HEAT EQUATION DERIVATIVE FORMULAS FOR VECTOR BUNDLES

BRUCE K. DRIVER^{†} AND ANTON THALMAIER^{∗}

Abstract. We use martingale methods to give Bismut type derivative formulas for differentials and co-differentials of heat semigroups on forms, and more generally for sections of vector bundles. The formulas are mainly in terms of Weitzenböck curvature terms, in most cases derivatives of the curvature are not involved. In particular, our results improve the formula in Driver [19] for logarithmic derivatives of the heat kernel measure on a Riemannian manifold. Our formulas also include the formulas in Elworthy and Li [26].

Contents

1. Introduction 1 2. General Stochastic and Geometric Notation 2 3. Local Martingales 9 4. The Fundamental Derivative Formulas 13 5. Applications for Compact M 18 6. Applications for Non-Compact M 32 7. Higher Derivative Formulas 34 8. Appendix I: Differential Geometric Notation and Identities 38 9. Appendix II: Semigroup Results 45 References 50

Keywords: Heat kernel measure, Malliavin Calculus, Bismut formula, integration by parts, Dirac operator, de Rham-Hodge Laplacian, Weitzenböck decomposition

AMS 2000 Subject Classification: Primary 60H30; Secondary 58J65

1. Introduction

Let M be an n-dimensional oriented Riemannian manifold (not necessarily complete) without boundary and E a smooth Hermitian vector bundle over M. Denote by Γ(E) the smooth sections of E. Further assume that L is a second order elliptic differential operator on Γ(E) whose principle symbol is the dual of the Riemannian metric on M tensored with the identity section of Hom(E). In this paper we derive stochastic calculus formulas for De^{tL}α and e^{tL}Dα where α ∈ Γ(E) and D is an appropriately chosen first order differential operator on Γ(E).

Date: March 14, 2000 File: bismut21.tex.

†This research was partially supported by NSF Grant DMS 96-12651 and DMS 9971036.

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

As an example of the kind of formula found in this paper, let us consider one representative special case. Namely suppose that M is a compact spin manifold, E = S is a spinor bundle over M, D is the Dirac operator on Γ(S) and L = −D^{2}. Let scal denote the scalar curvature of M. Then

(e^{−T D^{2}/2}Dα)(x) = (De^{−T D^{2}/2}α)(x)

= \frac{1}{T} 𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} γ_{B_{T}} //_{T}^{−1}α(X_{T} (x))] ,

where X_{t}(x) is a Brownian motion on M starting at x ∈ M, //_{t} is stochastic parallel translation along X.(x) in S relative to the spin connection, B_{t} is a T_{x}M-valued Brownian motion associated to X_{t}(x) and γ_{B_{T}} is 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 D^{2}e^{−T D^{2}/2}α by iterating a minor generalization of the previous formula. For example if 0 < T_{1} < T then

(D^{2}e^{−T D^{2}/2}α)(x)

= \frac{1}{T_{1}(T − T_{1})}𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} γ_{B_{T_{1}}} γ_{B_{T} −B_{T_{1}}} //_{T}^{−1} α(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 [35, 34].

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 I.

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 particular 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 context 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 II.

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 equation 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 on M, and ∇^{T M} be the Levi-Civita covariant derivative on T M. Let (Ω, F, {F_{t}}_{t≥0}, ℙ) be a filtered probability space satisfying the usual hypothesis, and for each x ∈ M let {X_{t}(x) : t < ζ(x)} be an M-valued Brownian motion on (Ω, F, {F_{t}}_{t≥0}, ℙ), starting from x, with possibly finite lifetime ζ(x). Recall that X_{t} = X_{t}(x) is said to be an M-valued Brownian motion provided that X is a diffusion process such that

M_{t}^{f} := f(X_{t}) − f(X_{0}) − \frac{1}{2} ∫_{0}^{t} Δf(X_{s}) ds on \{t < ζ(x)\}

is a real local martingale for every f ∈ C^{∞}(M). Here Δf denotes the Riemannian Laplacian of f.

2.2. Covariant Derivatives and Parallel Translation. Let E → M and Ẽ → M be two vector bundles over the Riemannian manifold M. Further assume that E and Ẽ are equipped with covariant derivatives, ∇^{E} and ∇Ẽ respectively. In the case that E and Ẽ are Riemannian vector bundles with fiber metrics (·, ·)_{E} and (·, ·)Ẽ respectively, we will always assume that ∇^{E} and ∇Ẽ are metric compatible covariant derivatives. Given a smooth curve σ : [0, ∞[ → M, let

//_{t}^{T M}(σ) : T_{σ(0)}M → T_{σ(t)}M, //_{t}^{E}(σ) : E_{σ(0)} → E_{σ(t)} and //Ẽ ; t(σ) : Ẽ_{σ(0)} → Ẽ_{σ(t)}

denote parallel translation along σ up to time t relative to ∇^{T M}, ∇^{E}, and ∇Ẽ respectively. The corresponding stochastic parallel translations along the Brownian motion X.(x) will simply be denoted by //_{t}^{T M} : T_{x}M → T_{X_{t}(x)}M, //_{t}^{E} : E_{x} →

E_{X_{t}(x)} and //Ẽ ; t : Ẽ_{x} → Ẽ_{X_{t}(x)} respectively.

Notation 2.1. In what follows, let ⟨·, ·⟩ denote the pairing between a vector space V and its dual space V^{∗}. For a linear operator C : V → W, let C^{tr} : W^{∗} → V^{∗} denote the transpose of C. If V and W are inner product spaces, C^{∗} : W → V will denote the adjoint of C relative to the inner products on V and W. The inner product on V will be denoted by (·, ·) or (·, ·)_{V}.

The covariant derivatives ∇^{T M}, ∇^{E}, and ∇Ẽ induce covariant derivatives on any vector bundle E over M which is constructed by taking tensor products of the bundles T M, E, Ẽ and their dual bundles. In order to simplify notation we will simply write ∇ for the induced covariant derivative on any such vector bundle E and similarly we will write //_{t} : E_{x} → E_{X_{t}(x)} for stochastic parallel translation and R for the curvature tensor relative to this covariant derivative.

Example 2.2. Suppose that E = T M ⊗ E^{∗} ⊗ Ẽ and that v ⊗ α ⊗ ξ ∈ E_{x}, then

//_{t} (v ⊗ α ⊗ ξ) = (//_{t}^{T M}v) ⊗ ((//_{t}^{E tr})^{−1}α) ⊗ (//Ẽ ; tξ) (2.1)

and for a, b ∈ T_{x}M,

R(a, b)(v ⊗ α ⊗ ξ) = R^{T M}(a, b)v ⊗ α ⊗ ξ
+ v ⊗ (−α ∘ R^{E}(a, b)) ⊗ ξ + v ⊗ α ⊗ RẼ(a, b)ξ,

where (//_{t}^{E tr})^{−1}α := α ∘ (//_{t}^{E})^{−1} and R^{T M}, R^{E}, and RẼ are the curvature tensors for ∇^{T M}, ∇^{E} and ∇Ẽ respectively. Our convention of denoting parallel translation and the curvature tensor as //_{t} and R respectively on all bundles associated to T M, E, and Ẽ leads to the strange looking identities, //_{t}^{tr} = //_{t}^{−1} and R^{tr} = −R.

For example, on E = T M ⊗ E^{∗} ⊗ Ẽ,

//_{t}^{tr} = (//_{t}^{T M})^{tr} ⊗ ((//_{t}^{E tr})^{−1})^{tr} ⊗ (//Ẽ ; t)^{tr}
= (//_{t}^{T^{∗}M})^{−1} ⊗ (//_{t}^{E} )^{−1} ⊗ (//Ẽ^{∗} ; t)^{−1}

which is //_{t}^{−1} on E^{∗} = T^{∗}M ⊗ E ⊗ Ẽ^{∗}. Similarly, R^{tr} = −R is a consequence of the fact that, in general, R^{E^{∗}} = −(R^{E})^{tr}.

Definition 2.3. The orthogonal frame bundle of M will be denoted by O(M). Given a point x ∈ M, the principle bundle O(M) may be realized as ∪_{m∈M}O_{m}(M), where

O_{m}(M) := {u : T_{x}M → T_{m}M | u is an isometry}.

(The base point x will be suppressed from the notation since different x’s lead to equivalent principle bundles.) Let π : O(M) → M be the fiber projection map defined by π(u) = m if u ∈ O_{m}(M) and let ϑ be the T_{x}M-valued one-form on O(M) defined by ϑ(ξ) = u^{−1}π_{∗}ξ for all ξ ∈ T_{u}O(M).

Definition 2.4 (Brownian Motion on T_{x}M). Associated to the Brownian motion X_{t}(x) on M is a T_{x}M-valued local martingale B_{t} defined on [0, ζ(x)[ by the Fisk Stratonovich stochastic integral,

B_{t} := ∫_{0}^{t} //_{s}^{−1}δX_{s}(x) := ∫_{0}^{t} ϑ(δ//_{s}^{T M}),

see [28], [22].

2.3. Laplacians and First Order Differential Operators. For a vector bundle E over M, let Γ(E) denote the smooth sections of E. In case of a Riemannian vector bundle E over M, let L^{2}(E) denote the square-integrable sections relative to the Riemannian volume measure on M and L^{2}-Γ(E) the square-integrable smooth sections of E.

Definition 2.5 (Horizontal Laplacians). The horizontal Laplacians □ : Γ(E) → Γ(E) and ˜□ : Γ(Ẽ) → Γ(Ẽ) are the second order differential operators given by

□a := ∑_{i=1}^{n} ∇_{e_{i}⊗e_{i}}^{2}a and ˜□ã := ∑_{i=1}^{n} ∇_{e_{i}⊗e_{i}}^{2} ã,

where {e_{i}}_{i=1}^{n} is any local orthonormal frame of T M and for a ∈ Γ(E)

∇_{e_{i}⊗e_{i}}^{2}a := (∇_{e_{i}}^{E})^{2}a − ∇_{∇_{e_{i}}^{T M} e_{i}}^{E}a (2.2)

with ∇_{e_{i}⊗e_{i}}^{2} ã being defined analogously. In other words, □ is given as the following composition,

□: Γ(E) ⟶^{∇^{E}} Γ(T^{∗}M ⊗ E) ⟶^{∇^{T^{∗}M⊗E}} Γ(T^{∗}M ⊗ T^{∗}M ⊗ E) \underrightarrow{tr} Γ(E).

Definition 2.6. A multiplication map from E to Ẽ is a smooth section m of the vector bundle Hom(T^{∗}M ⊗ E, Ẽ) ≅ T M ⊗ E^{∗} ⊗ Ẽ. To each multiplication map m we define a first order differential operator D_{m} : Γ(E) → Γ(Ẽ) by

D_{m}a = m∇a. (2.3)

Notation 2.7. For v ∈ T_{x}M, let m_{v} ∈ Hom(E_{x}, Ẽ_{x}) be given by

m_{v}ξ := m((v, ·) ⊗ ξ) ∈ Ẽ_{x}

for all ξ ∈ E_{x}. In the sequel we will typically describe m by describing m_{v} for v ∈ T M.

Definition 2.8 (Compatibility with ∇). A multiplication map m is said to be compatible with ∇ provided ∇m = 0, i.e.

∇Ẽ ; v(m_{X}a) = m_{(∇_{v}^{T M} X)}a + m_{X} (∇_{v}^{E}a)

for all X ∈ Γ(T M), a ∈ Γ(E), and v ∈ T M.

Since a ∈ T_{x}^{∗}M ⊗ E_{x} ≅ Hom(T_{x}M, E) may be written as a = ∑_{i=1}^{n}(e_{i}, ·) ⊗ a(e_{i}) where {e_{i}}_{i=1}^{n} is an orthonormal frame for T_{x}M, it follows that

ma = ∑_{i=1}^{n} m_{e_{i}}a(e_{i}). (2.4)

In particular, the operator D_{m} defined in Eq. (2.3) may be expressed as

(D_{m}a)_{x} = ∑_{i=1}^{n} m_{e_{i}}∇_{e_{i}}a, (2.5)

where a ∈ Γ(E) and {e_{i}}_{i=1}^{n} is an orthonormal frame for T_{x}M.

Definition 2.9. To each multiplication map m, there is a dual multiplication map m^{tr} which is the smooth section of the bundle Hom(T^{∗}M ⊗ Ẽ^{∗}, E^{∗}) determined by m_{v}^{tr} = (m_{v})^{tr} for all v ∈ T M. If E and Ẽ are Riemannian vector bundles, the adjoint multiplication map m^{∗} from Ẽ to E is determined by m_{v}^{∗} := (m_{v})^{∗} for all v ∈ T M.

Remark 2.10. It is easily checked that if m is compatible with ∇ then so is its dual m^{tr} and its adjoint m^{∗}. Moreover, if m is ∇-compatible, then //_{t}^{−1}m = m, //_{t}^{−1}m^{tr} = m^{tr} and //_{t}^{−1}m^{∗} = m^{∗}. More precisely

m_{v} = (//Ẽ ; t)^{−1} m_{//_{t}^{T M} v} //_{t}^{E},

m_{v}^{tr} = (//_{t}^{E})^{tr} m_{//_{t}^{T M} v}^{tr} (//_{t}^{Ẽ tr})^{−1},

and

m_{v}^{∗} = (//_{t}^{E})^{−1} m_{//_{t}^{T M} v}^{∗} //Ẽ ; t

for all v ∈ T_{x}M. Also recall that when ∇^{E} and ∇Ẽ are metric compatible covariant derivatives, then (//_{t}^{E})^{−1} = (//_{t}^{E})^{∗} and (//Ẽ ; t)^{−1} = (//Ẽ ; t)^{∗}.

In geometrically natural situations one is often presented with the following formalism.

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

1. The operators D := D_{m}, L, and L̃ obey the commutation relation:

DL = L̃D − ρ,

for some ρ ∈ Γ(Hom(E, Ẽ)).

2. The operators R := □ − L : Γ(E) → Γ(E) and R̃ := ˜□ − L̃ : Γ(Ẽ) → Γ(Ẽ) are zeroth order operators, i.e. R and R̃ are sections in Γ(End(E)) and Γ(End(Ẽ)) 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 8.3 of Appendix I .

