
In this sense we have (cid:3)= h .iLL2(eEt) H−bhe∇the∇hoiLri2z(To∗nMta⊗lEs)ubbundle of the G-invariant
∗
−∇ ∇
splittingofTO(M)and
h: π TM H֒ TO(M)
∗ ∼
−→ →
thehorizontalliftoftheG-connection;fibrewisethisbundleisomorphismreadsas
h : T M H , u O(M).
u π(u) ∼ u
−→ ∈
IntermsofthestandardhorizontalvectorfieldsH ,...,H onO(M),
1 n
H(u):=h (ue), u O(M),
i u i
∈
Bochner’shorizontalLaplacian∆hor,actingonsmoothfunctionsonO(M),isgivenas
n
∆hor= H2.
i
Xi=1
4 DERIVATIVEANDDIVERGENCEFORMULAE
Toformulatetherelationbetween(cid:3)and∆hor,itisconvenienttowritesectionsa Γ(E)as
∈
equivariantfunctionsF : O(M) V via F (u)=u 1a wherewereadu O(M)asan
a a − π(u)
→ ∈
isomorphismu: V E . Equivariancemeansthat
∼ π(u)
−→
F (ug)=g 1F (u), u O(M), g G=O(n).
a − a
∈ ∈
Lemma1.1(see[9],p.115). Fora Γ(E)andF thecorrespondingequivariantfunction
a
∈
onO(M),wehave
(HF )(u)=F (u), u O(M).
i a ∇ueia ∈
Hence
∆horFa=F(cid:3)a,
whereasabove
(cid:3): Γ(E) ∇ Γ(T∗M E) ∇ Γ(T∗M T∗M E)traceΓ(E).
−→ ⊗ −→ ⊗ ⊗ −→
Proof. Fix u O(M) and choose a curve γ in M such that γ(0)=π(u) and γ˙ =ue. Let
i
∈
t u(t)bethehorizontalliftofγtoO(M)suchthatu(0)=u. Notethatu˙(t)=h (γ˙(t)),
u(t)
7→
andinparticularu˙(0)=h (ue)=H(u). Hence,denotingtheparalleltransportalongγby
u i i
// =u(ε)u(0) 1,weget
ε −
F (u)=u 1 a
∇ueia − ∇uei π(u)
(cid:16) (cid:17)
// 1a a
=u 1lim ε− γ(ε)− γ(0)
−
ε 0 ε
↓
u(ε) 1a u(0) 1a
− γ(ε) − γ(0)
=lim −
ε 0 ε
↓
F (u(ε)) F (u(0))
=lim a − a
ε 0 ε
↓
=(H) F
i u a
=(HF )(u). (cid:3)
i a
NowconsiderdiffusionprocessesX onMgeneratedbytheoperator
t
L =∆+Z
whereZ Γ(TM)isasmoothvectorfield. SuchdiffusionsonMmaybeconstructedfrom
∈
thecorrespondinghorizontaldiffusionsonO(M)generatedby
∆hor+Z¯
wherethevectorfieldZ¯ isthehorizontalliftofZ toO(M),i.e. Z¯ =h (Z ),u O(M).
u u π(u)
∈
Moreprecisely,westartfromtheStratonovichstochasticdifferentialequationonO(M),
n
(1.1) dU = H(U) dBi+Z¯(U)dt, U =u O(M)
t i t ◦ t t 0 ∈
Xi=1
where B is a Brownian motion on Rn sped up by 2, that is dBidBj =2δ dt. Then for
t t t ij
X =π(U),thefollowingequationholds:
t t
n
(1.2) dX = Ue dBi+Z(X)dt, X =x:=πu.
t t i◦ t t 0
Xi=1
DERIVATIVEANDDIVERGENCEFORMULAE 5
TheBrownianmotionBisthemartingalepartoftheanti-development ϑofX,whereϑ
U
denotesthecanonical1-formϑonO(M),i.e. R
ϑ (e)=u 1e , e T O(M).
