Heat equation derivative formulas for vector bundles

Heat Equation Derivative Formulas for Vector Bundles

Bruce K. Driver¹

Department of Mathematics-0112,University of California at San Diego,La Jolla, California92093-0112

E-mail: drivermath.ucsd.edu

and Anton Thalmaier²

Institut fur Angewandte Mathematik,Universitat Bonn,Wegelerstr.6,53115Bonn,Germany E-mail: antonwiener.iam.uni-bonn.de

Communicated by L.Gross

Received April 5, 2000; accepted April 28, 2000

We use martingale methods to give Bismut type derivative formulas for differen- tials and co-differentials of heat semigroups on forms, and more generally for sec- tions of vector bundles. The formulas are mainly in terms of Weitzenbock curvature terms; in most cases derivatives of the curvature are not involved. In particular, our results improve B. K. Driver's formula in (1997,J.Math.Pures Appl.(9)76, 703737) for logarithmic derivatives of the heat kernel measure on a Riemannian manifold. Our formulas also include the formulas of K. D. Elworthy and X.-M. Li (1998,C.R.Acad.Sci.Paris Ser.I Math.327, 8792). 2001 Academic Press

Key Words:heat kernel measure; Malliavin calculus; Bismut formula; integration by parts; Dirac operator; de RhamHodge Laplacian; Weitzenbock decomposition.

Contents.

1. Introduction.

2. General stochastic and geometric notation.

3. Local martingales.

4. The fundamental derivative formulas.

5. Applications for compact M.

6. Applications for non-compact M.

7. Higher derivative formulas.

Appendix A: Differential geometric notation and identities.

Appendix B: Semigroup results.

References.

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0022-12360135.00

Copyright2001 by Academic Press All rights of reproduction in any form reserved.

1This research was partially supported by NSF Grants DMS 96-12651 and DMS 9971036.

2Research supported by Deutsche Forschungsgemeinschaft and SFB 256 (University of Bonn).

1. INTRODUCTION

Let M be an n-dimensional oriented Riemannian manifold (not necessarily complete) without boundary andEa smooth Hermitian vector bundle overM. Denote by1(E) the smooth sections ofE. Further assume that Lis a second order elliptic differential operator on 1(E) whose prin- ciple symbol is the dual of the Riemannian metric onMtensored with the identity section of Hom(E). In this paper we derive stochastic calculus for- mulas for De^tL: and e^tLD: where :#1(E) and D is an appropriately chosen first order differential operator on 1(E).

As an example of the kind of formula found in this paper, let us consider one representative special case. Namely suppose that Mis a compact spin manifold, E=S is a spinor bundle over M, D is the Dirac operator on 1(S) andL=&D². Let scal denote the scalar curve of M. Then

(e^&TD22D:)(x)=(De^&TD22:)(x)

=1

TE[e^{&(18)0Tscal(Xt(x))dt}#_BT^&1_T :(X_T(x))],

whereX_t(x) is a Brownian motion onMstarting atx#M, _tis stochastic parallel translation alongX_v(x) in S relative to the spin connection, B_t is aT_xM-valued Brownian motion associated to X_t(x) and#_BTis the Clifford multiplication of B_T on S_x. This result is described in more detail in Section 5.2 below.

It is also possible to get a formula for D²e^&TD22: by iterating a minor generalization of the previous formula. For example if 0<T₁<Tthen

(D²e^&TD22:)(x)

= 1

T₁(T&T₁)E[e^{&(18)0Tscal(Xt(x))dt}#_BT

1#_BT&BT

1^&1_T :(X_T(x))];

see Theorem 7.4 below.

There are a number of other related approaches to derivative formulas in the literature; see for example Norris [44], Elworthy and Li [24, 26], Stroock and Turetsky [54, 55] and Hsu [34, 35].

Let us end the introduction with a short outline of the paper. Section 2 introduces the basic stochastic and differential geometric notation along with the standing Assumption 1 used throughout the paper. Geometric examples satisfying Assumption 1 are also presented here. More details on the notation and the examples of Section 2 may be found in Appendix A.

Section 3 introduces a number of local martingales associated to the geometric data of Section 2. Theorem 3.7 and Corollary 3.9 of this section are fundamental to the rest of the paper. In Section 4 we describe some general heat equation derivative formulas under the assumption that the local martingales introduced in Section 3 are in fact martingales, see in par- ticular Eqs. (4.22) and (4.23).