Example 2.11 (Exterior Bundle Examples). Let ΛT^{∗}M := ⊕_{k=0}^{n}Λ^{k}T^{∗}M denote the exterior bundle over M, Ω^{k}(M) denote the sections of Λ^{k}T^{∗}M and Ω(M) := ⊕_{k=0}^{n}Ω^{k}(M) be the space of differential forms over M. Let d denote the exterior differential on Ω(M), d^{∗} be the co-differential and

Δ = −(d + d^{∗})^{2} = − (d^{∗}d + dd^{∗})

be the de Rham-Hodge Laplacian on Ω(M). We now give three related examples satisfying Assumption 1 with ρ = 0.

1. Let E := Λ^{k}T^{∗}M, Ẽ := Λ^{k+1}T^{∗}M and let m_{v} = C_{v} be the exterior product or creation operator which is defined by C_{v}α := (v, ·) ∧ α for v ∈ T_{x}M, α ∈ Λ^{k}T_{x}^{∗}M and x ∈ M. Then

D_{C} = d_{k} = d|Ω^{k}(M), L := Δ_{k} := Δ|Ω^{k}(M), L̃ := Δ_{k+1} = Δ|Ω^{k+1}(M)

satisfy Assumption 1 with ρ = 0.

2. Let E := Λ^{k}T^{∗}M, Ẽ := Λ^{k−1}T^{∗}M and m_{v} = A_{v} be the interior product or annihilation operator which is defined by A_{v}α := α(v, ·, …, ·) for all v ∈ T_{x}M, α ∈ Λ^{k}T_{x}^{∗}M and x ∈ M. Then

D_{A} = −d_{k}^{∗} = −d^{∗}|Ω^{k}(M), L = Δ_{k} = Δ|Ω^{k}(M), L̃ = Δ_{k−1} = Δ|Ω^{k−1}(M)

satisfy Assumption 1 with ρ = 0.

3. Let E = Ẽ := ΛT^{∗}M, L = L̃ := Δ and let m = γ be the “Clifford” multiplication defined by γ = C − A. Then

D_{γ} = d + d^{∗}, L = L̃ = Δ

satisfy the Assumption 1 with ρ = 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 S → M be a Euclidean vector bundle over M (with fiber inner product denoted by (·, ·)_{S}) equipped with a metric compatible covariant derivative ∇^{S}. Let E = Ẽ = ΛT^{∗}M ⊗ S and let m be the Clifford multiplication γ determined by γ = C − A, where as above C_{v} := (v, ·) ∧ a and A_{v}a = a(v, ·, …, ·) for all v ∈ T_{x}M, a ∈ Λ^{k}T_{x}^{∗}M ⊗ S and x ∈ M (see Section 8.1 of Appendix I for our conventions). Then D_{γ} = d_{∇} + d_{∇}^{∗} where d_{∇} and d_{∇}^{∗} are the covariant differential and co-differential on A(E) := Γ(ΛT^{∗}M ⊗ S). Then L = L̃ := −D_{γ}^{2} = −(d_{∇} + d_{∇}^{∗})^{2} and γ satisfy Assumption 1 with ρ = 0.

Example 2.13 (Dirac Operator on a Spin Manifold). Assume now that M is a spin manifold and S → M a spinor bundle over M. Let ∇^{S} denote the spin connection on S and let S̃ = S. Further let m_{v}α = γ_{v}α denote the Clifford action of v ∈ T_{x}M on α ∈ S_{x}. Then D_{γ} is the Dirac operator on Γ(S) and L = L̃ := −D_{γ}^{2} satisfy Assumption 1 with ρ = 0 and R = R̃ = \frac{1}{4} scal, where scal is the scalar curvature of M. 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 S is tensored with an auxiliary Riemannian/Hermitian vector bundle F over M to give a Dirac operator of the form

D: Γ(S ⊗ F) ⟶^{∇^{S⊗F}} Γ(T^{∗}M ⊗ S ⊗ F) ⟶^{γ⊗1} Γ(S ⊗ F), (2.6)

where as in the previous example γ denotes Clifford multiplication. Then again L = L̃ = −D^{2} satisfy Assumption 1 with ρ = 0.

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

Ẽ := T^{∗}M ⊗ E ≅ Hom(T M, E)

and m : T^{∗}M ⊗ E → Ẽ = T^{∗}M ⊗ E be the identity map considered as a multiplication map from E to Ẽ. (Notice that D_{m} = ∇^{E}.) Given R ∈ Γ(End(E)) let

(2.7)  R̃ = Ric^{tr} ⊗1_{E} − 2R^{E}· + 1_{T^{∗}M} ⊗ R ∈ Γ(End(Ẽ))

and

ρ = ∇ · R^{E} + ∇^{End (E)}R ∈ Γ(Hom(E, Ẽ)), (2.8)

where for η ∈ Ẽ_{x} = T_{x}^{∗}M ⊗ E_{x} ≅ Hom(T_{x}M, E_{x}), v ∈ T_{x}M, a ∈ E_{x}, and {e_{i}} again an orthonormal frame for T_{x}M,

(2.9)  (R^{E} · η) (v) = ∑_{i=1}^{n} R^{E}(v, e_{i})η(e_{i}),

(2.10)  (∇ · R^{E} a) (v) ≡ ∑_{i=1}^{n}(∇_{e_{i}}R^{E})(e_{i}, v)a,

(∇^{End (E)}R a)(v) = (∇_{v}^{End (E)}R)a, (2.11)

and Ric is the Ricci tensor of M,

Ric v ≡ ∑_{i=1}^{n} R^{T M}(v, e_{i})e_{i}. (2.12)

Then L := □ − R, L̃ = ˜□ − R̃, ρ, and m = id satisfy Assumption 1.

Proof. By Eq. (8.18) of Appendix I,

∇_{v}^{E}La = ∇_{v}^{E}(□ − R)a = ∇_{v}^{E}□a − (∇_{v}^{End (E)}R)a − R∇_{v}^{E}a

= ( ˜□∇^{E}a) (v) − ∇_{Ric v}a + 2 ∑_{i=1}^{n} R^{E}(v, e_{i})∇_{e_{i}}a

− (∇ · R^{E} a)(v) − (∇_{v}^{End (E)}R)a − R∇_{v}^{E}a

= (L̃∇^{E}a)(v) − (ρa) (v).

2.5. The Adjoint of D_{m}. We will end this section by computing the formal adjoint of D_{m} when m is compatible with ∇.

Lemma 2.16. Suppose that E and Ẽ are vector bundles with fiber metrics (·, ·)_{E} and (·, ·)Ẽ and that ∇^{E} and ∇Ẽ are metric compatible covariant derivatives on E and Ẽ respectively. Also assume that m is a ∇-compatible multiplication map and let m^{∗} be the adjoint multiplication map. Then the operator D_{m^{∗}} : Γ(Ẽ) → Γ(E) is the formal adjoint of −D_{m}. More precisely,

∫_{M}(D_{m}S, T)Ẽ d vol = − ∫_{M}(S, D_{m^{∗}}T)_{E} d vol

for all smooth sections S ∈ Γ(E) and T ∈ Γ(Ẽ) such that S ⊗ T has compact support.

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

(m_{v}S, T)Ẽ = (X, v)_{T M}

for all v ∈ T M. Let {e_{i}}_{i=1}^{n} be a local orthonormal frame on M, then

e_{i}(X, e_{i})_{T M} = e_{i} (m_{e_{i}}S, T)Ẽ = (∇_{e_{i}} (m_{e_{i}}S) , T)Ẽ + (m_{e_{i}}S, ∇_{e_{i}}T)Ẽ
= (m_{∇_{e_{i}} e_{i}}S, T )Ẽ + (m_{e_{i}}∇_{e_{i}}S, T)Ẽ + (S, m_{e_{i}}^{∗}∇_{e_{i}}T )_{E}
= (X, ∇_{e_{i}}e_{i})_{T M} + (m_{e_{i}}∇_{e_{i}}S, T)Ẽ + (S, m_{e_{i}}^{∗}∇_{e_{i}}T )_{E} .

Hence

div(X) = ∑_{i=1}^{n}(∇_{e_{i}}X, e_{i})_{T M} = ∑_{i=1}^{n}[e_{i}(X, e_{i})_{T M} − (X, ∇_{e_{i}}e_{i})_{T M}]

= ∑_{i=1}^{n} [(m_{e_{i}}∇_{e_{i}}S, T)Ẽ + (S, m_{e_{i}}^{∗}∇_{e_{i}}T )_{E}]
= (D_{m}S, T)Ẽ + (S, D_{m^{∗}}T)_{E} .

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

∫_{M} div(X) d vol = ∫_{M} d (A_{X} vol) = 0.

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

∫_{M}⟨D_{m}S, T ⟩ d vol = − ∫_{M}⟨S, D_{mtr}T ⟩ d vol

for all smooth sections S ∈ Γ(E) and T ∈ Γ(Ẽ^{∗}) such that S ⊗ T has compact support.

3. Local Martingales

In this section, suppose that m is a multiplication map and L = □ − R and L̃ = ˜□ − R̃ are second order differential operators on Γ(E) and Γ(Ẽ) satisfying Assumption 1, i.e. ρ := L̃D_{m} − D_{m}L ∈ Γ(Hom(E, Ẽ)). Let Q_{t} ∈ End(E_{x}) and Q̃_{t} ∈ End(Ẽ_{x}) denote the solutions to the ordinary differential equations,

\frac{d}{dt}Q_{t} = −\frac{1}{2} Q_{t}R_{//_{t}} with Q_{0} = id_{E_{x}} (3.1)

and

\frac{d}{dt}Q̃_{t} = −\frac{1}{2} Q̃_{t}R̃_{//_{t}} with Q̃_{0} = idẼ_{x}, (3.2)

where R_{//_{t}} := (//_{t}^{E})^{−1}R //_{t}^{E} and R̃_{//_{t}} := (//Ẽ ; t)^{−1}R̃ //Ẽ ; t are linear operators on E_{x} and Ẽ_{x} respectively.

Notation 3.1. A time dependent section {a_{t}}_{0≤t<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 L̃ be as in Assumption 1 and Q and Q̃ be defined by Eqs. (3.1) and (3.2) respectively. Suppose further that {a_{t}}_{0≤t<T} is a smooth time dependent section of E which satisfies the backwards heat equation,

\frac{∂}{∂t}a_{t} + \frac{1}{2}La_{t} = 0. (3.3)

For t in the stochastic interval [0, ζ(x) ∧ T [, let

N_{t} := Q_{t} //_{t}^{−1}a_{t}(X_{t}(x)) (3.4)

and

Ñ_{t} := Q̃_{t} //_{t}^{−1}Da_{t}(X_{t}(x)). (3.5)

Then the stochastic differentials of N_{t} and Ñ_{t} are given by

dN_{t} = Q_{t} //_{t}^{−1}∇_{//_{t}^{T M} dB_{t}}a_{t}(X_{t}(x)) (3.6)

and

dÑ_{t} = Q̃_{t} //_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x)) + \frac{1}{2}Q̃_{t} //_{t}^{−1} (ρa_{t}) (X_{t}(x))dt, (3.7)

where ρ := L̃D_{m} − D_{m}L ∈ Γ(Hom(E, Ẽ)) as in Assumption 1.

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

dN_{t} = Q_{t} //_{t}^{−1}∇_{//_{t} dB_{t}}a_{t}(X_{t}(x))

+ \frac{1}{2} ( \begin{matrix}−Q_{t} R_{//_{t}} //_{t}^{−1}a_{t}(X_{t}(x)) + Q_{t} //_{t}^{−1}□a_{t}(X_{t}(x)) \\ −Q_{t} //_{t}^{−1}La_{t}(X_{t}(x)) \end{matrix} ) dt

= Q_{t} //_{t}^{−1}∇_{//_{t} dB_{t}}a_{t}(X_{t}(x)),

where in the last equality we used the identity,

−R_{//_{t}} //_{t}^{−1} + //_{t}^{−1}□ = //_{t}^{−1} (□ − R) = //_{t}^{−1}L.

Similarly,

dÑ_{t} = Q̃_{t} //_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x))

+ \frac{1}{2} ( \begin{matrix}−Q̃_{t} R̃_{//_{t}} //_{t}^{−1}Da_{t}(X_{t}(x)) + Q̃_{t} //_{t}^{−1} ˜□Da_{t}(X_{t}(x)) \\ −Q̃_{t} //_{t}^{−1}DLa_{t}(X_{t}(x)) \end{matrix} ) dt

= Q̃_{t} //_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x)) + \frac{1}{2}Q̃_{t} //_{t}^{−1} (ρa) (X_{t}(x))dt

because

−R̃_{//_{t}} //_{t}^{−1}D + //_{t}^{−1} ˜□D − //_{t}^{−1}DL = //_{t}^{−1}(L̃D − DL) = //_{t}^{−1}ρ.

3.1. Dual Pairs.

Definition 3.3. A pair of adapted continuous processes ℓ_{t} ∈ E_{x}^{∗} and ℓ̃_{t} ∈ Ẽ_{x}^{∗} is called a dual pair if Z_{t} = ⟨Ñ_{t}, ℓ̃_{t}⟩ − ⟨N_{t}, ℓ_{t}⟩ is a local martingale.

Theorem 3.4. Suppose that ℓ_{t} ∈ E_{x}^{∗} and ℓ̃_{t} ∈ Ẽ_{x}^{∗} are continuous semimartingales such that

dℓ_{t} = α_{t} dB_{t} + β_{t} dt

and

dℓ̃_{t} = α̃_{t} dB_{t} + β̃_{t} dt,

where α_{t} ∈ Hom(T_{x}M, E_{x}^{∗}), α̃_{t} ∈ Hom(T_{x}M, Ẽ_{x}^{∗}), β_{t} ∈ E_{x}^{∗} and β̃_{t} ∈ Ẽ_{x}^{∗} are predictable processes. Then ℓ and ℓ̃ is a dual pair provided

(3.8)  α̃ ≡ 0,
(3.9)  \frac{1}{2}ρ^{tr}//_{t} Q̃_{t}^{tr} ℓ̃_{t} = //_{t} Q_{t}^{tr} β_{t},  and
(3.10)  m_{//_{t} v}^{tr} //_{t} Q̃_{t}^{tr} β̃_{t} = //_{t} Q_{t}^{tr} α_{t}v  for each v ∈ T_{x}M.