u − π(u) u
∈
Inparticular,forF C (O(M)),resp. f C (M),wehave
∞ ∞
∈ ∈
n
d(F U)= (HF)(U) dBi+(Z¯F)(U)dt
◦ t i t ◦ t t
Xi=1
n
(1.3) = (HF)(U)dBi+ ∆hor+Z¯ (F)(U)dt,
i t t t
Xi=1 (cid:16) (cid:17)
respectively
n
d(f X)= (df)(Ue) dBi+(Zf)(X)dt
◦ t t i ◦ t t
Xi=1
n
= (df)(Ue)dBi+(∆+Z)(f)(X)dt.
t i t t
Xi=1
Typically,solutionsto (1.2) are definedupto some maximallifetime ζ(x) whichmay be
finite. Thenwehave,almostsurely,
ζ(x)< X ast ζ(x)
t
∞ ⊂ →∞ ↑
(cid:8) (cid:9) (cid:8) (cid:9)
whereonthe right-handside, thesymbol denotesthepointatinfinityin theone-point
∞
compactificationof M. Itcanbeshownthatthemaximallifetimeofsolutionstoequation
(1.1)andto(1.2)coincide,seee.g.[12].
Incaseofanon-triviallifetimethesubsequentstochasticequationsshouldbereadfor
t<ζ(x).
Proposition1.2. Let // : E E be paralleltransportin E along X, inducedby the
t X0 → Xt
paralleltransportonM,
// =U U 1: T M T M.
t t 0− X0 → Xt
Then,fora Γ(E),wehave
∈
n
d // 1a(X) = // 1 a dBi+// 1( a)(X)dt,
t− t t− ∇Utei ◦ t t− ∇Z t
(cid:16) (cid:17) Xi=1 (cid:16) (cid:17)
respectivelyinItoˆ form,
n
d // 1a(X) = // 1 a dBi+// 1((cid:3)a+ a)(X)dt.
t− t t− ∇Utei t t− ∇Z t
(cid:16) (cid:17) Xi=1 (cid:16) (cid:17)
Moresuccinctly,thelasttwoequationsmaybewrittenas
d // 1a(X) =// 1 a,
t− t t− ∇◦dXt
(cid:16) (cid:17)
respectively
d // 1a(X) =// 1 α+// 1((cid:3)a)(X)dt.
t− t t− ∇dXt t− t
(cid:16) (cid:17)
6 DERIVATIVEANDDIVERGENCEFORMULAE
Proof. Wehave// 1a(X)=U U 1a(X)=U F (U).ItiseasilycheckedthatZ¯F =F .
t− t 0 t− t 0 a t a ∇Za
Thus,weobtainfromequation(1.3)
n
dF (U)= (HF )(U)dBi+ ∆horF +Z¯F (U)dt
a t i a t t a a t
Xi=1 (cid:16) (cid:17)
n
=Xi=1(cid:16)F∇Uteia(cid:17)(Ut)dBit+(cid:16)F(cid:3)a+F∇Za(cid:17)(Ut)dt
n
= U 1 a (X)dBi+U 1((cid:3)a+ a)(X)dt. (cid:3)
− ∇Utei t t t− ∇Z t
Xi=1 (cid:16) (cid:17)
Corollary1.3. FixT >0andleta Γ(E)solvetheequation
t
∈
∂
a =(cid:3)a + a on[0,T] M.
t t Z t
∂t ∇ ×
Then
// 1a (X), 0 t<T ζ(x),
t− T t t ≤ ∧
−
isalocalmartingale.
Proof. Indeedwehave
d(// 1a (X))m=// 1 (cid:3)a + a + ∂a (X)dt=0,
t− T−t t t− T − t ∇ Z t ∂ t T − t! t
=0
wherem=denotesequalitymodulodiffe|rentialsof{lozcalmartin}gales. (cid:3)
WearenowgoingtolookatoperatorsLRonΓ(E)whichdifferfrom(cid:3)byazero-order
term,inotherwords,
(1.4) (cid:3) LR=R whereR Γ(EndE).