Section 5 illustrates our results for compact M. Dirac operators are covered in Section 5.2 and the differential d and co-differential d* are treated in Section 5.3. Heat equation derivative formulas involving covariant derivatives are covered in Section 5.4, and Section 5.5 in the con- text of 1-forms. In Section 6 we show how to relax the compactness assumption on M. Some technical heat semi-group properties used in the section are gathered in Appendix B.

Section 7 is devoted to higher derivative formulas. The ideas are illustrated in Theorems 7.4 and 7.7. Theorem 7.4 gives a formula for the square of the Dirac operator on spinor valued solutions to the heat equa- tion while Theorem 7.7 gives a formula for the Hessian of a solution to the scalar heat equation.

2. GENERAL STOCHASTIC AND GEOMETRIC NOTATION

2.1. Brownian Motion on M

Let M be an n-dimensional oriented Riemannian manifold (not necessarily complete) without boundary, ( } , } ) be the Riemannian metric onM, and {^TMbe the LeviCivita covariant derivative onTM. Let (0, F, [F_t]_t0, P) be a filtered probability space satisfying the usual hypothesis, and for eachx#Mlet[X_t(x):t<`(x)] be anM-valued Brownian motion on (0, F, [F<sub>t</sub>]<sub>t0</sub>, P), starting from x, with possibly finite lifetime `(x).

Recall that X_t=X_t(x) is said to be an M-valued Brownian motion provided that Xis a diffusion process such that

M_t^f:=f(X_t)&f(X₀)&¹₂

|

is a real local martingale for every f#C(M). Here 2f denotes the Riemannian Laplacian of f.

2.2. Covariant Derivatives and Parallel Translation

Let EÄM and EÄM be two vector bundles over the Riemannian manifold M. Further assume that E and E are equipped with covariant

derivatives,{^Eand {^Erespectively. In the case that EandE are Rieman- nian vector bundles with fiber metrics ( } , } )_E and ( } , } )_Erespectively, we will always assume that {^E and {^E are metric compatible covariant derivatives. Given a smooth curve _: [0,[ÄM, let

^TM_t (_):T_{_(0)}MÄT_{_(t)}M, ^E_t(_):E_{_(0)}ÄE_{_(t)} and ^E_t(_):E_{_(0)}ÄE_{_(t)}

denote parallel translation along_ up to time t relative to {^TM, {^E, and {^Erespectively. The correspondingstochastic parallel translationsalong the Brownian motion X_v(x) will simply be denoted by ^TM_t :T_xMÄT_Xt(x)M, ^E_t :E_xÄE_Xt(x) and^E_t: E_xÄE_Xt(x) respectively.

Notation 2.1. In what follows, let(} , }) denote the pairing between a vector space Vand its dual spaceV*. For a linear operatorC:VÄW, let C^tr: W*ÄV* denote the transpose of C. If V and W are inner product spaces, C*: WÄV will denote the adjoint of C relative to the inner products on V and W. The inner product on Vwill be denoted by ( } , } ) or ( } , } )_V.

The covariant derivatives {^TM, {^E, and{^E induce covariant derivatives on any vector bundle E over M which is constructed by taking tensor products of the bundlesTM, E, Eand their dual bundles. In order to sim- plify notation we will simply write{ for the induced covariant derivative on any such vector bundleEand similarly we will write_t:E_xÄE_Xt(x)for stochastic parallel translation andRfor the curvature tensor relative to this covariant derivative.

Example 2.2. Suppose that E=TME*E and thatv:!#E_x, then

_t(v:!)=(^TM_t v)((^E_t ^tr)^&1:)(^E_t !) (2.1)

and for a, b#T_xM,

R(a,b)(v:!)=R^TM(a,b)v:!

+v(&:bR^E(a,b))!+v:R^E(a,b)!, where (^E_t ^tr)^&1::=:b(^E_t)^&1andR^TM,R^E, andR^Eare the curvature ten- sors for {^TM, {^Eand {^E respectively. Our convention of denoting parallel translation and the curvature tensor as_tandRrespectively on all bundles

associated to TM, E, and E leads to the strange looking identities,

^tr_t =^&1_t andR^tr=&R. For example, onE=TME*E,

_t^tr=(^TM_t )^tr((_t^Etr)^&1)^tr(^E_t)^tr

=(^T*M_t )^&1(_t^E)^&1(^E*_t )^&1

which is ^&1_t on E*=T*MEE*. Similarly, R^tr=&R is a conse- quence of the fact that, in general, R^E*=&(R^E)^tr.