Proof. Let Z_{t} = ⟨Ñ_{t}, ℓ̃_{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} ≃ ⟨dÑ_{t}, ℓ̃_{t}⟩ + ⟨Ñ_{t}, dℓ̃_{t}⟩ + ⟨dÑ_{t}, dℓ̃_{t}⟩ − ⟨N_{t}, dℓ_{t}⟩ − ⟨dN_{t}, dℓ_{t}⟩
≃ ⟨\frac{1}{2}Q̃_{t} //_{t}^{−1} (ρa_{t}) (X_{t}(x)), ℓ̃_{t}⟩ dt

+ ⟨Ñ_{t}, β̃_{t}⟩ dt + ⟨Q̃_{t} //_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x)), α̃_{t} dB_{t}⟩

− ⟨N_{t}, β_{t}⟩ dt − ⟨Q_{t} //_{t}^{−1}∇_{//_{t} dB_{t}}a_{t}(X_{t}(x)), α_{t} dB_{t}⟩

= \frac{1}{2}⟨Q̃_{t} //_{t}^{−1} (ρa_{t}) (X_{t}(x)), ℓ̃_{t}⟩ dt

+ ∑_{i=1}^{n}⟨Q̃_{t} //_{t}^{−1}m_{//_{t} e_{i}}∇_{//_{t} e_{i}}a_{t}(X_{t}(x)), β̃_{t}⟩ dt

+ ∑_{i=1}^{n}⟨Q̃_{t} //_{t}^{−1}∇_{//_{t} e_{i}}Da_{t}(X_{t}(x)), α̃_{t}e_{i}⟩ dt − ⟨Q_{t} //_{t}^{−1}a_{t}(X_{t}(x)), β_{t}⟩ dt

− ∑_{i=1}^{n}⟨Q_{t} //_{t}^{−1}∇_{//_{t} e_{i}}a_{t}(X_{t}(x)), α_{t}e_{i}⟩ dt.

Keeping in mind that //_{t} = (//_{t}^{−1})^{tr} (i.e. //_{t}^{E^{∗}} = ((//_{t}^{E})^{−1})^{tr}) we find, by comparing terms involving a, ∇_{//_{t} e_{i}}a, and ∇_{//_{t} e_{i}}Da, that dZ_{t} ≃ 0 provided that Eqs. (3.8), (3.9) and (3.10) are satisfied.

The upshot of the previous theorem is that to make a dual pair we should choose α̃ = 0 (i.e. dℓ̃_{t} = β̃_{t} dt),

β_{t} = \frac{1}{2}(Q_{t}^{tr})^{−1}//_{t}^{−1}ρ^{tr}//_{t} Q̃_{t}^{tr}ℓ̃_{t}  and
α_{t}v = (Q_{t}^{tr})^{−1} //_{t}^{−1}m_{//_{t} v}^{tr}//_{t} Q̃_{t}^{tr}β̃_{t} for each v ∈ T_{x}M.

Because the processes Q_{t}^{tr} and Q̃_{t}^{tr} will arise often in the sequel, it is convenient to introduce the following notation.

Notation 3.5. Let Q_{t} := Q_{t}^{tr} ∈ End(E_{x}^{∗}) and Q̃_{t} := Q̃_{t}^{tr} ∈ End(Ẽ_{x}^{∗}). Taking the transposes of Eqs. (3.1) and (3.2) shows that Q and Q̃ solve the following ordinary differential equations:

(3.11) \frac{d}{dt}Q_{t} = −\frac{1}{2}R_{//_{t}}^{tr} Q_{t} with Q_{0} = id_{E_{x}^{∗}}

and

(3.12) \frac{d}{dt}Q̃_{t} = −\frac{1}{2} R̃_{//_{t}}^{tr} Q̃_{t} with Q̃_{0} = idẼ_{x}^{∗} .

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) s → ℓ_{s} is absolutely continuous and ∫_{0}^{T} |dℓ_{s}/ds|_{V}^{2} ds < ∞, where | · |_{V} denotes any one of the equivalent norms on V. If in addition there is a p ∈ [1, ∞) such that

𝔼 [(∫_{0}^{T} |dℓ_{s}/ds|_{V}^{2} ds)^{p/2}] < ∞,

then we say that {ℓ_{s}}_{s∈[0,T [} is an L^{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 ℓ̃ for a finite energy process with values in Ẽ_{x}^{∗}.

Theorem 3.7. Let a, N, and Ñ be as in Proposition 3.2, ℓ_{t} ∈ Ẽ_{x}^{∗} be a finite energy process and define the E_{x}^{∗}-valued process,

U_{t}^{ℓ} := ∫_{0}^{t} Q_{s}^{−1} //_{s}^{−1} m_{//_{s}dB_{s}}^{tr}//_{s}Q̃_{s}ℓ′_{s} + \frac{1}{2} ∫_{0}^{t} Q_{s}^{−1} ρ_{//_{s}}^{tr} Q̃_{s}ℓ_{s} ds, (3.13)

where ρ_{//_{s}} := //_{s}^{−1}ρ(X_{s}(x))//_{s} and

(3.14) ρ_{//_{s}}^{tr} = //_{s}^{tr}ρ^{tr}(X_{s}(x)) (//_{s}^{−1})^{tr} = //_{s}^{−1}ρ^{tr}(X_{s}(x))//_{s}.

(See the remarks at the end of Example 2.2 explaining the identity //_{s}^{tr} = //_{s}^{−1}.) Then

Z_{t}^{ℓ} := ⟨Ñ_{t}, ℓ_{t}⟩ − ⟨N_{t}, U_{t}^{ℓ}⟩ (3.15) is a local martingale on [0, ζ(x) ∧ T [ and

dZ_{t}^{ℓ} = ⟨Q̃_{t}//_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x)), ℓ_{t}⟩ − ⟨Q_{t} //_{t}^{−1}∇_{//_{t}dB_{t}}a_{t}(X_{t}(x)), U_{t}^{ℓ}⟩

− ⟨N_{t}, Q_{t}^{−1}//_{t}^{−1} m_{//_{t} dB_{t}}^{tr} //_{t} Q̃_{t} ℓ′_{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_{t}^{ℓ} is a local martingale, we need only keep track of those terms in dZ_{t}^{ℓ} which depend linearly on dB_{t}. Hence Eq. (3.16) follows from the identity

dZ_{t}^{ℓ} = ⟨dÑ_{t}, ℓ_{t}⟩ + ⟨Ñ_{t}, dℓ_{t}⟩ + ⟨dÑ_{t}, dℓ_{t}⟩
− ⟨dN_{t}, U_{t}^{ℓ}⟩ − ⟨N_{t}, dU_{t}^{ℓ}⟩ − ⟨dN_{t}, dU_{t}^{ℓ}⟩

and Eqs. (3.6), (3.7) and (3.13).

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

m_{v}^{tr} = //_{s}^{tr} m_{//_{s}v}^{tr}(//_{s}^{−1})^{tr} = //_{s}^{−1} m_{//_{s}v}^{tr}//_{s}

for all v ∈ T_{x}M. Hence, the formula for U_{t}^{ℓ} in Eq. (3.13) simplifies to

U_{t}^{ℓ} = ∫_{0}^{t} Q_{s}^{−1} m_{dB_{s}}^{tr} Q̃_{s} ℓ′_{s} + \frac{1}{2} ∫_{0}^{t} Q_{s}^{−1} ρ_{//_{s}}^{tr} Q̃_{s} ℓ_{s} ds. (3.17)

Remark 3.8. Suppose that ℓ_{t} ∈ Ẽ_{x} is a finite energy process. Under the further assumption that E and Ẽ are Riemannian vector bundles and m is compatible with ∇, the results of Theorem 3.7 remains true with U_{t}^{ℓ} and Z_{t}^{ℓ} defined by

U_{t}^{ℓ} := ∫_{0}^{t}(Q_{s}^{−1})^{∗}m_{dB_{s}}^{∗} Q̃_{s}^{∗} ℓ′_{s} + \frac{1}{2} ∫_{0}^{t}(Q_{s}^{−1})^{∗}ρ_{//_{s}}^{∗} Q̃_{s}^{∗} ℓ_{s} ds (3.18)

and

Z_{t}^{ℓ} := (Ñ_{t}, ℓ_{t})Ẽ − (N_{t}, U_{t}^{ℓ})_{E}, (3.19)

where ρ_{//_{s}}^{∗} := //_{s}^{−1}ρ^{∗}(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 Ẽ_{x}^{∗}-valued finite energy process and a, N are as in Proposition 3.2. Also define:

N̂_{t} := //_{t}^{−1}Da_{t}(X_{t}(x)), (3.20)

Û_{t}^{k} := ∫_{0}^{t} Q_{s}^{−1}m_{dB_{s}}^{tr}(k′_{s} + \frac{1}{2}R̃_{//_{s}}^{tr}k_{s}) + \frac{1}{2} ∫_{0}^{t} Q_{s}^{−1}ρ_{//_{s}}^{tr}k_{s} ds (3.21)

and

Ẑ_{t}^{k} := ⟨N̂_{t}, k_{t}⟩ − ⟨N_{t}, Û_{t}^{k}⟩. (3.22)

Then Ẑ_{t}^{k} is a local martingale on [0, ζ(x) ∧ T [ whose Itô differential is given by

dẐ_{t}^{k} = ⟨//_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x)), k_{t}⟩ − ⟨Q_{t} //_{t}^{−1}∇_{//_{t}dB_{t}}a_{t}(X_{t}(x)), Û_{t}^{k}⟩

− ⟨Q_{t}^{−1}N_{t}, m_{dB_{t}}^{tr}(k′_{t} + \frac{1}{2}R̃_{//_{t}}^{tr}k_{t})⟩. (3.23)

Proof. Comparing Eqs. (3.5) and (3.20) shows that Ñ_{t} = Q̃_{t}N̂_{t}. Define the finite energy process ℓ_{s} by ℓ_{s} := (Q̃_{s})^{−1}k_{s} so that k_{s} = Q̃_{s}ℓ_{s.} Because

k′_{s} = Q̃_{s}ℓ′_{s} − \frac{1}{2}R̃_{//_{s}}^{tr} Q̃_{s}ℓ_{s} = Q̃_{s}ℓ′_{s} − \frac{1}{2}R̃_{//_{s}}^{tr}k_{s}, (3.24)

it follows from Eq. (3.17) that Û_{t}^{k} = U_{t}^{ℓ}. This identity and

⟨Ñ_{t}, ℓ_{t}⟩ = ⟨Q̃_{t}N̂_{t}, (Q̃_{t})^{−1}k_{t}⟩ = ⟨N̂_{t}, k_{t}⟩

shows that Ẑ_{t}^{k} = Z_{t}^{ℓ} as well. By Theorem 3.7 Ẑ_{t}^{k} = Z_{t}^{ℓ} is a local martingale on [0, ζ(x) ∧ T [ and by Eq. (3.16) and Eq. (3.24) the differential of Ẑ_{t}^{k} is given by

dẐ_{t}^{k} = ⟨Q̃_{t}//_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x)), (Q̃_{t})^{−1}k_{t}⟩

− ⟨Q_{t} //_{t}^{−1}∇_{//_{t}dB_{t}}a_{t}(X_{t}(x)), Û_{t}^{k}⟩

− ⟨N_{t}, Q_{t}^{−1}m_{dB_{t}}^{tr}(k′_{t} + \frac{1}{2}R̃_{//_{t}}^{tr}k_{t})⟩

which is the same as Eq. (3.23).

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 m is compatible with the covariant derivative ∇. Thus the assumptions which are in force in the remainder of the paper are:

Assumption 2 (Standing Assumptions). Let E and Ẽ be vector bundles endowed with covariant derivatives. Assume that L, L̃, 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 Q̃ be given by (3.1) and (3.2). Also let τ be a stopping time bounded by T < ∞ such that τ < ζ(x) and let ℓ_{t} ∈ Ẽ_{x}^{∗} be a finite energy process on the stochastic interval [0, τ]. Assume that τ and ℓ have been chosen such that

𝔼|⟨Q̃_{τ} //_{τ}^{−1}Da_{τ} (X_{τ} (x)), ℓ_{τ} ⟩| < ∞, 𝔼|⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), U_{τ}^{ℓ}⟩| < ∞,

where U^{ℓ} is defined in Eq. (3.17). Further assume (Z^{ℓ})_{t}^{τ} := Z_{t∧τ}^{ℓ} is a martingale where Z_{t}^{ℓ} is the local martingale in Eq. (3.15). Then

𝔼[⟨Da_{0}(x), ℓ_{0}⟩] = 𝔼[⟨Q̃_{τ} //_{τ}^{−1}Da_{τ} (X_{τ} (x)), ℓ_{τ} ⟩] − 𝔼[⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), U_{τ}^{ℓ}⟩]. (4.1)

Therefore, 1. if ℓ_{τ} = 0 and ℓ_{0} = ξ ∈ Ẽ_{x}^{∗} then

⟨Da_{0}(x), ξ⟩ = −𝔼 [⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), U_{τ}^{ℓ}⟩] , (4.2)

2. or if ℓ_{0} = 0 and ℓ_{τ} = ξ ∈ Ẽ_{x}^{∗} (where ξ may be random here) then

𝔼 [⟨Q̃_{τ} //_{τ}^{−1}Da_{τ} (X_{τ} (x)), ξ⟩] = 𝔼 [⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), U_{τ}^{ℓ}⟩] . (4.3)

Proof. Since we have assumed that the expression (Z^{ℓ})^{τ} in Eq. (3.15) of Theorem 3.7 is a martingale, it follows that 𝔼 Z_{0}^{ℓ} = 𝔼 (Z^{ℓ})_{T}^{τ} = 𝔼 Z_{τ}^{ℓ}, i.e.