− ∈
Thus,bydefinition,theactionR : E E islinearforeachx M.
x x x
→ ∈
Example1.4. AtypicalexampleisE=ΛpT MandAp(M)=Γ(ΛpT M)with p 1.The
∗ ∗
≥
deRham-HodgeLaplacian
∆(p)= (d d+dd ):Ap(M) Ap(M)
∗ ∗
− →
thentakestheform
∆(p)α=(cid:3)α Rα
−
whereRisgivenbytheWeitzenbo¨ckdecomposition.Inthespecialcasep=1,oneobtains
Rα=Ric(α♯, )whereRic: TM TM RistheRiccitensor.
· ⊕ →
Definition1.5. Fix x M andletX beadiffusiontoL =∆+Z,startingat x. LetQ be
t t
∈
theAut(E )-valuedprocessdefinedbythefollowinglinearpathwisedifferentialequation
x
d
Q = QR , Q =id ,
dt t − t //t 0 Ex
where
R :=// 1 R // End(E )
//t t− ◦ Xt◦ t ∈ x
and// isparalleltransportinEalongX.
t
DERIVATIVEANDDIVERGENCEFORMULAE 7
Proposition1.6. LetLR =(cid:3) R beasin equation(1.4)and X bea diffusiontoL =
t
−
∆+Z,startingatx. Then,foranya Γ(E),
∈
n
d Q// 1a(X) = Q// 1 a dBi+Q// 1((cid:3)a+ a Ra)(X)dt.
t t− t t t− ∇Utei t t t− ∇Z − t
(cid:16) (cid:17) Xi=1 (cid:16) (cid:17)
Proof. Letn :=// 1a(X). Then
t t− t
d(Qn)=(dQ)n +Q dn
t t t t t t
= Q// 1R // 1n dt+Q dn
− t t− Xt t− t t t
= Q// 1(Ra)(X)dt+Q dn.
− t t− t t t
TheclaimthusfollowsfromProposition1.2. (cid:3)
Corollary1.7. FixT >0andletX(x)beadiffusiontoL =∆+Z,startingatx. Suppose
t
thata solves
t
∂
a =((cid:3) R+ )a on[0,T] M,
t Z t
∂t − ∇ ×
at|t=0=a∈Γ(E).
Then 
(1.5) N :=Q// 1a (X(x)), 0 t<T ζ(x),
t t t− T−t t ≤ ∧
isalocalmartingale,startingata (x). Inparticular,ifζ(x)= andifequation(1.5)isa
T
∞
truemartingaleon[0,T],wearriveattheformula
a (x)=E Q // 1a(X (x)) , a Γ(E).
T T T− T ∈
(cid:2) (cid:3)
Proof. Indeed,wehave
dN m=Q// 1 ((cid:3)+ R)a + ∂a (X)dt=0. (cid:3)
t t t− ∇ Z − T − t ∂ t T − t! t
=0
| {z }
Remark1.8. Notethat
d
Q = QR , withQ =id ,
dt t − t //t 0 Ex
impliestheobviousestimate
t
Q exp R(X (x))ds
t op s
k k ≤ −Z0 !
whereR(x)=inf R v,w : v,w E , v 1and w 1 .
x x
{h i ∈ k k≤ k k≤ }
1.2. Commutationformulae. Inthesequel,weconsiderthespecialcaseE=T M. Thus
∗
Γ(E)isthespaceofdifferential1-formson M. Theresultsofthissectionapplytovector
fields as well, by identifying vector fields V Γ(TM) and 1-forms α Γ(T M) via the
∗
∈ ∈
metric:
V V♭, α α#.
←→ ←→
Let Z Γ(TM) be a vector field on M. Then the divergence of Z, denoted by divZ
C (M∈),isdefinedbydivZ:= trace(v Z).Therefore ∈
∞ v
7→∇
n
(divZ)(x)= X,v
h∇vi ii
Xi=1
8 DERIVATIVEANDDIVERGENCEFORMULAE
foranyorthonormalbasis v n forT M. Forcompactlysupported f wehave
{ i}i=1 x
Z, f = divZ,f .
h ∇ iL2(TM) −h iL2(M)
TheadjointZ ofZisgivenbytherelation
∗
Z f = Zf (divZ)f, f C (M).