Definition 2.3. The orthogonal frame bundle ofMwill be denoted by O(M). Given a pointx#M, the principle bundleO(M) may be realized as m#MO_m(M), where

O_m(M) :=[u:T_xMÄT_mM|uis an isometry].

(The base point x will be suppressed from the notation since different x's lead to equivalent principle bundles.) Let?:O(M)ÄMbe the fiber projec- tion map defined by ?(u)=m if u#O_m(M) and let be the T_xM-valued one-form onO(M) defined by (!)=u^&1?

*!for all !#T_uO(M).

Definition 2.4 (Brownian Motion on T_xM). Associated to the Brownian motionX_t(x) onMis aT_xM-valued local martingaleB_t defined on [0,`(x)[ by the Fisk Stratonovich stochastic integral,

B_t:=

|

|

see [22, 28].

2.3. Laplacians and First Order Differential Operators

For a vector bundle E over M, let 1(E) denote the smooth sections of E. In case of a Riemannian vector bundle Eover M, let L²(E) denote the square-integrable sections relative to the Riemannian volume measure on MandL²-1(E) the square-integrable smooth sections ofE.

Definition 2.5 (Horizontal Laplacians). The horizontal Laplacians g: 1(E)Ä1(E) and g : 1(E)Ä1(E) are the second order differential operators given by

ga:=:

n

i=1

{²_eieia and g a~:=:

n

i=1

{²_eieia~,

where [e_i]ⁿ_i=1 is any local orthonormal frame ofTMand for a#1(E) {²_eieia:=({^E_ei)²a&{^E_{

ei

TMeia (2.2)

with{²_eieia~ being defined analogously. In other words, gis given as the following composition,

g:1(E)w^{Ä^E 1(T*ME) wwÄ^{T*ME 1(T*MT*ME)wÄ^tr 1(E).

Definition 2.6 A multiplication map from E to E is a smooth sec- tion m of the vector bundle Hom(T*ME,E)$TME*E. To each multiplication map m we define a first order differential operator D_m: 1(E)Ä1(E) by

D_ma=m{a. (2.3)

Notation 2.7. Forv#T_xM, let m_v# Hom(E_x,E_x) be given by m_v!:=m((v, } )!) #E_x

for all!#E_x. In the following we will typically describemby describingm_v forv#TM.

Definition 2.8 (Compatibility with{). A multiplication mapmis said to becompatible with{provided {m=0, i.e.

{^E_v(m_Xa)=m_({

v

TMX)a+m_X({^E_va)

for allX#1(TM), a#1(E), and v#TM.

Since a#T*_xMEx$Hom(TxM,E) may be written as a=ⁿ_i=1 (e_i, } )a(ei) where [e_i]ⁿ_i=1 is an orthonormal frame for T_xM, it follows that

ma=:

n

i=1

m_eia(e_i). (2.4)

In particular, the operatorD_m defined in Eq. (2.3) may be expressed as (D_ma)_x=:

n

i=1

m_ei{_eia, (2.5)

where a#1(E) and [e_i]ⁿ_i=1is an orthonormal frame for T_xM.

Definition 2.9. To each multiplication map m, there is a dual multi- plication mapm^trwhich is the smooth section of the bundle Hom(T*M E*, E*) determined by m^tr_v=(m_v)^trfor allv#TM. IfEandEare Rieman- nian vector bundles, the adjoint multiplication map m* from E to E is determined by m_v* :=(m_v)* for allv#TM.

Remark 2.10. It is easily checked that ifmis compatible with{then so is its dual m^tr and its adjoint m*. Moreover, if m is {-compatible, then ^&1_t m=m, ^&1_t m^tr=m^trand^&1_t m*=m*. More precisely

m_v=(^E_t)^&1m

t TMv^E_t , m^tr_v=(^E_t)^trm^tr

t

TMv(^E_t ^tr)^&1, and

m_v*=(^E_t)^&1m*

t TMv^E_t

for all v#T_xM. Also recall that when {^E and {^E are metric compatible covariant derivatives, then (^E_t)^&1=(^E_t)* and (^E_t)^&1=(^E_t)*.

In geometrically natural situations one is often presented with the follow- ing formalism.

Assumption 1. We suppose that m is a multiplication operator and L and L are given second order differential operators on 1(E) and 1(E) respectively which satisfy the following conditions.

1. The operatorsD:=D_m, L, andLobey the commutation relation:

DL=LD&\, for some\#1(Hom(E,E)).