𝔼 ⟨Ñ_{τ} , ℓ_{τ} ⟩ − 𝔼 ⟨Ñ_{0}, ℓ_{0}⟩ = 𝔼 ⟨N_{τ} , U_{τ}^{ℓ}⟩

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

Remark 4.2. Suppose that ℓ_{t} ∈ Ẽ_{x} is a finite energy process. Under the further assumption that E and Ẽ are Riemannian vector bundles and m is compatible with ∇, Theorem 4.1 remains valid with all pairings ⟨·, ·⟩ replaced by the appropriate inner products (·, ·)_{E} or (·, ·)Ẽ and all transposes replaced by adjoints and U^{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 τ be a stopping time bounded by T < ∞ such that τ < ζ(x) and let k_{t} ∈ Ẽ_{x}^{∗} be a finite energy process on the stochastic interval [0, τ]. Assume that τ and k have been chosen such that

𝔼|⟨//_{τ}^{−1}Da_{τ} (X_{τ} (x)), k_{τ} ⟩| < ∞, 𝔼|⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), Û_{τ}^{k}⟩| < ∞,

where Û^{k} is defined in Eq. (3.21). Further assume (Ẑ^{k})_{t}^{τ} := Ẑ_{t∧τ}^{k} is a martingale where Ẑ_{t}^{k} is the local martingale in Eq. (3.22). Then

𝔼[⟨Da_{0}(x), k_{0}⟩] = 𝔼[⟨//_{τ}^{−1}Da_{τ} (X_{τ} (x)), k_{τ} ⟩] − 𝔼[⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), Û_{τ}^{k}⟩]. (4.4)

Therefore, 1. if k_{τ} = 0 and k_{0} = ξ ∈ Ẽ_{x}^{∗} then

⟨Da_{0}(x), ξ⟩ = −𝔼 [⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), Û_{τ}^{k}⟩] , (4.5)

2. or if k_{0} = 0 and k_{τ} = ξ ∈ Ẽ_{x}^{∗} (where ξ may be random here) then

𝔼 [⟨//_{τ}^{−1}Da_{τ} (X_{τ} (x)), ξ⟩] = 𝔼 [⟨Q_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x)), Û_{τ}^{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.

The formulas appearing in Theorem 3.7 take on a simpler form when m is compatible with ∇ and ℓ_{t} is of the form ℓ_{t} = ℓ̄_{t}ξ where ℓ̄_{t} is an ℝ-valued finite energy process and ξ ∈ Ẽ^{∗}. To write out these formula, it is convenient to introduce the process V ℓ̄ ; t ∈ Hom(E_{x}, Ẽ_{x}) by

V ℓ̄ ; t = ∫_{0}^{t} ℓ̄′_{s}Q̃_{s}m_{dB_{s}}Q_{s}^{−1} + \frac{1}{2} ∫_{0}^{t} ℓ̄_{s}Q̃_{s}ρ_{//_{s}}Q_{s}^{−1}ds, (4.7)

where ρ_{//_{s}} := //_{s}^{−1}ρ(X_{s}(x))//_{s} and the Ẽ_{x}-valued process

Z̄ℓ̄ ; t:= ℓ̄_{t}Ñ_{t} − V ℓ̄ ; tN_{t}. (4.8)

Corollary 4.4. Let a be a solution to Eq. (3.3) and Q and Q̃ be given by (3.1) and (3.2). Also let τ be a stopping time bounded by T < ∞ such that τ < ζ(x) and let ℓ̄_{t} be an ℝ-valued finite energy process on the stochastic interval [0, τ]. Assume that τ and ℓ̄ have been chosen such that

𝔼 |ℓ̄_{τ} Q̃_{τ} //_{τ}^{−1}Da_{τ} (X_{τ} (x))| < ∞, 𝔼 |V ℓ̄ ; τQ_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x))| < ∞,

where V ℓ̄ is defined in Eq. (4.7). Further assume (Z̄ℓ̄)_{t}^{τ} := Z̄ℓ̄ ; t∧τ is a martingale where Z̄ℓ̄ ; t is the local martingale in Eq. (4.8). Then

Da_{0}(x) = 𝔼 [ℓ̄_{τ} Q̃_{τ} //_{τ}^{−1}Da_{τ} (X_{τ} (x))] − 𝔼 [V ℓ̄ ; τQ_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x))] . (4.9)

Moreover, 1. if ℓ̄_{τ} = 0 and ℓ̄_{0} = 1 then

Da_{0}(x) = −𝔼 [V ℓ̄ ; τQ_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x))] , (4.10)

2. or if ℓ̄_{0} = 0 and ℓ̄_{τ} = 1 then

𝔼 [Q̃_{τ} //_{τ}^{−1}Da_{τ} (X_{τ} (x))] = 𝔼 [V ℓ̄ ; τQ_{τ} //_{τ}^{−1}a_{τ} (X_{τ} (x))] . (4.11)

In the remainder of Section 4, we will illustrate the type of formulas that can be derived from Theorem 4.1. For now we will just assume that the hypothesis of Theorem 4.1 or Corollary 4.4 are satisfied. The subsequent sections will address the issue of verifying these hypotheses in a number of different contexts.

In order to apply Theorem 4.1, it is necessary to have a solution a to Eq. (3.3). There are basically two choices which will be used in the sequel: i) a_{t} = e^{(T −t)L/2}α where α ∈ Γ(E) and ii) a_{t} = α where Lα = 0. The second case is formally a special case of the first.

4.1. Feynman-Kac Formulas. For this subsection, we will assume that the lifetime ξ(x) of the Brownian motion is infinite. We begin by recalling the Feynman-Kac representation for e^{tL} and e^{tL̃}.

Proposition 4.5 (Feynman-Kac). Suppose that α and α̃ are smooth sections of E and Ẽ such that there exist smooth solutions to the partial differential equations

\frac{∂}{∂t}u(t) = \frac{1}{2}Lu(t) with u(0) = α (4.12)

and

\frac{∂}{∂t}ũ(t) = \frac{1}{2}L̃ũ(t) with ũ(0) = α̃. (4.13)

Further assume that there is a solution to Eq. (4.13) when α̃ = Dα. We will write e^{tL/2}α for u(t), respectively e^{tL̃/2}α̃ for ũ(t), and e^{tL̃/2}Dα for the solution to Eq. (4.13) when α̃ = Dα. Let a_{t} = u(T − t) = e^{(T −t)L/2}α, respectively. ã_{t} = ũ(T − t) = e^{(T −t)L̃/2}α̃, and let N_{t} and Ñ_{t} be defined by Eqs. (3.4) and (3.5). If the local martingales N_{t} and

t → Q̃_{t} //_{t}^{−1}α̃(X_{t}(x))

are in fact martingales for 0 ≤ t ≤ T, then

(e^{T L/2}α)(x) = 𝔼 [Q_{T} //_{T}^{−1}α(X_{T} (x))] , (4.14)

(e^{T L̃/2}α̃)(x) = 𝔼[Q̃_{T} //_{T}^{−1}α̃(X_{T} (x))]. (4.15)

Under the further assumptions:

i) ∫_{0}^{T} 𝔼 |Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}N_{T} | dt < ∞ (4.16)

(with ρ_{//_{t}} as in Eq. (3.14)) and ii) the local martingale

t → ∫_{0}^{t} Q̃_{r} //_{r}^{−1}∇_{//_{r}dB_{r}}Da_{r}(X_{r}(x)) (4.17)

is a martingale on [0, T], then

(e^{T L̃/2}Dα)(x)

= (De^{T L/2}α)(x) + \frac{1}{2}𝔼 [(∫_{0}^{T} Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}dt) Q_{T} //_{T}^{−1}α(X_{T} (x))] . (4.18)

Remarks 4.6. i) Equation (4.18) is a stochastic version of Duhamel’s principle. ii) The hypotheses of the previous proposition are easily satisfied when M is compact, see Proposition 5.1 below.

Proof. Since N_{t} is assumed to be a martingale,

(e^{T L/2}α)(x) = 𝔼 [N_{0}] = 𝔼 [N_{T}] = 𝔼 [Q_{T} //_{T}^{−1}α(X_{T} (x))]

which proves Eq. (4.14). The proof of Eq. (4.15) is the same, just put “tildes” on everything.

Taking expectations of Eq. (3.7) of Proposition 3.2 and using Eq. (4.15) with α̃ = Dα, we find that

(e^{T L̃/2}Dα)(x) = 𝔼 [Q̃_{T} //_{T}^{−1}Dα(X_{T} (x))] = 𝔼 [Ñ_{T}]

= 𝔼 [Ñ_{0}] + \frac{1}{2}𝔼 ∫_{0}^{T} Q̃_{t} //_{t}^{−1} (ρa_{t}) (X_{t}(x)) dt

= (De^{T L/2}α)(x) + \frac{1}{2}𝔼 ∫_{0}^{T} Q̃_{t} //_{t}^{−1}ρ(X_{t}(x))//_{t}Q_{t}^{−1}Q_{t}//_{t}^{−1}a_{t}(X_{t}(x)) dt

= (De^{T L/2}α)(x) + \frac{1}{2}𝔼 ∫_{0}^{T} Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}N_{t} dt. (4.19)

Note that Eq. (4.16), along with the martingale property of N_{t}, implies as well that

∫_{0}^{T} 𝔼 |Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}N_{t}| dt < ∞. (4.20)

To simplify notation, let Γ_{t} := Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}. From the bounds in Eqs. (4.16) and (4.20) and Fubini’s Theorem, it follows that

𝔼 |Γ_{t}N_{T} | < ∞ and 𝔼 |Γ_{t}N_{t}| < ∞ (4.21)

for almost every t ∈ [0, T]. At any t where Eq. (4.21) holds, one shows using the martingale property of N_{t} that 𝔼 [Γ_{t}N_{t}] = 𝔼 [Γ_{t}N_{T}]. (This is done by first truncating Γ_{t} and then passing to the limit.) Since 𝔼 [Γ_{t}N_{t}] = 𝔼 [Γ_{t}N_{T}] for almost every t, we find that

𝔼 ∫_{0}^{T} Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}N_{t} dt = ∫_{0}^{T} 𝔼 [Γ_{t}N_{t}] dt = ∫_{0}^{T} 𝔼 [Γ_{t}N_{T}] dt,

which along with Eq. (4.19) implies Eq. (4.18) since

N_{T} = Q_{T} //_{T}^{−1}a_{T} (X_{T} (x)) = Q_{T} //_{T}^{−1}α(X_{T} (x)).

4.2. Semigroup Derivative Formulas. Let a_{t} = e^{(T −t)L/2}α where α ∈ Γ(E) and e^{tL/2} is the semigroup generated by a suitable extension of L. Taking τ = T in Theorem 4.1, we find the following derivative formula from Eq. (4.2),

⟨De^{T L/2}α(x), ξ⟩ = −𝔼 [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩] for any ξ ∈ Ẽ_{x}^{∗} (4.22)

where ℓ is an appropriate finite energy process with values in Ẽ_{x}^{∗} such that ℓ_{T} = 0 and ℓ_{0} = ξ. Similarly, using Proposition 4.5 and Eq. (4.3),

⟨e^{T L̃/2}Dα(x), ξ⟩ = 𝔼[⟨Q̃_{T} //_{T}^{−1}Dα(X_{T} (x)), ξ⟩]

= 𝔼[⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩] for any ξ ∈ Ẽ_{x}^{∗}, (4.23)

where now ℓ_{T} = ξ and ℓ_{0} = 0.

Remark 4.7. It is interesting to notice that by choosing ℓ_{T} to be random, we may also use Eq. (4.3) of Theorem 4.1 to get a formula for e^{T L̂/2}Dα(x) for any operator L̂ on Γ(Ẽ) of the form L̂ = ˜□ − R̂. To do this, let

\frac{d}{dt}Q̂_{t} = −\frac{1}{2}Q̂_{t} R̂_{//_{t}} with Q̂_{0} = idẼ_{x}, (4.24)

where R̂_{//_{t}} := (//Ẽ ; t)^{−1}R̂ //Ẽ ; t, and choose ℓ such that ℓ_{0} = 0 and ℓ_{T} = (Q̂_{T} Q̃_{T}^{−1})^{tr}ξ with ξ ∈ Ẽ_{x}^{∗} non-random. Then by the Feynman-Kac formula (Proposition 4.5)

⟨e^{T L̂/2}Dα(x), ξ⟩ = 𝔼[⟨Q̂_{T} //_{T}^{−1}Da_{T} (X_{T} (x)), ξ⟩]
= 𝔼[⟨Q̃_{T} //_{T}^{−1}Da_{T} (X_{T} (x)), (Q̂_{T} Q̃_{T}^{−1})^{tr}ξ⟩]
(4.25)  = 𝔼[⟨Q̃_{T} //_{T}^{−1}Da_{T} (X_{T} (x)), ℓ_{T} ⟩].

This equation along with Eq. (4.3) and a_{T} = α implies that

⟨e^{T L̂/2}Dα(x), ξ⟩ = 𝔼[⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩], (4.26)

where ℓ is an Ẽ_{x}^{∗}-valued process such that ℓ_{0} = 0 and ℓ_{T} = (Q̂_{T} Q̃_{T}^{−1})^{tr}ξ.

Remark 4.8. Eqs. (4.22) and (4.23) provide stochastic formulas for De^{T L/2}a and e^{T L̃/2}Dα, not containing derivatives of the section α. These and related formulas rely on the fact that one of the local martingales given by Eq. (3.15), or Eq. (3.19), or Eq. (3.22), is a martingale for certain choices of a finite energy process {ℓ_{t}}, respectively {k_{t}}. Nevertheless there is an essential difference between formula (4.22) and (4.23). To get the formula for De^{T L/2}α, we need to know that the local martingale

Z_{t}^{ℓ} = ⟨Ñ_{t}, ℓ_{t}⟩ − ⟨N_{t}, U_{t}^{ℓ}⟩,

as given by Eq. (3.15), is a martingale for a finite energy process ℓ such that ℓ_{0} = ξ and ℓ_{T} = 0. As we shall see (Section 6), this can always be achieved (independently of whether M is compact or complete), for instance, by taking ℓ_{s} = 0 already for s ≥ τ ∧ T where τ is the first exit time of X(x) from some relatively compact neighborhood of x. On the other hand, to get the formula for e^{T L̃/2}Dα, the martingale property for Z^{ℓ} is exploited where now ℓ_{0} = 0 and ℓ_{T} = ξ, which no longer is a local problem. It implies the second equality in (4.23), which combined with the Feynman-Kac representation for e^{T L̃/2}Dα gives both equalities in Eq. (4.23). As will be seen in Section 5, the martingale property in this case is easily checked if M is compact (for most applications, completeness of M is sufficient), but it is not automatically satisfied in general. For instance, sticking to the case ρ = 0 for simplicity, and under the assumption that Z^{ℓ} is a martingale also for ℓ_{s} ≡ ξ, we get the formula

⟨(De^{T L̃/2}α)(x), ξ⟩ = 𝔼[⟨Q̃_{T} //_{T}^{−1}Dα(X_{T} (x)), ξ⟩]

which combined with Eq. (4.23) shows that De^{T L̃/2}α = e^{T L̃/2}Dα, see also Eq. (4.18). The validity of such commutation rules is known to be an intriguing question for (non-complete) Riemannian manifolds.