∗ ∞
− − ∈
Ifeither f orhiscompactlysupported,thisimplies
Zf,h = f,Z h .
h iL2(M) h ∗ iL2(M)
Similarly,forα Γ(T M),welet
∗
∈
α #
(divα)(x)=trace TxM−∇→Tx∗M−→TxM .
ThusdivY =divY♭ and divα=divα#. (cid:0)That is, if δ=d denotes(cid:1)the usual codifferential
∗
thendivα= δα. Finally,wedefine
−
Ric (X,Y):=Ric(X,Y) Z,Y , X,Y Γ(TM).
Z X
−h∇ i ∈
Notation1.9. For the sake of convenience, we read bilinear forms on M, such as Ric ,
Z
likewiseassectionsofEnd(T M)orEnd(TM),e.g.
∗
Ric (α):=Ric (α♯, ), α T M,
Z Z ∗
· ∈
Ric (v):=Ric (v, )♯, v TM.
Z Z
· ∈
Ifthereisnoriskofconfusion,wedonotdistinguishinnotation. Inparticular,depending
onthecontext,(Ric ) maybearandomsectionofEnd(T M)orofEnd(TM).
Z // ∗
t
Lemma1.10(Commutationrules). LetZ Γ(TM).
∈
(1) Forthedifferentiald,wehave
d ∆+Z = (cid:3) Ric + d;
Z Z
− ∇
(2) forthecodifferentiald = (cid:0)div,we(cid:1) ha(cid:0)ve (cid:1)
∗
−
∆+Z d =d (cid:3) Ric + ,
∗ ∗ ∗ − ∗Z ∇∗Z
wheretheformaladjoint(cid:0)of (a(cid:1)ctingon(cid:0) 1-forms)is (cid:1)α= α (divZ)α.
∇Z ∇∗Z −∇Z −
Proof. Indeed,foranysmoothfunction f wehave
d ∆+Z f =d d df+(df)Z
∗
−
(cid:0) (cid:1) =∆(cid:0)(1)df+ Zdf+ (cid:1).Z, f
∇ h∇ ∇ i
=((cid:3)+ )(df) Ric ( , f)
Z Z
∇ − · ∇
= (cid:3) Ric + (df).
Z Z
− ∇
Theformulain(2)isthenjustdualto(cid:0)(1). (cid:1) (cid:3)
1.3. Aformulaforthedifferential. Now,letX(x)beadiffusionto∆+ZonM,starting
t
atX (x)=x,U ahorizontalliftof X toO(M)and B=U ϑthemartingalepartofthe
0 t 0 U
anti-developmentofXt(x)toTxM. LetQt betheAut(Tx∗M)R-valuedprocessdefinedby
d
Q = Q (Ric )
dt t − t Z //t
withQ =id ,let
0 Tx∗M
P f(x)=E 1 f(X(x))
t t<ζ(x) t
{ }
h i
DERIVATIVEANDDIVERGENCEFORMULAE 9
betheminimalsemigroupgeneratedby∆+Z on M,actingonboundedmeasurablefunc-
tions f.
Fix T >0 and let ℓ be an adapted process with paths in the Cameron-Martin space
t
L1,2([0,T];T M). ByCorollary1.7
x
(1.6) N :=Q// 1(dP f), t<T ζ(x),
t t t− T−t ∧
islocalmartingale.Therefore
t
N(ℓ) Q // 1(dP f)(ℓ˙ )ds
t t −Z0 s s− T−s s
isalocalmartingale.Byintegrationbyparts
t 1 t
Q // 1(dP f)(ℓ˙ )ds (P f)(X(x)) Qtr(ℓ˙ ),dB
Z0 s s− T−s s −2 T−t t Z0 h s s si
isalsoalocalmartingaleandtherefore
1 t
(1.7) Q// 1(dP f)(ℓ) (P f)(X(x)) Qtrℓ˙ ,dB
t t− T−t t −2 T−t t Z0 h s s si
isalocalmartingale,startingat(dP f)(ℓ ). Choosingℓ sothat(1.7)isatruemartingale
T 0 t
on[0,T]withℓ =vandℓ =0,weobtaintheformula
0 T
1 T
(1.8) (dP f)(v)= E 1 f(X (x)) Qtrℓ˙ ,dB .