2. The operators R:=g&L: 1(E)Ä1(E) and R :=g &L:

1(E)Ä1(E) are zeroth order operators, i.e. R and R are sections in 1(End(E)) and 1(End(E)) respectively.

2.4. Examples

Let us now gives some examples where Assumption 1 is satisfied. More detailed comments about these examples and the general setup may be found in Section A.3 of Appendix A.

Example 2.11 (Exterior Bundle Examples). Let 4T*M:=ⁿ_k=0 4^kT*M denote the exterior bundle overM, 0^k(M) denote the sections of 4^kT*Mand0(M) :=ⁿ_k=00^k(M) be the space of differential forms over M. Letddenote the exterior differential on 0(M),d* be the co-differential and

2=&(d+d*)²=&(d*d+dd*)

be the de RhamHodge Laplacian on 0(M). We now give three related examples satisfying Assumption 1 with \=0.

1. Let E:=4^kT*M, E:=4^k+1T*Mand let m_v=C_vbe the exterior product or creation operator which is defined by C_v::=(v, } )7: for v#T_xM, :#4^kT*_xMand x#M. Then

D_C=d_k=d|0^k(M), L:=2_k:=2|0^k(M), L:=2_k+1=2|0^k+1(M)

satisfy Assumption 1 with \=0.

2. Let E:=4^kT*M, E:=4^k&1T*M and m_v=A_v be the interior product or annihilation operator which is defined byA_v::=:(v, } , ..., } ) for allv#T_xM, :#4^kT*_xMandx#M. Then

D_A= &d_k*=&d* |0^k(M), L=2_k=2|0^k(M), L=2_k&1=2|0^k&1(M)

satisfy Assumption 1 with \=0.

3. Let E=E:=4T*M, L=L:=2 and let m=# be the ``Clifford'' multiplication defined by#=C&A. Then

D_#=d+d*, L=L=2 satisfy the Assumption 1 with\=0.

Item 3 of the last example generalizes to differential forms with values in a vector bundle. This is described in the next example.

Example 2.12 (Vector-valued Forms). Let SÄMbe a Euclidean vec- tor bundle overM(with fiber inner product denoted by ( } , } )_S) equipped with a metric compatible covariant derivative{^S. LetE=E=4T*MS and letm be the Clifford multiplication #determined by#=C&A, where as above C_va:=(v, } )7a and A_va=a(v, } , ..., } ) for all v#T_xM, a#4^kT*_xMS andx#M(see Section A.1 of Appendix A for our conven- tions). Then D_#=d_{+d*_{ where d_{ and d*_{ are the covariant differential and co-differential on A(E) :=1(4T*MS). Then L=L:= &D²_#=

&(d_{+d*_{)² and#satisfy Assumption 1 with \=0.

Example 2.13 (Dirac Operator on a Spin Manifold). Assume now that Mis a spin manifold and SÄMa spinor bundle over M. Let {^S denote the spin connection on S and let S=S. Further let m_v:=#_v: denote the Clifford action of v#T_xM on :#S_x. Then D_# is the Dirac operator on 1(S) and L=L:=&D²_# satisfy Assumption 1 with \=0 and R=R =

1

4 scal, where scal is the scalar curvature ofM. For details on this and the next example, see (for instance) Theorem 3.52 on p. 126 of [4].

Example 2.14 (Twisted Dirac Operators). The spinor bundle Sis ten- sored with an auxiliary RiemannianHermitian vector bundleF overMto give a Dirac operator of the form

D:1(SF) wwÄ^{SF 1(T*MSF) wwÄ^#1 1(SF), (2.6) where as in the previous example # denotes Clifford multiplication. Then againL=L=&D² satisfy Assumption 1 with \=0.

Proposition 2.15 ({ on a General Vector Bundle). Let EÄM be a vector bundle with covariant derivative{^E,R^Ebe the curvature tensor of{^E,

E:=T*ME$Hom(TM,E)

and m:T*MEÄE=T*ME be the identity map considered as a multi- plication map from E to E. (Notice that D_m={^E.)GivenR#1(End(E))let R =Ric^tr1_E&2R^E} +1_T*MR#1(End(E)) (2.7) and

\={}R^E+{^End(E)R#1(Hom(E,E)), (2.8)

where for '#E_x=T*_xME_x$Hom(TxM,E_x), v#T_xM, a#E_x, and [e_i] again an orthogonal frame for T_xM,

(R^E}')(v)=:

n

i=1

R^E(v,e_i)'(e_i), (2.9)

({}R^Ea)(v)#:

n

i=1

({_eiR^E)(e_i,v)a, (2.10)

({^End(E)Ra)(v)=({^End(E)_v R)a, (2.11)

and Ric is the Ricci tensor of M,

Ricv#:

n

i=1

R^TM(v,e_i)e_i. (2.12)

Then L:=g&R, L=g &R , \, and m=id satisfy Assumption1.