4.3. Harmonic Section Derivative Formula. Suppose that a ∈ Γ(E) is a Lharmonic section (i.e. La = 0) defined locally in a neighborhood V of x. Let τ be the first exit time of X_{t}(x) from some relatively compact neighborhood of x which is contained in V. If {ℓ_{t}}_{0≤t<τ} is a bounded L^{1}-finite energy process (i.e. (∫_{0}^{τ} |ℓ′(s)|^{2}ds)^{1/2} ∈ L^{1}) such that ℓ_{0} = ξ ∈ Ẽ_{x}^{∗} and ℓ_{τ} = 0, then

⟨Da(x), ξ⟩ = −𝔼[⟨Q_{τ} //_{τ}^{−1}a(X_{τ} (x)), U_{τ}^{ℓ}⟩] (4.27)

where U^{ℓ} is the process defined in Eq. (3.17).

5. Applications for Compact M

In order to avoid technical complications we will first demonstrate some applications of the previous results under the assumption that M is a compact manifold without boundary. Applications in the case that M is not compact will be given in the next section. One key consequence of the compactness of M is that ζ(x) = ∞ a.s. for all x ∈ M. Again recall that Assumption 2 (at the beginning of Section 4) is in force throughout the remainder of this paper.

By standard elliptic P.D.E. theory and the Minakshisundaram-Pleijel method for constructing heat kernels we have the following facts. The heat equation

∂_{t}a_{t} = La_{t} with a_{0} = α ∈ Γ(E) (5.1)

has a unique solution which we write as P_{t}α. The linear operator P_{t} : Γ(E) → Γ(E) extends continuously to a one-parameter semigroup on L^{2}(E), see for example Chapter 2 of [4]. The L^{2} generator of this semigroup is the closure L̄ of L. To verify this last assertion, let L̂ be the L^{2} generator of P_{t} and L^{†} denote the formal adjoint of L. Then using the result in [4], L^{†} generates (in the same way L generated P) the adjoint semigroup P_{t}^{∗}. Therefore, for a ∈ D(L̂) and b ∈ Γ(E), we have that

(L̂a, b) = \frac{d}{dt}|_{0+}(P_{t}a, b) = \frac{d}{dt}|_{0+}(a, P_{t}^{∗}b) = (a, L^{†}b).

This shows that L ⊂ L̄ ⊂ L̂ ⊂ (L^{†})^{∗}. However, by basic elliptic regularity theory, D((L^{†})^{∗}) = D(L̄) = H^{2} – the Sobolev space with two derivatives in L^{2}. Therefore, L̄ = L̂ = (L^{†})^{∗}. From now on we will write e^{tL} for P_{t}.

Proposition 5.1. Suppose that α and α̃ are L^{2} sections of E and Ẽ respectively, then the Feynman-Kac formula in Eqs. (4.14) and (4.15) of Proposition 4.5 are valid. If we further assume that α is an H^{1} section of E (i.e. α is an L^{2} section of E with one weak derivative in L^{2}), then Eq. (4.18) is valid as well.

Proof. Using a continuity argument, it suffices to prove Eqs. (4.14), (4.15) and (4.18) under the assumption that α and α̃ are smooth sections of E and Ẽ respectively. Since M is compact, Q_{t}, Q_{t}^{−1} and Q̃_{t} are bounded by a e^{Kt}, where K is a non-random constant depending on the bounds on R and R̃. Using these facts and the assumed smoothness of α and α̃, it is easy to see that the assumptions in Proposition 4.5 are satisfied.

Remark 5.2. A simple consequence of Eq. (4.18) is that De^{tL/2}α = e^{tL̃/2}Dα when ρ = 0. Of course this may be proved directly as well. Indeed, for α ∈ Γ(E), a_{t} = De^{tL/2}α and b_{t} = e^{tL̃/2}Dα are both solutions to the heat equation \frac{d}{dt}a_{t} = \frac{1}{2} L̃a_{t} with initial condition a_{t}|_{t=0} = Dα. Uniqueness of solutions to the L̃ heat equation gives a_{t} = b_{t}, i.e.

De^{tL/2}α = e^{tL̃/2}Dα for all α ∈ Γ(E).

By continuity, the previous equation extends to all H^{1} sections α of E.

For a more general account on the elliptic theory of the heat equation (5.1), including elliptic boundary problems in the case of compact manifolds with boundary, the reader is referred to Agranovich [1], Grubb [33], as well as Seeley [47, 48, 49].

5.1. Corollaries of Theorem 4.1.

Corollary 5.3. Suppose that α is a bounded measurable section of E and ξ ∈ Ẽ_{x}^{∗}. Let {ℓ_{t}}_{0≤t≤T} be an L^{1}-finite energy process with values in Ẽ_{x}^{∗} (see Definition 3.6) such that ℓ_{T} = 0. Then, with ξ := ℓ_{0} ∈ Ẽ_{x}^{∗},

⟨De^{T L/2}α(x), ξ⟩ = −𝔼 [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩] , (5.2)

where U^{ℓ} is given by Eq. (3.17). More generally, letting p ∈ (1, ∞) and q = p/(p−1) be the conjugate exponent of p, if {ℓ_{t}}_{0≤t≤T} is an L^{q}-finite energy process and α is an L^{p} section of E, then Eq. (5.2) is still valid.

Proof. We will apply Theorem 4.1 with a_{t}(x) := (e^{(T −t)L/2}α)(x). By the Burkholder-Davis-Gundy inequality, there exists a constant C, depending only on q ∈ [1, ∞), K and T, such that

𝔼 |∫_{0}^{T} Q_{s}^{−1}m_{dB_{s}}^{tr} Q̃_{s}ℓ′_{s}|^{q} ≤ C 𝔼 [(∫_{0}^{T} |ℓ′_{t}|^{2}dt)^{q/2}]

and also

𝔼 |∫_{0}^{T} Q_{s}^{−1} ρ_{//_{s}}^{tr} Q̃_{s} ℓ_{s} ds|^{q} ≤ C 𝔼 [(∫_{0}^{T} |ℓ_{s}|^{2} ds)^{q/2}]

≤ C (ℓ_{0}^{q} + 𝔼 [(∫_{0}^{T} |ℓ′_{s}|^{2} ds)^{q/2}]) .

These equations together show that U_{T}^{ℓ} is L^{q}-integrable.

First suppose that α is a bounded measurable section of E. As in the proof of Proposition 5.1, the local martingales N and Ñ of Proposition 3.2 and Z^{ℓ} of Theorem 3.7 are already martingales. In case |De^{tL/2}α| is not bounded on (0, T]×M it may be necessary to first modify ℓ such that ℓ(T − ε) = 0 for some small ε > 0 and then to take the limit as ε → 0, see [58] for details.

Finally, if p ∈ (1, ∞) and {ℓ_{t}}_{0≤t≤T} is an L^{q}-finite energy process, then both sides of Eq. (5.2) depend continuously on α ∈ L^{p}(E), hence it suffices to prove Eq. (5.2) when α is smooth. But for smooth α it is easy to verify the hypothesis of Theorem 4.1.

Corollary 5.4. Suppose that α ∈ Γ(E) is an L-harmonic section (i.e. Lα = 0) and ξ ∈ Ẽ_{x}^{∗}. Let {ℓ_{t}}_{0≤t≤T} be an L^{1}-finite energy process with values in Ẽ_{x}^{∗} such that ℓ_{T} = 0 and ℓ_{0} = ξ ∈ Ẽ_{x}^{∗}. Then

⟨Dα(x), ξ⟩ = −𝔼 [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩] , (5.3)

where again U^{ℓ} is given by Eq. (3.17).

Proof. This is a consequence of Theorem 4.1 with a_{t} := α or directly from the previous Corollary upon noting that e^{T L/2}α = α since Lα = 0.

Corollary 5.5. Suppose that α is a C^{1} section of E and ξ ∈ Ẽ_{x}^{∗}. Let {ℓ_{t}}_{0≤t≤T} be an L^{1}-finite energy process with values in Ẽ_{x}^{∗} such that ℓ_{0} = 0 and ℓ_{T} = ξ ∈ Ẽ_{x}^{∗}. Under Assumption 2,

⟨e^{T L̃/2}Dα(x), ξ⟩ = 𝔼 [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩] , (5.4)

where U^{ℓ} is given by Eq. (3.17). If instead we choose ℓ as above except with ℓ_{0} = ξ ∈ Ẽ_{x}^{∗} and ℓ_{T} = 0, then

⟨e^{T L̃/2}Dα(x), ξ⟩ = 𝔼 [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), W_{T}^{ℓ} ⟩] , (5.5)

where

W_{T}^{ℓ} := \frac{1}{2} (∫_{0}^{T} Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}dt)^{tr} ξ − U_{T}^{ℓ} .

Proof. Eq. (5.4) follows from Eq. (4.3) of Theorem 4.1 with a_{t} := e^{(T −t)L/2}α. Eq. (5.5) is a consequence of Eq. (4.2) of Theorem 4.1 and Eq. (4.18) which imply,

−𝔼[⟨Q_{T} //_{T}^{−1}a_{T} (X_{T} (x)), U_{T}^{ℓ} ⟩] = ⟨De^{T L/2}α(x), ξ⟩

= ⟨(e^{T L̃/2}Dα)(x), ξ⟩ − \frac{1}{2}𝔼 [⟨(∫_{0}^{T} Q̃_{t} ρ_{//_{t}}Q_{t}^{−1}dt) Q_{T} //_{T}^{−1}α(X_{T} (x)), ξ⟩] .

Remark 5.6. Suppose that E and Ẽ are Hermitian vector bundles with metric compatible covariant derivatives. Under these conditions, the results of Corollaries 5.3, 5.4, and 5.5 may be rewritten by replacing the dual space Ẽ_{x}^{∗} by Ẽ_{x} and the dual pairings ⟨·, ·⟩ by the appropriate Hermitian metrics and then using Eq. (3.18) to define the process U_{t}^{ℓ}.

5.2. Formulas for Dirac Operators. Let us specialize Corollaries 5.3 and 5.5 to the case where E = Ẽ = S is a spin bundle over M. Recall that S is Riemannian vector bundle with metric compatible spin connection ∇^{S}, see Example 2.13. In this case R = R̃ = \frac{1}{4}scal, ρ = 0,

Q_{T} = Q̃_{T} = e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} id_{S_{x}}

and

U_{T}^{ℓ} = − ∫_{0}^{T} γ_{dB_{s}}ℓ′_{s},

wherein we have used the fact that γ^{∗} = −γ. Corollary 5.3 (see Remark 5.6) becomes

(De^{−T D^{2}/2}α(x), ξ)_{S_{x}} = 𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} (//_{T}^{−1}α(X_{T} (x)), ∫_{0}^{T} γ_{dB_{s}}ℓ′_{s})_{S_{x}}]

where ℓ is an L^{1}-finite energy process with values in S_{x} such that ℓ_{T} = 0 and ℓ_{0} = ξ ∈ S_{x}. Taking ℓ_{t} = h_{t}ξ in the previous equation (where ξ ∈ S_{x} and h_{t} is an L^{1}-finite energy process with values in ℝ) gives

(De^{−T D^{2}/2}α) (x) = −𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} (∫_{0}^{T} h′_{s}γ_{dB_{s}}) //_{T}^{−1}α(X_{T} (x))] (5.6)

wherein we have used the fact that γ_{v}^{∗} = −γ_{v}. Choosing h_{t} = 1 − t/T in this equation implies

(De^{−T D^{2}/2}α) (x) = \frac{1}{T} 𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} γ_{B_{T}} //_{T}^{−1}α(X_{T} (x))] . (5.7)

Similarly, using Eq. (5.4) of Corollary 5.5 (with ξ ∈ S_{x}),

(e^{−T D^{2}/2}Dα(x), ξ)_{S_{x}} = −𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} (//_{T}^{−1}α(X_{T} (x)), ∫_{0}^{T} γ_{dB_{s}}ℓ′_{s})_{S_{x}}]

where ℓ is an L^{1}-finite energy process with values in S_{x} such that ℓ_{T} = ξ and ℓ_{0} = 0. Taking ℓ_{t} = tξ/T in the previous equation gives the same formula for e^{−T D^{2}/2}Dα(x) as in the right hand side of Eq. (5.7). This of course should be the case since [D, L] = 0.

5.3. Application to d and d^{∗}. Let E = Ẽ = ΛT^{∗}M, L = L̃ = Δ and P_{t} := e^{tΔ̄/2} where Δ̄ is the self-adjoint extension of Δ. An application of Corollary 5.3 with m being either the interior product A or the exterior product C (see Example 2.11 and Definition 8.1 in Appendix I in Section 8 for the notation) gives the following theorem.

Theorem 5.7. Let M be a compact manifold, a be a bounded measurable section of ΛT^{∗}M and v ∈ ΛT_{x}M for some x ∈ M. Then

(5.8)  ⟨(dP_{T} a)_{x}, v⟩ = −𝔼 [⟨ //_{T}^{−1}a(X_{T} (x)), Q_{T} ∫_{0}^{T} Q_{t}^{−1} (A_{dB_{t}}Q_{t} ℓ′_{t})⟩]

(5.9) ⟨(d^{∗}P_{T} a)_{x}, v⟩ = −𝔼 [⟨ //_{T}^{−1}a(X_{T} (x)), Q_{T} ∫_{0}^{T} Q_{t}^{−1} (dB_{t} ∧ Q_{t} ℓ′_{t})⟩] ,

where ℓ is any L^{1}-finite energy process with values in ΛT_{x}M such that ℓ_{0} = v and ℓ_{T} = 0, Q_{t} is the solution to the differential equation

(5.10) \frac{d}{dt}Q_{t} = −\frac{1}{2}R_{//_{t}}^{tr} Q_{t} with Q_{0} = id_{ΛT_{x}M},

R_{//_{t}}^{tr} = //_{t}^{−1}R^{tr}//_{t} and R^{tr} is the Weitzenböck curvature term described in Eq. (8.16) of Lemma 8.9, Appendix I.