T −2 " {T<ζ(x)} T Z0 h s s si#
Forfurtherdetails,see[14,15]. Denotingby p(x,y)thesmoothheatkernelassociatedto
t
∆+Z,sinceformula(1.8)holdsforallsmoothfunctions f ofcompactsupport,itimplies
Bismut’sformula
1 τ T
(dlogp ( ,y)) (v)= E ∧ Qtrℓ˙ ,dB X (x)=y .
T · x −2 "Z0 h s s si(cid:12) T #
(cid:12)
(cid:12)
Theargumentleadingtoformula(1.8)isbasedonthefactthatthelocalmartingale(1.7)is
atruemartingale.Sincetheconditiononℓ isimposedontheleftendpoint,thiscanalways
t
beachieved,bytakingℓ =0for s τ T whereτisthefirstexittimeofsomerelatively
s
≥ ∧
compactneighbourhoodof x. No boundson the geometryare needed; also explosionin
finite times of the underlyingdiffusion can be allowed. For the problem of constructing
appropriate finite energy processes ℓ with the property ℓ = 0 for s τ T, see [15],
s s
≥ ∧
resp.[16,Lemma4.3].
Imposingin(1.7)howevertheconditionsℓ =0andℓ =vwouldleadtoaformulafor
0 T
E Q // 1(df) (v)
T T− XT(x)
h i
not involving derivatives of f, which clearly requires strong assumptions. If the local
martingale(1.6)isatruemartingale,wegettheformula
(dP f) (v)=E Q // 1(df) (v) .
T x T T− XT(x)
h i
Forsuchaformulatohold,obviouslyX(x)needstobenon-explosive.
t
10 DERIVATIVEANDDIVERGENCEFORMULAE
1.4. Aformulaforthecodifferential. Recallthat,accordingtoLemma1.10,wehave
(1.9) ∆+Z+divZ div=div (cid:3)+ Ric +divZ .
∇Z− ∗Z
−
(cid:0) (cid:1) (cid:0) (cid:1)
Forabounded1-formαsupposeα satisfies
t
d
(1.10) α = (cid:3)+ Ric +divZ α
dt t ∇Z− ∗Z t
(cid:16) − (cid:17)
with α =α, where divZ acts fibrewise as a multiplication operator, and that Θ is the
0 t
Aut(T M)-valuedprocesswhichsolves
x
d
Θ = (Ric divZ) Θ
dt t − ∗Z− //t t
−
withΘ =id . HereRic istheadjointtoRic actingasendomorphismofT M,see
0 TxM ∗Z Z x
Notation1.9. − −
Remark1.11. WehaveΘ =Qtr ifwesetR:=Ric divZ End(T M)anddefineQ
t t ∗Z− ∈ ∗ t
viaDefinition1.5. −
Proposition1.12. FixT >0. LetX(x)beadiffusionto∆+ZonM,startingatx.
t
(i) Then
t
(divα )(X(x))exp (divZ)(X (x))ds
T−t t Z0 s !
isalocalmartingale,startingatdivα .
T
(ii) Supposeh isanadaptedprocesswithpathsinL1,2([0,T];R).Then
t
1 t
(1.11) divα h + α //Θ h˙ (divZ)(X (x))h Θ 1// 1dB
T−t t 2 T−t t tZ0 (cid:16) s− s s(cid:17) −s s− s!
isalocalmartingale,startingatdivα h .
T 0
Proof. (i)Takingintoaccountthecommutationrule(1.9)andtheevolutionequation(1.10)
ofα,weget
t
∂ divα =div∂α
t t t t
(1.12) =div((cid:3)+∇Z−Ric∗Z+divZ)αt
−
=(∆+Z+divZ)divα.
t
TheclaimthenfollowsfromItoˆ’sformula.
(ii)Toverifytheseconditem,set
t
A :=exp (divZ)(X (x))ds
t s
Z !
0
anddefineℓ :=A 1h. Usingthefactthatα (//Θ)isalocalmartingale,indeed
t −t t T t t t
−
n
d α (//Θ) = ( α )(//Θ)dBi
T t t t ∇Utei T t t t t
(cid:0) − (cid:1) Xi=1 −