Proof. By Eq. (A.18) of Appendix A,

{^E_vLa={^E_v(g&R)a={^E_vga&({^End(E)_v R)a&R{^E_va

=(g {^Ea)(v)&{_Ricva+2 :

n

i=1

R^E(v,e_i){_eia

&({}R^Ea)(v)&({^End(E)_v R)a&R{^E_va

=(L{^Ea)(v)&(\a)(v). K 2.5. The Adjoint of D_m

We will end this section by computing the formal adjoint ofD_mwhenm is compatible with{.

Lemma 2.16. Suppose that E and E are vector bundles with fiber metrics ( } , } )_E and ( } , } )_E and that {^E and {^E are metric compatible covariant derivatives on E and E respectively. Also assume that m is a {-compatible multiplication map and let m* be the adjoint multiplication map. Then the operator D_m*: 1(E)Ä1(E)is the formal adjoint of &D_m. More precisely,

|

|

for all smooth sections S#1(E)and T#1(E)such that ST has compact support.

Proof. For S and T fixed as above, let X be the compactly supported vector field onMdetermined by

(m_vS,T)_E=(X,v)_TM

for allv#TM. Let [e_i]ⁿ_i=1 be a local orthonormal frame on M, then e_i(X,e_i)_TM=e_i(m_eiS,T)_E=({_ei(m_eiS),T)_E+(m_eiS,{_eiT)_E

=(m_{

eieiS,T)_E+(m_ei{_eiS,T)_E+(S,m*_ei{_eiT)_E

=(X,{_eie_i)_TM+(m_ei{_eiS,T)_E+(S,m*_ei{_eiT)_E. Hence

div(X)=:

n

i=1

({_eiX,e_i)_TM=:

n

i=1

[e_i(X,e_i)_TM&(X,{_eie_i)_TM]

=:

n

i=1

[(m_ei{_eiS,T)_E+(S,m*_ei{_eiT)_E]

=(D_mS,T)_E+(S,D_m*T)_E.

The lemma now follows by integrating this last expression over M and using Stokes' theorem to conclude that

|

|

Remark 2.17. An analogous proof shows that if m is a {-compatible multiplication map, then

|

|

for all smooth sections S#1(E) and T#1(E*) such that ST has compact support.

3. LOCAL MARTINGALES

In this section, suppose that mis a multiplication map and L=g&R and L=g &R are second order differential operators on 1(E) and 1(E) satisfying Assumption 1, i.e. \:=LD_m&D_mL#1(Hom(E,E)). Let Q_t# End(E_x) and Q_t# End(E_x) denote the solutions to the ordinary dif- ferential equations,

d

dtQ_t=&1

2Q_tR_t with Q₀=id_Ex (3.1) and

d

dtQ_t=&1

2Q_tR_t with Q₀=id_Ex, (3.2) whereR_t:=(^E_t)^&1R^E_t and R_t:=(^E_t)^&1R ^E_t are linear operators on E_xand E_xrespectively.

Notation 3.1. A time dependent section [a_t]_0t<T of E is said to be smooth if (t,x)[a_t(x) is infinitely differentiable for (t,x) # ]0,T[_M with derivatives extending continuously to [0,T[_M.

Proposition 3.2. Let m, L and L be as in Assumption1 and Q and Q be defined by Eqs. (3.1) and (3.2) respectively. Suppose further that

[a_t]_0t<T is a smooth time dependent section of E which satisfies the backwards heat equation,

ta_t+1

2La_t=0. (3.3)

For t in the stochastic interval [0,`(x)7T[, let

N_t:=Q_t^&1_t a_t(X_t(x)) (3.4)

and

N_t:=Q_t^&1_t Da_t(X_t(x)). (3.5)

Then the stochastic differentials of N_t and N_t are given by

dN_t=Q_t^&1_t {_t^TM_dBta_t(X_t(x)) (3.6)

and

dN_t=Q_t^&1_t {_tdBtDa_t(X_t(x))+¹₂Q_t^&1_t (\a_t)(X_t(x))dt, (3.7) where \:=LD_m&D_mL#1(Hom(E,E))as in Assumption1.