Similarly an application of Corollary 5.5 with m being either the interior product A or the exterior product C implies the following theorem.

Theorem 5.8. Let M be a compact manifold, a be a C^{1} section (H^{1} would do) of ΛT^{∗}M and v ∈ ΛT_{x}M for some x ∈ M. Then

(5.11)  ⟨(P_{T} da)_{x}, v⟩ = 𝔼 [⟨//_{T}^{−1}a(X_{T} (x)), Q_{T} ∫_{0}^{T} Q_{t}^{−1} (A_{dB_{t}}Q_{t} ℓ′_{t})⟩]

(5.12)  ⟨(P_{T} d^{∗}a)_{x}, v⟩ = 𝔼 [⟨//_{T}^{−1}a(X_{T} (x)), Q_{T} ∫_{0}^{T} Q_{t}^{−1} (dB_{t} ∧ Q_{t} ℓ′_{t})⟩] ,

where ℓ is any L^{1}-finite energy process with values in ΛT_{x}M such that ℓ_{0} = 0 and ℓ_{T} = v and Q_{t} solves Eq. (5.10) as above.

This theorem also follows from Theorem 5.7 because dP_{T} a = P_{T} da and d^{∗}P_{T} a = P_{T} d^{∗}a for C^{1} sections a of ΛT^{∗}M, see Remark 9.9 below.

In the special case where ℓ′ ≡ constant, the formulas (5.9) and (5.10) have been derived by Elworthy and Li [26] from a non-intrinsic formula by the techniques of filtering out redundant noise, as developed in Elworthy and Yor [27]; see [23] for a general account on this.

Specializing Eq. (5.8) to zero forms gives the following Bismut type formula, see [24], [58], [56].

Corollary 5.9. Let f : M → ℝ be bounded and measurable, x ∈ M and v ∈ T_{x}M. Then, for any L^{1}-finite energy process ℓ with values T_{x}M such that ℓ_{0} = v, and ℓ_{T} = 0, then

⟨(dP_{T} f)_{x}, v⟩ = −𝔼 [f (X_{T} (x)) ∫_{0}^{T} (Q_{t} ℓ′_{t}, dB_{t})]

where Q_{t} is the Aut(T_{x}M)-valued process satisfying the differential equation:

(5.13) \frac{d}{dt}Q_{t} = −\frac{1}{2}Ric_{//_{t}}Q_{t} with Q_{0} = id_{T_{x}M} .

Proof. Letting a = f ∈ Λ^{0}(T^{∗}M) in Eq. (5.8) and using the fact that //_{t} and Q_{t} act as the identity on 0-forms we find that

⟨(dP_{T} f)_{x}, v⟩ = −𝔼 [f(X_{T} (x)) ∫_{0}^{T} A_{dB_{t}}Q_{t} ℓ′_{t}]

= −𝔼 [f(X_{T} (x)) ∫_{0}^{T} (Q_{t} ℓ′_{t}, dB_{t})] ,

where Q_{t} is the restriction of the solution of Eq. (5.10) to T_{x}M. By Eq. (8.17) of Appendix I, R^{tr}|T M = Ric, and thus Q_{t} restricted to T_{x}M solves Eq. (5.13).

The following theorem is a special case of Eq. (5.9) of Theorem 5.7 and improves a result in [19] (see Corollary 5.18 below) by giving a formula for 𝔼 [∇ · Y (X_{T} (x))] which does not contain derivatives of the curvature tensor.

Theorem 5.10. Let M be a compact Riemannian manifold, Q_{t} denote the solution to Eq. (5.13), Y be a smooth vector field on M, ∇ · Y denote the divergence of Y, T > 0 and ℓ be a real-valued L^{1}-finite energy process. If ℓ_{0} = 0 and ℓ_{T} = 1, then

𝔼[∇ · Y (X_{T} (x))] = −𝔼 [(//_{T}^{−1}Y (X_{T} (x)), Q_{T} ∫_{0}^{T} ℓ′_{t} Q_{t}^{−1}dB_{t} )] . (5.14)

Proof. Let a = (Y, ·) ∈ Ω^{1}(M). By Eq. (8.9) of the Appendix

d^{∗}a = − ∑_{i} A_{e_{i}}∇_{e_{i}}a = − ∑_{i} A_{e_{i}}(∇_{e_{i}}Y, ·) = −∇ · Y.

Applying Eq. (5.12) with v = 1 ∈ Λ^{0}(T_{x}M) implies,

𝔼[∇ · Y (X_{T} (x))] = − ⟨(P_{T} d^{∗}a)_{x}, 1⟩
= − 𝔼 [⟨//_{T}^{−1}a(X_{T} (x)), Q_{T} ∫_{0}^{T} Q_{t}^{−1}ℓ′_{t} dB_{t}⟩]

where we have used the fact that R^{tr} = 0 on Λ^{0}(T M) and R^{tr} = Ric on Λ^{1}(T M) and hence that Q_{t}|Λ^{0}(T_{x}M) = id and Q_{t}|Λ^{1}(T_{x}M) solves Eq. (5.13). This proves Eq. (5.14) because //_{T}^{−1}a(X_{T} (x)) = (//_{T}^{−1}Y (X_{T} (x)), ·).

Note that compactness of M is not essential here: the formulas in Theorem 5.8 only require the martingale property of (3.15), or (3.19), for some ℓ such that ℓ_{0} = v and ℓ_{T} = 0. As indicated in Remark 4.8, this can always be achieved and gives a formula for (dP_{T} a)(x), respectively. (d^{∗}P_{T} a)(x), as long as P_{T} a is well-defined, i.e. Q_{T}^{−1}//_{T}^{−1}a(X_{T} (x)) ∈ L^{1}. In particular, a need not be differentiable. Also the finite lifetime of the Brownian motion only effects the stochastic representation of P_{T} a, see Section 9.1 of Appendix II, but not the given argument. (See Section 6 for precise statements in this direction.) In the situation of Theorem 5.10 this shows that the right-hand side of Eq. (5.14) is just −d^{∗}P_{T} a where a = (Y, ·). To verify however that

d^{∗}P_{T} a = −𝔼 [(∇ · Y)(X_{T} (x))] (≡ P_{T} d^{∗}a) (5.15)

requires assumptions (in particular, differentiability of a): it precisely reflects the property that the local martingale (3.5) is actually a martingale, from where (5.15) follows by taking expectations.

5.3.1. An Integrated Logarithmic Gradient Estimate for the Heat Kernel on M. As an application of formula (5.14) we get the following integrated estimate for the gradient of the logarithmic derivative of the heat kernel on M. Of course this result may be derived by partial differential equation techniques as well.

Theorem 5.11. Let M be a compact Riemannian manifold without boundary. For every q ∈ [1, ∞[ there is a constant C_{q} < ∞ such that for all t ≤ 1,

(∫_{M} |∇_{z} log p_{t}(x, z)|^{q} p_{t}(x, z) vol(dz))^{1/q} ≤ C_{q} e^{2Kt}t^{−1/2}, (5.16)

where K is a bound on the Ricci curvature (Ric) of M.

Remark 5.12. This estimate with q = 1 may be used to show that pinned Brownian motion on a compact manifold is a semimartingale.

Proof. Without loss of generality, we may assume that q ≥ 2. Since

𝔼[∇ · Y (X_{T} (x))] = ∫_{M} p_{T} (x, z)∇ · Y (z) vol(dz)
= − ∫_{M} (Y (z), ∇_{z} log p_{T} (x, z))_{T M} p_{T} (x, z) vol(dz),

Eq. (5.14) is equivalent to:

∫_{M} (Y (z), ∇_{z} log p_{T} (x, z))_{T M} p_{T} (x, z) vol(dz)

= 𝔼 [(//_{T}^{−1}Y (X_{T} (x)), Q_{T} ∫_{0}^{T} ℓ′_{t}Q_{t}^{−1}dB_{t} )] ,

where ℓ is a real-valued L^{1}-finite energy process such that ℓ_{0} = 0 and ℓ_{T} = 1.

If K is a bound on Ric and 1/p + 1/q = 1, Hölder’s and the Burkholder-Davis-Gundy inequalities imply

|∫_{M} (Y (z), ∇_{z} log p_{T} (x, z))_{T M} p_{T} (x, z) vol(dz)|

≤ e^{KT} 𝔼 [|Y (X_{T} (x))| |∫_{0}^{T} ℓ′_{t} Q_{t}^{−1}dB_{t}|]

≤ e^{KT} (𝔼|Y (X_{T} (x))|^{p})^{1/p} (𝔼 |∫_{0}^{T} ℓ′_{t} Q_{t}^{−1}dB_{t}|^{q})^{1/q}

≤ C_{q}e^{KT} (𝔼|Y (X_{T} (x))|^{p})^{1/p} (𝔼 |∫_{0}^{T} tr [(Q_{t}^{−1})^{∗} Q_{t}^{−1}] ℓ_{t}^{′ 2} dt|^{q/2})^{1/q}

≤ C_{q}e^{2KT} (𝔼|Y (X_{T} (x))|^{p})^{1/p} (𝔼 |∫_{0}^{T} ℓ_{t}^{′ 2} dt|^{q/2})^{1/q}

for some constant C_{q} < ∞. Choosing ℓ_{t} := t/T in this inequality shows,

|∫_{M} (Y (z), ∇_{z} log p_{T} (x, z))_{T M} p_{T} (x, z) vol(dz)|
≤ T^{−1/2}C_{q}e^{2KT} (𝔼|Y (X_{T} (x))|^{p})^{1/p}
= T^{−1/2}C_{q}e^{2KT} (∫_{M} |Y (z)|^{p} p_{T} (x, z) vol(dz))^{1/p} .

Now choose Y (z) := |∇_{z} log p_{T} (x, z)|^{q−2}∇_{z} log p_{T} (x, z) to get

∫M |∇_{z} log p_{T} (x, z)|^{q} p_{T} (x, z) vol(dz)

≤ T^{−1/2}C_{q}e^{2KT} (∫_{M} |∇_{z} log p_{T} (x, z)|^{p(q−1)} p_{T} (x, z) vol(dz))^{1/p}

= T^{−1/2}C_{q}e^{2KT} (∫_{M} |∇_{z} log p_{T} (x, z)|^{q} p_{T} (x, z) vol(dz))^{1−1/q} .

Solving this equation for (∫_{M} |∇_{z} log p_{T} (x, z)|^{q}p_{T} (x, z) vol(dz))^{1/q} proves Eq. (5.16).

5.4. Formulas for ∇e^{T L/2}. In this subsection, we will write out the results in Corollaries 5.3, 5.5 and Theorem 4.3 when D = ∇. Let E → M be a Riemannian vector bundle with metric compatible covariant derivative ∇^{E}, R ∈ Γ(End(E)), Ẽ := T^{∗}M ⊗ E, m = id (the identity multiplication map), D_{m} = ∇^{E}, and L = □ − R. We also define R̃ ∈ Γ(End(Ẽ)) and ρ ∈ Γ(Hom(E, Ẽ)) by

R̃ = Ric^{tr} ⊗ id_{E} −2R^{E} · + id_{T^{∗}M} ⊗R,

and

ρ = ∇ · R^{E} + (∇^{End (E)}R).

By Proposition 2.15, L := □ − R, L̃ = ˜□ − R̃, ρ, and m satisfy Assumption 2. We have the following immediate consequences of Corollaries 5.3 and 5.5.

Theorem 5.13. Suppose that α is a bounded measurable section of E and ξ ∈ Ẽ_{x}^{∗} = T_{x}M ⊗ E_{x}^{∗} and {ℓ_{t}}_{0≤t≤T} a L^{1}-finite energy process with values in Ẽ_{x}^{∗} (see Definition 3.6) such that ℓ_{T} = 0 and ℓ_{0} = ξ ∈ Ẽ_{x}^{∗}. Then

⟨∇e^{T L/2}α(x), ξ⟩ = −𝔼 [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩] , (5.17)

where U^{ℓ} is given by Eq. (3.17). If we further assume that α is an H^{1} section of E and {ℓ_{t}}_{0≤t≤T} is an L^{1}-finite energy process with values in Ẽ_{x}^{∗} such that ℓ_{0} = 0, then

⟨e^{T L̃/2}∇α(x), ξ⟩ = 𝔼 [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩] . (5.18)

Remark 5.14. We may extend Eq. (5.18) using Remark 4.7 as follows. Let R̂ be an arbitrary section of End(Ẽ), L̂ = ˜□ − R̂ and

\frac{d}{dt}Q̂_{t} = −\frac{1}{2}Q̂_{t} R̂_{//_{t}} with Q̂_{0} = idẼ_{x}, (5.19)

where as before R̂_{//_{t}} := (//Ẽ ; t)^{−1}R̂ //Ẽ ; t. Then

⟨e^{T L̂/2}∇α(x), ξ⟩ = 𝔼[⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩]

where ℓ_{t} is any L^{1}-finite energy process such that ℓ_{0} = 0 and ℓ_{T} = (Q̂_{T} Q̃_{T}^{−1})^{tr}ξ.

By using Theorem 4.3, we may get another (more explicit) formula for ∇e^{T L/2}. This theorem will be given after the following preparatory Lemma.

Lemma 5.15. The transpose of the multiplication map m = id is the “annihilation” operator A, where A_{v} : Ẽ^{∗} = T M ⊗ E^{∗} → E^{∗} is determined by

A_{v}(w ⊗ α) = (v, w)α for all v, w ∈ T_{x}M, α ∈ E_{x}^{∗}, x ∈ M.