Proof. The proof is an application of Ito^'s lemma and the commutation relations in Assumption 1. In more detail we have,

dN_t=Q_t^&1_t {_tdBta_t(X_t(x))

+¹₂

\

+

=Q_t^&1_t {_tdBta_t(X_t(x)),

where in the last equality we used the identity,

&R_t^&1_t +^&1_t g=^&1_t (g&R)=^&1_t L.

Similarly,

dN_t=Q_t^&1_t {_tdBtDa_t(X_t(x))

+¹₂

\

+

=Q_t^&1_t {_tdBtDa_t(X_t(x))+¹₂Q_t^&1_t (\a)(X_t(x))dt because

&R_t^&1_t D+^&1_t g D&^&1_t DL=^&1_t (LD&DL)=^&1_t \. K

3.1. Dual Pairs

Definition 3.3. A pair of adapted continuous processes l_t#E*_x and l_t #E*_xis called adual pairifZ_t=( N_t,l_t)&( N_t,l_t) is a local martingale.

Theorem 3.4. Suppose thatl_t#E*_x andl_t#E*_xare continuous semimar- tingales such that

dl_t=:_tdB_t+;_tdt and

dl_t=:~_tdB_t+;_tdt,

where :_t # Hom(T_xM,E*_x), :~_t# Hom(T_xM,E*_x), ;_t#E*_x and ;_t#E*_x are predictable processes. Then l and l is a dual pair provided

:~#0, (3.8)

1

2\^tr_tQ^tr_t l_t=_tQ^tr_t ;_t, and (3.9) m^tr_tvtQ^tr_t ;_t=_tQ^tr_t :_tv for each v#T_xM. (3.10) Proof. Let Z_t=( N_t,l_t)&( N_t,l_t) and write dX&dY if X&Y is a local martingale. Computing dZ_t using Proposition 3.2 gives

dZ_t&(dN_t,l )+( N_t,dl_t)+( dN_t,dl_t)&( N_t,dl_t)&( dN_t,dl_t)

&(¹₂Q_t^&1_t (\a_t)(X_t(x)),l_t) dt

+( N_t,;_t) dt+( Q_t^&1_t {_tdBtDa_t(X_t(x)),:~_tdB_t)

&( N_t,;_t) dt&( Q_t^&1_t {_tdBta_t(X_t, (x)),:_tdB_t)

=¹₂( Q_t^&1_t (\a_t)(X_t(x)),l_t) dt +:

n

i=1

( Q_t^&1_t m_tei{_teia_t(X_t(x)),;_t) dt

+:

n

i=1

( Q_t^&1_t {_teiDa_t(X_t(x)),:~_te_i) dt

&( Q_t^&1_t a_t(X_t(x)),;_t) dt

&:

n

i=1

( Q_t^&1_t {_teia_t(X_t(x)),:_te_i) dt.

Keeping in mind that_t=(^&1_t )^tr(i.e.^E*_t =((^E_t)^&1)^tr) we find, by com- paring terms involving a, {_teia, and {_teiDa, that dZ_t&0 provided that Eqs. (3.8), (3.9) and (3.10) are satisfied. K

The upshot of the previous theorem is that to make a dual pair we should choose :~=0 (i.e. dl_t=;_tdt),

;_t=¹₂(Q^tr_t)^&1_t^&1\^tr_tQ^tr_t l_t and

:_tv=(Q^tr_t)^1&1_t m^tr_tvtQ^tr_t ;_t for each v#T_xM.

Because the processes Q^tr_t and Q^tr_t will arise often in the sequel, it is convenient to introduce the following notation.

Notation 3.5. Let Q_t:=Q^tr_t # End(E*_x) and Q_t:=Q^tr_t # End(E*_x). Taking the transposes of Eqs. (3.1) and (3.2) shows thatQandQ solve the follow- ing ordinary differential equations:

d

dtQ_t=&1

2R^tr_tQ_t with Q₀=id_E*x (3.11) and

d

dtQ_t=&1

2R ^tr_tQ_t with Q₀=id_E*x. (3.12) Definition 3.6 (Finite Energy Process). Let V be a finite dimensional vector space. A V-valued process [l_s]_{s# [0,T[} is said to be a finite energy process provided l is adapted and (on a set of measure one) sÄl_s is absolutely continuous and^T₀ |dl_sds|²_Vds<, where | } |_Vdenotes any one of the equivalent norms onV. If in addition there is a p# [1,) such that

E

_\ ^|

+

&

then we say that [l_s]_{s# [0,T[} is anL^p-finite energy process.

We have the following immediate consequence of Theorem 3.4. In this theorem and in the rest of the paper we will write l rather than l for a finite energy process with values inE*_x.

Theorem 3.7. Let a, N, and N be as in Proposition3.2, l_t #E*_x be a finite energy process and define the E*_x-valued process,

U^l_t :=

|

|

where \_s:=^&1_s \(X_s(x))_sand

\^tr_s=^tr_s\^tr(X_s(x))(_s^&1)^tr=^&1_s \^tr(X_s(x))_s. (3.14) (See the remarks at the end of Example2.2 explaining the identity ^tr_s= ^&1_s .)Then

Z^l_t :=( N_t,l_t)&( N_t,U^l_t) (3.15) is a local martingale on [0,`(x)7T[ and

dZ^l_t=( Q_t^&1_t {_tdBtDa_t(X_t(x)),l_t)

&( Q_t^&1_t {_tdBta_t(X_t(x)),U^l_t)

&( N_t,Q^&1_t ^&1_t m^tr_tdBttQ_tl$_t). (3.16) Proof. Because of Theorem 3.4 and the comments after its proof, we need only prove Eq. (3.16). Since we already know that Z^l_t is a local mar- tingale, we need only keep track of those terms in dZ^l_t which depend linearly on dB_t. Hence Eq. (3.16) follows from the identity

dZ^l_t=( dN_t,l_t)+( N_t,dl_t)+( dN_t,dl_t)

&( dN_t,U^l_t)&( N_t,dU^l_t)&( dN_t,dU^l_t) and Eqs. (3.6), (3.7) and (3.13). K

By Remark 2.10, if the multiplication operator m is compatible with{, then

m^tr_v=^tr_sm^tr_sv(_s^&1)^tr=^&1_s m^tr_svs

for allv#T_xM. Hence, the formula forU^l_t in Eq. (3.13) simplifies to U^l_t=

|

|

Remark 3.8. Suppose that l_t#E_x is a finite energy process. Under the further assumption that Eand Eare Riemannian vector bundles and mis compatible with{, the results of Theorem 3.7 remains true withU^l_t andZ^l_t defined by

U^l_t :=

|

|

and

Z^l_t :=(N_t,l_t)_E&(N_t,U^l_t)_E, (3.19) where\*_s:=^&1_s \*(X_s(x))_s.

We will finish this section with another equivalent version of Theorem 3.7.

Corollary 3.9. Suppose that m is compatible with { (for simplicity), k_t is an E*_x-valued finite energy process and a and N are as in Proposition 3.2. Also define:

N_t :=^&1_t Da_t(X_t(x)), (3.20)

U^k_t :=

|

|

and

Z^k_t :=( N_t,k_t)&( N_t,U^k_t). (3.22) Then Z^k_t is a local martingale on[0,`(x)7T[whose Ito^ differential is given by

dZ^k_t=( ^&1_t {_tdBtDa_t(X_t(x)),k_t)

&( Q_t^&1_t {_tdBta_t(X_t(x)),U^k_t)

&( Q^&1_t N_t,m^tr_dBt(k$_t+¹₂R ^tr_tk_t)). (3.23) Proof. Comparing Eqs. (3.5) and (3.20) shows that N_t=Q_tN_t. Define the finite energy processl_sby l_s:=(Q_s)^&1k_sso thatk_s=Q_sl_s. Because

k$_s=Q_sl$_s&₂¹R ^tr_sQ_sl_s=Q_sl$_s&¹₂R ^tr_sk_s, (3.24) it follows from Eq. (3.17) thatU^k_t :=U^l_t. This identity and

(N_t,l_t)=(Q_tN_t, (Q_t)^&1k_t)=( N_t,k_t)

shows that Z^k_t=Z^l_t as well. By Theorem 3.7 Z^k_t=Z^l_t is a local martingale on [0,`(x)7T[ and by Eq. (3.16) and Eq. (3.24) the differential ofZ^k_t is given by

dZ^k_t=(Q_t^&1_t {_tdBtDa_t(X_t(x)), (Q_t)^&1k_t)

&( Q_t^&1_t {_tdBta_t(X_t(x)),U^k_t)

&( N_t,Q^&1_t m^tr_dBt(k$_t+¹₂R ^tr_tk_t)) which is the same as Eq. (3.23). K

4. THE FUNDAMENTAL DERIVATIVE FORMULAS

In this section, we will give a number of derivative formulas under the assumption that the local martingales in Proposition 3.2 and Theorem 3.7 are in fact martingales. In the later sections we will verify this hypothesis in a number of cases. To simplify notation, we will from now on assume that the multiplication map mis compatible with the covariant derivative {. Thus the assumptions which are in force in the remainder of the paper are:

Assumption 2 (Standing Assumptions). Let EandE be vector bundles endowed with covariant derivatives. Assume that L, L, and m are given satisfying Assumption 1 and that m is compatible with the covariant derivatives.