Also

(5.20)  R̃^{tr} = Ric ⊗ id_{E^{∗}} +2 ∑_{i=1}^{n} e_{i} ⊗ R^{E^{∗}}(·, e_{i}) + id_{T M} ⊗R^{tr},

where {e_{i}}_{i=1}^{n} is a local orthonormal frame and e_{i} ⊗ R^{E^{∗}}(·, e_{i}) ∈ Γ(End(Ẽ^{∗})) is determined by

e_{i} ⊗ R^{E^{∗}}(·, e_{i})(v ⊗ α) = e_{i} ⊗ R^{E^{∗}}(v, e_{i})α = −e_{i} ⊗ α ∘ R^{E}(v, e_{i})

for v ∈ T_{x}M and α ∈ E_{x}^{∗}. Moreover,

(5.21) ρ^{tr} = (∇ · R^{E})^{tr} + (∇^{End (E)}R)^{tr} ∈ Γ(Hom(T M ⊗ E^{∗}, E^{∗})),

where

(∇ · R^{E})^{tr} (v ⊗ α) = α ∘ (∇ · R^{E})(v) = α ∘ ((∇_{e_{i}}R^{E})(e_{i}, v)) (5.22)

and

(5.23) (∇^{End (E)}R)^{tr} (v ⊗ α) = α ∘ (∇_{v}^{End (E)}R).

Proof. Let v, w ∈ T_{x}M, ξ ∈ E_{x} and α ∈ E_{x}^{∗}. Since m_{v} : E_{x} → Ẽ_{x} = T_{x}^{∗}M ⊗ E_{x} is given by m_{v}ξ = (v, ·) ⊗ ξ, it follows that

⟨m_{v}ξ, w ⊗ α⟩ = (v, w)⟨ξ, α⟩ = ⟨ξ, (v, w)α⟩ = ⟨ξ, A_{v}(w ⊗ α)⟩

which shows that m_{v}^{tr} = A_{v}. Now let β ∈ Hom(T_{x}M, E_{x}) ≅ T_{x}^{∗}M ⊗ E_{x}. Then Eq. (5.20) follows by taking transposes of Eq. (2.7) along with the computation,

⟨(R^{E}·)^{tr}(v ⊗ α), β⟩ = ⟨v ⊗ α, R^{E} · β⟩

= ∑_{i=1}^{n}⟨α, R^{E}(v, e_{i})β(e_{i})⟩

= ∑_{i=1}^{n}⟨e_{i} ⊗ R^{E}(v, e_{i})^{tr}α, β⟩

= − ∑_{i=1}^{n}⟨e_{i} ⊗ R^{E^{∗}}(v, e_{i})α, β⟩.

Eqs. (5.21), (5.22) and (5.23) are proved similarly:

⟨(∇ · R^{E})ξ, v ⊗ α⟩ = ∑_{i=1}^{n}⟨(∇_{e_{i}}R^{E})(e_{i}, v)ξ, α⟩

= ∑_{i=1}^{n}⟨ξ, α ∘ ((∇_{e_{i}}R^{E})(e_{i}, v))⟩,

and

⟨(∇^{End (E)}R)ξ, v ⊗ α⟩ = ⟨(∇_{v}^{End (E)}R)ξ, α⟩
= ⟨ξ, (∇^{End (E)}R)^{tr}(v ⊗ α)⟩ = ⟨ξ, α ∘ (∇_{v}^{End (E)}R)⟩.

Theorem 5.16. Let R ∈ Γ(End(E)), Q be as in Eq. (3.1), L = □−R, and m = id. Suppose that α is a bounded measurable section of E, ℓ_{s} is a T_{x}M-valued L^{1}-finite energy process and

V_{t}^{ℓ} := ∫_{0}^{t} ((ℓ′_{s} + \frac{1}{2} Ric_{//_{s}} ℓ_{s}, dB_{s}) − R_{//_{s}}^{E}(ℓ_{s}, δB_{s}) + \frac{1}{2}(ℓ_{s}, \overleftarrow{dB}_{s})R_{//_{s}}) Q_{s}^{−1},

where δB_{s}, and \overleftarrow{dB}_{s} denotes the Fisk-Stratonovich and backwards Itô differential respectively. (More precisely, if X is another semimartingale, then X_{s} δB_{s} = X_{s} dB_{s} + \frac{1}{2}dX_{s}dB_{s} and X_{s} \overleftarrow{dB}_{s} = X_{s} dB_{s} +dX_{s}dB_{s}.) If ℓ_{0} = v ∈ T_{x}M and ℓ_{T} = 0 then

∇_{v}e^{T L/2}α(x) = −𝔼 [V_{T}^{ℓ} Q_{T} //_{T}^{−1}α(X_{T} (x))] , (5.24)

and assuming in addition that a ∈ Γ(E), if ℓ_{0} = 0 and ℓ_{T} = v ∈ T_{x}M then

(e^{T ˜□/2}∇α)_{v} = 𝔼 [V_{T}^{ℓ} Q_{T} //_{T}^{−1}α(X_{T} (x))] . (5.25)

Proof. Let a_{t} := e^{(T −t)L/2}α, ξ ∈ E_{x}^{∗}, ℓ_{s} be an L^{1}-finite energy path with values in T_{x}M and k_{s} = ℓ_{s} ⊗ ξ. In order to apply Theorem 4.3 we need to work out Û_{t}^{k} defined in Eq. (3.21). Using Lemma 5.15,

m_{dB_{s}}^{tr} (k′_{s} + \frac{1}{2}R̃_{//_{s}}^{tr}k_{s}) = A_{dB_{s}} (ℓ′_{s} ⊗ ξ + \frac{1}{2}R̃_{//_{s}}^{tr}(ℓ_{s} ⊗ ξ))
= (ℓ′_{s}, dB_{s})ξ + \frac{1}{2}(Ric_{//_{s}} ℓ_{s}, dB_{s})ξ

+ ∑_{i=1}^{n}(e_{i}, dB_{s}) R_{//_{s}}^{E^{∗}}(ℓ_{s}, e_{i})ξ + \frac{1}{2}(ℓ_{s}, dB_{s}) R_{//_{s}}^{tr}ξ

= (ℓ′_{s} + \frac{1}{2} Ric_{//_{s}} ℓ_{s}, dB_{s})ξ + R_{//_{s}}^{E^{∗}}(ℓ_{s}, dB_{s})ξ

+ \frac{1}{2}(ℓ_{s}, dB_{s}) R_{//_{s}}^{tr}ξ.

Similarly,

ρ_{//_{s}}^{tr}k_{s} = (∇ · R^{E})_{//_{s}}^{tr} (ℓ_{s})ξ + (∇^{End (E)}R)_{//_{s}}^{tr}(ℓ_{s})ξ

and hence from Eq. (3.21),

Û_{t}^{k} = ∫_{0}^{t} Q_{s}^{−1} ((ℓ′_{s} + \frac{1}{2} Ric_{//_{s}} ℓ_{s}, dB_{s})ξ + R_{//_{s}}^{E^{∗}}(ℓ_{s}, dB_{s})ξ + \frac{1}{2}(ℓ_{s}, dB_{s})R_{//_{s}}^{tr}ξ)

+ \frac{1}{2} ∫_{0}^{t} Q_{s}^{−1} ((∇ · R^{E})_{//_{s}}^{tr} (ℓ_{s})ξ + (∇^{End (E)}R)_{//_{s}}^{tr}(ℓ_{s})ξ) ds

= ∫_{0}^{t} Q_{s}^{−1} ((ℓ′_{s} + \frac{1}{2} Ric_{//_{s}} ℓ_{s}, dB_{s})ξ + R_{//_{s}}^{E^{∗}}(ℓ_{s}, δB_{s})ξ + \frac{1}{2}(ℓ_{s}, \overleftarrow{dB}_{s})R_{//_{s}}^{tr}ξ)
= (V_{t}^{ℓ})^{tr}ξ,

where in the last equality recall that R^{E^{∗}}(v, w) = −(R^{E}(v, w))^{tr}. This computation along with Eq. (4.5) of Theorem 4.3 gives:

If ℓ_{T} = 0 and ℓ_{0} = v ∈ T_{x}M then

⟨∇_{v}e^{T L/2}α(x), ξ⟩ = ⟨∇e^{T L/2}α(x), v ⊗ ξ⟩
= −𝔼[⟨Q_{T} //_{T}^{−1}a_{T} (X_{T} (x)), (V_{T}^{ℓ})^{tr}ξ⟩]

which implies Eq. (5.24) since ξ ∈ E_{x}^{∗} is arbitrary. Similarly if ℓ_{T} = v ∈ T_{x}M and ℓ_{0} = 0, then by Eq. (4.6) of Theorem 4.3

𝔼[⟨//_{T}^{−1}∇α(X_{T} (x)), v ⊗ ξ⟩] = 𝔼[⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), (V_{T}^{ℓ})^{tr}ξ⟩].

By Eq. (4.15) of Proposition 4.5,

⟨(e^{T ˜□/2}∇α)_{v}, ξ⟩ : = ⟨(e^{T ˜□/2}∇α)(x), v ⊗ ξ⟩
= 𝔼[⟨//_{T}^{−1}∇α(X_{T} (x)), v ⊗ ξ⟩].

Combining the last two equations proves Eq. (5.25).

Corollary 5.17. Let α ∈ L^{2}(E) and T > 0 and Q_{t} be the End(T_{x}M)-valued process defined as the solution to the ordinary differential equation

Q̇_{t} + \frac{1}{2} Ric_{//_{t}} Q_{t} = 0 with Q_{0} = id_{T_{x}M} .

If h_{T} = 0 and h_{0} = v ∈ T_{x}M then

∇_{v}e^{T □/2}α = −𝔼 [{∫_{0}^{T} [(Q_{t}ḣ_{t}, dB_{t}) + R_{//_{t}}^{E} (δB_{t}, Q_{t}h_{t})]} //_{T}^{−1}α(X_{T})] . (5.26)

Assume in addition that α ∈ Γ(E). If h_{T} = Q_{T}^{−1}v ∈ T_{x}M and h_{0} = 0, then

(e^{T ˜□/2}∇α)_{v} = 𝔼 [{∫_{0}^{T} [(Q_{t}ḣ_{t}, dB_{t}) + R_{//_{t}}^{E} (δB_{t}, Q_{t}h_{t})]} //_{T}^{−1}α(X_{T})] (5.27)

where Ẽ = T^{∗}M ⊗ E.

Proof. This follows from an application of Theorem 5.16 with R = 0 and ℓ_{t} := Q_{t}h_{t}.

As an application of Theorem 5.16 we may recover the following result in [19] (see Theorem 4.1 and Corollary 4.3). A better version of this formula has already appeared in Theorem 5.10 above.

Corollary 5.18. Suppose that Y is a C^{1} vector field on M and h is an L^{1}-finite energy path with values in ℝ such that h_{0} = 0 and h_{T} = 1, then

𝔼[(∇ · Y) (X_{T} (x))] = 𝔼 (//_{T}^{−1}Y (X_{T} (x)), ∫_{0}^{T} (h′_{s}dB_{s} − \frac{1}{2}h_{s} Ric_{//_{s}} \overleftarrow{dB}_{s})) . (5.28)

Proof. By Theorem 5.16 with E = T M, R = 0 (and hence Q = id) and ℓ_{0} = 0, we have

𝔼 [//_{T}^{−1} (∇_{//_{T} ℓ_{T}} α) (X_{T} (x))] = 𝔼 [V_{T}^{ℓ} //_{T}^{−1}α(X_{T} (x))] , (5.29)

where

V_{t}^{ℓ} := ∫_{0}^{t}(ℓ′_{s} + \frac{1}{2} Ric_{//_{s}} ℓ_{s}, dB_{s}) − ∫_{0}^{t} R_{//_{s}}^{T M}(ℓ_{s}, δB_{s}).

Let {e_{i}}_{i=1}^{n} be an orthonormal basis for T_{x}M. Replace ℓ in Eq. (5.29) by ℓ_{s}^{i} := h_{s}e_{i}, take the inner product with e_{i} and then sum on i to find:

𝔼[(∇ · Y) (X_{T} (x))] = ∑_{i=1}^{n} 𝔼[(∇_{//_{T} e_{i}}Y (X_{T} (x)), //_{T} e_{i})]

= ∑_{i=1}^{n} 𝔼 [(//_{T}^{−1}∇_{//_{T} ℓ_{T}^{i}} Y (X_{T} (x)), e_{i})]

= ∑_{i=1}^{n} 𝔼 [(V_{T}^{ℓ^{i}}//_{T}^{−1}Y (X_{T} (x)), e_{i})]

= ∑_{i=1}^{n} 𝔼 [(//_{T}^{−1}Y (X_{T} (x)), (V_{T}^{ℓ^{i}})^{∗}e_{i})] .

This finishes the proof since

∑_{i=1}^{n}(V_{T}^{ℓ^{i}})^{∗}e_{i} = ∫_{0}^{T} ∑_{i=1}^{n} ((h′_{s}e_{i} + \frac{1}{2}h_{s} Ric_{//_{s}} e_{i}, dB_{s})e_{i} + R_{//_{s}}^{T M}(h_{s}e_{i}, δB_{s})e_{i})

= ∫_{0}^{T} ((h′_{s} + \frac{1}{2}h_{s} Ric_{//_{s}})dB_{s} − h_{s} Ric_{//_{s}} δB_{s})

and

\frac{1}{2} h_{s} Ric_{//_{s}} dB_{s} − h_{s} Ric_{//_{s}} δB_{s}

= \frac{1}{2} h_{s} Ric_{//_{s}} dB_{s} − h_{s} Ric_{//_{s}} dB_{s} − \frac{1}{2} h_{s} d Ric_{//_{s}} dB_{s}

= −\frac{1}{2} h_{s} Ric_{//_{s}} dB_{s} − \frac{1}{2}h_{s} d Ric_{//_{s}} dB_{s}

= −\frac{1}{2} h_{s} Ric_{//_{s}} \overleftarrow{dB}_{s}.