All of our examples in Section 3 satisfy this assumption. The following theorem contains the basic derivative formulas in this paper.

Theorem 4.1 (Basic Derivative Formula I). Let a be a solution to Eq. (3.3)and Q and Q be given by (3.1) and (3.2). Also let { be a stopping time bounded by T< such that {<`(x) and let l_t#E*_x be a finite energy process on the stochastic interval [0,{]. Assume that { and l have been chosen such that

E|( Q_{^&1_{ Da_{(X_{(x)),l_{)| <, E|( Q_{^&1_{ a_{(X_{(x)),U^l_{)| <, where U^l is defined in Eq. (3.17). Further assume (Z^l)^{_t:=Z^l_t7{ is a mar- tingale where Z^l_t is the local martingale in Eq. (3.15).Then

E[( Da₀(x),l₀)]=E[( Q_{^&1_{ Da_{(X_{(x)),l_{)]

&E[( Q_{^&1_{ a_{(X_{(x)),U^l_{)]. (4.1) Therefore,

1. if l_{=0and l₀=!#E*_x then

( Da₀(x),!)=&E[(Q_{^&1_{ a_{(X_{(x)),U^l_{)], (4.2) 2. or ifl₀=0 and l_{=!#E*_x(where ! may be random here) then

E[( Q_{^&1_{ Da_{(X_{(x)),!)]=E[( Q_{^&1_{ a_{(X_{(x)),U^l_{)]. (4.3) Proof. Since we have assumed that the expression (Z^l)^{in Eq. (3.15) of Theorem 3.7 is a martingale, it follows thatEZ^l₀=E(Z^l)^{_T=EZ^l_{, i.e.

E( N_{,l_{)&E( N₀,l₀)=E(N_{,U^l_{)

which along with the definitions of N, N in Proposition 3.2 proves Eq. (4.1). K

Remark 4.2. Suppose that l_t#E_x is a finite energy process. Under the further assumption that Eand Eare Riemannian vector bundles and mis compatible with {, Theorem 4.1 remains valid with all pairings (} , }) replaced by the appropriate inner products ( } , } )_Eor ( } , } )_Eand all trans- poses replaced by adjoints andU^l defined as in Eq. (3.18) above.

Theorem 4.3 (Basic Derivative Formula II). Let a be a solution to Eq. (3.3)and Q be given by (3.1).Also let { be a stopping time bounded by T< such that {<`(x) and let k_t#E*_x be a finite energy process on the stochastic interval[0,{].Assume that { and k have been chosen such that

E|( ^&1_{ Da_{(X_{(x)),k_{)| <, E|( Q_{^&1_{ a_{(X_{(x)),U^k_{)| <, where U^k is defined in Eq. (3.21). Further assume (Z^k)^{_t:=Z^k_t7{ is a mar- tingale where Z^k_t is the local martingale in Eq. (3.22). Then

E[( Da₀(x),k₀)]=E[( ^&1_{ Da_{(X_{(x)),k_{)]

&E[( Q_{^&1_{ a_{(X_{(x)),U^k_{)]. (4.4) Therefore,

1. if k_{=0and k₀=!#E*_xthen

( Da₀(x),!)=&E[(Q_{^&1_{ a_{(X_{(x)),U^k_{)], (4.5) 2. or if k₀=0 and k_{=!#E*_x(where ! may be random here) then

E[( ^&1_{ Da_{(X_{(x)),!)]=E[(Q_{^&1_{ a_{(X_{(x)),U^k_{)]. (4.6) Proof. The proof is the same as Theorem 4.1 except that we use Corollary 3.9 in place of Theorem 3.7. K

The formulas appearing in Theorem 3.7 take on a simpler form when m is compatible with{ andl_t is of the forml_t=l_t! wherel_t is anR-valued finite energy process and !#E*. To write out these formula, it is con- venient to introduce the processV^l_t # Hom(E_x,E_x) by

V^l_t=

|

|

where \_s:=^&1_s \(X_s(x))_s and the E_x-valued process

Z^l_t :=l_tN_t&V^l_tN_t. (4.8)