Remark 5.19. The backward’s Itô differential in Eq. (5.28) may be expressed as

(5.30) h_{s} Ric_{//_{s}} \overleftarrow{dB}_{s} = h_{s} Ric_{//_{s}} dB_{s} + \frac{1}{2}h_{s} //_{s}^{−1} ∇scal(X_{s}(x)) ds,

where ∇scal denotes the gradient of the scalar curvature of M. The proof proceeds as follows:

h_{s} Ric_{//_{s}} \overleftarrow{dB}_{s} = h_{s} Ric_{//_{s}} dB_{s} + h_{s} d Ric_{//_{s}} dB_{s}
= h_{s} Ric_{//_{s}} dB_{s} + ∑_{i=1}^{n} h_{s} //_{s}^{−1} (∇_{//_{s}e_{i}} Ric) //_{s}e_{i} ds
(5.31)  = h_{s} Ric_{//_{s}} dB_{s} + h_{s} //_{s}^{−1}∇ · Ric ds

where ∇ · Ric is the vector field on M given by

∇ · Ric := ∑_{j=1}^{n} (∇_{e_{j}}Ric) e_{j}

where {e_{j}}_{j=1}^{n} is a local orthonormal frame on M. On the other hand for v ∈ T_{x}M,

v scal = ∑_{i=1}^{n} v (Ric e_{i}, e_{i}) = ∑_{i=1}^{n}(∇_{v} Ric e_{i}, e_{i})

= ∑_{i,j=1}^{n} ((∇_{v}R^{T M})(e_{i}, e_{j})e_{j}, e_{i})

which by the Bianchi identity (d_{∇}R^{T M} = 0) may be written as

v scal = − ∑_{i,j=1}^{n} ((∇_{e_{j}}R^{T M})(v, e_{i})e_{j}, e_{i}) − ∑_{i,j=1}^{n}((∇_{e_{i}}R^{T M})(e_{j}, v)e_{j}, e_{i})

= ∑_{i,j=1}^{n} ((∇_{e_{j}}R^{T M})(v, e_{i})e_{i}, e_{j}) + ∑_{i,j=1}^{n}((∇_{e_{i}}R^{T M})(v, e_{j})e_{j}, e_{i})

= 2 ∑_{j=1}^{n} ((∇_{e_{j}}Ric)v, e_{j}) = 2 ∑_{j=1}^{n} ((∇_{e_{j}}Ric)e_{j}, v)
= 2 (∇ · Ric, v).

Hence ∇ · Ric = ∇scal/2 which combined with Eq. (5.31) proves Eq. (5.30).

5.5. Formulas for ∇e^{tΔ/2} on Ω^{1}(M). Using Example 2.11, Theorem 5.13 we may write formulas for ∇e^{tΔ/2} on Ω^{k}(M) for 0 ≤ k ≤ n. Rather than doing this in general, we will content ourselves with the case k = 1. So in this section let E = T^{∗}M, Ẽ = T^{∗}M ⊗ T^{∗}M, and Δ = −d^{∗}d − dd^{∗} on Ω^{1}(M).

Proposition 5.20. Suppose that α is a bounded measurable differential 1-form (i.e. section of T^{∗}M) and ξ ∈ Ẽ_{x}^{∗} = T_{x}M ⊗ T_{x}M. Let {ℓ_{t}}_{0≤t≤T} be an L^{1}-finite energy process with values in T_{x}M ⊗T_{x}M (see Definition 3.6) such that ℓ_{T} = 0 and ℓ_{0} = ξ. Then

⟨∇e^{T Δ/2}α(x), ξ⟩ = −𝔼[⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{T}^{ℓ} ⟩], (5.32)

where

U_{t}^{ℓ} = ∫_{0}^{t} Q_{s}^{−1}A_{dB_{s}} Q̃_{s} ℓ′_{s} + \frac{1}{2} ∫_{0}^{t} Q_{s}^{−1}ρ_{//_{s}}^{tr} Q̃_{s}ℓ_{s} ds, (5.33)

ρ^{tr} ∈ Γ(Hom(T M ⊗ T M, T M)) is given by

ρ^{tr}(v ⊗ w) = (∇_{v}Ric) w − (∇ · R^{T M}) (v)w, (5.34)

Q and Q̃ are defined by Eqs. (3.11) and (3.12) with R^{tr} = Ric and

(5.35)  R̃^{tr} = Ric ⊗ id_{T M} +2 ∑_{i=1}^{n} e_{i} ⊗ R^{T M}(·, e_{i}) + id_{T M} ⊗ Ric,

where (see Lemma 5.15)

e_{i} ⊗ R^{T M}(·, e_{i})(v ⊗ w) = e_{i} ⊗ R^{T M}(v, e_{i})w

for v, w ∈ T_{x}M.

Proof. By Eq. (8.17) of Lemma 8.9, we have Δ = □ − R where R = Ric^{tr}. Let E = T^{∗}M, Ẽ = T^{∗}M ⊗T^{∗}M and m, ρ and R̃ be as in Proposition 2.15. By Lemma 5.15 above, m_{v}^{tr} = A_{v}, R̃^{tr} is given by Eq. (5.35) and

ρ^{tr} = (∇ · R^{T^{∗}M})^{tr} + (∇^{End (T^{∗}M)} Ric^{tr})^{tr}.

Since R^{T^{∗}M}(v, w) = − (R^{T M}(v, w))^{tr} and in general ∇_{v} (A^{tr}) = (∇_{v}A)^{tr}, the previous equation is the same as Eq. (5.34) above. Finally Eq. (5.32) follows from Corollary 5.3.

We may get another formula by using Theorem 5.16 in place of Corollary 5.3.

Proposition 5.21. Suppose α is a bounded measurable 1-form on M and ℓ_{s} is a T_{x}M-valued L^{1}-finite energy process such that ℓ_{0} = v ∈ T_{x}M and ℓ_{T} = 0. Then

∇_{v}e^{T Δ/2}α(x) = −𝔼[V_{T}^{ℓ} Q_{T} //_{T}^{−1}α(X_{T} (x))]

where

V_{t}^{ℓ} := ∫_{0}^{t} ((ℓ′_{s} + \frac{1}{2} Ric_{//_{s}} ℓ_{s}, dB_{s}) − R_{//_{s}}^{T^{∗}M}(ℓ_{s}, δB_{s}) + \frac{1}{2}(ℓ_{s}, \overleftarrow{dB}_{s})Ric_{//_{s}}^{tr}) Q_{s}^{−1}

and Q solves

\frac{dQ_{s}}{ds} = −Q_{s} Ric_{//_{s}}^{tr} with Q_{0} = id_{T_{x}^{∗}M} .

See Theorem 5.16 for the meaning of δB_{s} and \overleftarrow{dB}_{s}.

Remark 5.22. The expression for V_{t}^{ℓ} may be written solely in terms of Itô differentials using

R_{//_{s}}^{T^{∗}M}(ℓ_{s}, δB_{s}) = R_{//_{s}}^{T^{∗}M}(ℓ_{s}, dB_{s}) + \frac{1}{2} ∑_{i=1}^{n}(∇_{//_{s}e_{i}}R^{T^{∗}M})_{//_{s}}(ℓ_{s}, e_{i}) ds

= R_{//_{s}}^{T^{∗}M}(ℓ_{s}, dB_{s}) − \frac{1}{2}(∇ · R^{T^{∗}M})_{//_{s}}(ℓ_{s}) ds

and

(ℓ_{s}, \overleftarrow{dB}_{s})Ric_{//_{s}}^{tr} = (ℓ_{s}, dB_{s})Ric_{//_{s}}^{tr} + ∑_{i=1}^{n}(ℓ_{s}, e_{i}) (∇_{//_{s}e_{i}}Ric)_{//_{s}}^{tr} ds
= (ℓ_{s}, dB_{s})Ric_{//_{s}}^{tr} + (∇_{//_{s}ℓ_{s}}Ric)_{//_{s}}^{tr} ds.

6. Applications for Non-Compact M

For E = ΛT^{∗}M let again Δ = □ − R be the de Rham-Hodge Laplacian on Γ(E) where R ∈ Γ(End E) denotes the Weitzenböck curvature term, see Proposition 8.7 of Appendix I. Further, let R = min Spec R, i.e.

R(x) = min {⟨R_{x}v, v⟩ : v ∈ E_{x}, |v| = 1} , (6.1)

and consider the scalar semigroup P_{t}^{R} as defined in Appendix II, Sect. 9.1.

Now let Δ̂ be the Friedrichs extension of Δ|Γ_{c}(E) and P_{t}a = e^{t/2Δ̂}a be the semigroup on L^{2}(E) generated by Δ̂/2 and Q_{t} denote the solution to Eq. (3.1). Then,

P_{t}a(x) = 𝔼 [Q_{t} //_{t}^{−1}a(X_{t}(x)) 1_{{t<ζ(x)}}] (6.2)

for all a ∈ L^{2}(E) with P_{t}^{R}|a|(x) < ∞, see Appendix II, Theorem 9.4.

On a complete manifold, by the spectral theorem, one has dP_{t}a = P_{t}da, and dual to this, d^{∗}P_{t}a = P_{t}d^{∗}a, see Section 9.2 in Appendix II below. If we drop completeness then these equations are no longer true, even if M is BM-complete, see [59]. But we will show that there always exist Bismut type formulas for dP_{t}a and d^{∗}P_{t}a, not involving derivatives of a, independently whether a is smooth or not.

Theorem 6.1. Let M be a Riemannian manifold, a ∈ L^{2}(ΛT^{∗}M) and x ∈ M such

that P_{t}^{R}|a|(x) < ∞, further let τ be the first exit time of X(x) from some relatively compact neighborhood of x and T^{∗} = (T − ε) ∧ τ for some arbitrary small ε > 0. Then for any v ∈ ΛT_{x}M the following formulas hold:

⟨(dP_{T} a)_{x}, v⟩ = −𝔼 [⟨ //_{T}^{−1}a(X_{T} (x))1_{{T <ζ(x)}}, Q_{T} ∫_{0}^{T} Q_{t}^{−1} (A_{dB_{t}}Q_{t} ℓ′_{t})⟩]

⟨(d^{∗}P_{T} a)_{x}, v⟩ = −𝔼 [⟨ //_{T}^{−1}a(X_{T} (x))1_{{T <ζ(x)}}, Q_{T} ∫_{0}^{T} Q_{t}^{−1} (dB_{t} ∧ Q_{t} ℓ′_{t})⟩]

for any bounded finite energy process ℓ with values in ΛT_{x}M such that ℓ_{0} = v, ℓ_{t} = 0 for all t ≥ T^{∗}, and the property that 𝔼[(∫_{0}^{T^{∗}} |ℓ′_{s}|^{2}ds)^{1/2}] < ∞. If, in addition, a ∈ L^{2}(ΛT^{∗}M) is bounded on this neighborhood, one can take ε = 0.

Proof. We fix a relatively compact neighborhood U of x. Then |(P_{T −t}a)(X_{t})|, |(dP_{T −t}a)(X_{t})| and |(d^{∗}P_{T −t}a)(X_{t})|, where X ≡ X.(x) denotes our Brownian motion starting from x, as well as Q_{t} and Q_{t}^{tr}, are all bounded on the stochastic interval [0, T^{∗}]. This shows that the local martingales in Proposition 3.2 and in Theorem 3.7 are uniformly integrable martingales when stopped at t = T^{∗}. Taking expectations at time 0 and T^{∗} leads to

⟨(dP_{T} a)_{x}, v⟩ = −𝔼 [⟨Q_{T^{∗}}//_{T^{∗}}^{−1}(P_{T −T^{∗}}a)(X_{T^{∗}}), ∫_{0}^{T^{∗}} Q_{t}^{−1} (A_{dB_{t}}Q_{t}^{tr} ℓ′(t))⟩] .

Note that

Q_{T^{∗}} //_{T^{∗}}^{−1}(P_{T −T^{∗}}a)(X_{T^{∗}}) = 𝔼^{F_{T^{∗}}} [Q_{T} //_{T}^{−1}a(X_{T}) 1_{{T <ζ(x)}}]

which is by definition a bounded F_{T^{∗}}-measurable random variable. This gives the first formula of the Theorem, the second one is derived in a completely analogous way.

If a is bounded, eventually by modifying ℓ, we assume first that ℓ ≡ 0 already on [T ∧ τ, T] ∩ [T − ε, T] for some small ε > 0. Finally, this restriction can again be removed in the resulting formulas by letting ε tend to 0, see [58] for technical details.

6.1. Bismut’s Formula. The following example is taken from [58]. There are similar formulas for (du)_{x} if u is harmonic on some domain about x, see [58], [60].

Theorem 6.2. Let f: M → ℝ be a bounded measurable function, x ∈ M and v ∈ T_{x}M. Then, for any bounded finite energy process {ℓ_{t}}_{t∈[0,∞[} with values in

T_{x}M such that 𝔼[(∫_{0}^{τ∧T} |ℓ′_{s}|^{2} ds)^{1/2}] < ∞, and the property that ℓ_{0} = v, ℓ_{s} = 0 for all s ≥ τ ∧ T, the following formula holds:

⟨(dP_{T} f)_{x}, v⟩ = −𝔼 [f (X_{T} (x)) 1_{{T <ζ(x)}} ∫_{0}^{τ∧T} (Q_{s}ℓ′_{s}, dB_{s})] (6.3)

where τ is the first exit time of X(x) from some relatively compact open neighborhood D of x and Q is the process defined in Eq. (5.13) of Corollary 5.9.

On the other hand showing, for instance, that

⟨(dP_{T} f)_{x}, v⟩ = 𝔼 [⟨Q_{T} //_{T}^{−1}(df)_{X_{T} (x)}, v⟩ 1_{{T <ζ(x)}}]

is a quite different matter: it requires the martingale property of

Q_{t} //_{t}^{−1}(dP_{T −t}f)_{X_{t}(x)} 1_{{t<ζ(x)}} , 0 ≤ t ≤ T,

which comes down to a question of differentiation under the expectation. In particular, it is necessary for f to be differentiable.

6.2. Dirichlet Problem for Harmonic Forms. We conclude this section by specializing our results in case of the Dirichlet problem for harmonic forms on bounded domains. In particular, we present stochastic formulas for differentials and co-differential of harmonic forms on manifolds with boundary. These formulas can be used to prove local Harnack type estimates for harmonic forms in the same way as has been done for harmonic functions in [60].

Theorem 6.3. Let M be a compact Riemannian manifold with non-empty boundary ∂M, a ∈ Γ(ΛT^{∗}M), x ∈ M^{\}∂M and τ be the first time the Brownian motion X(x) hits the boundary ∂M. (For simplicity, a is assumed to be smooth up to the boundary of M.) Let Q_{t} denote the solution to Eq. (5.10) of Theorem 5.7 defined on the stochastic interval [0, τ] and Q_{t} = Q_{t}^{tr}. If a is harmonic (i.e. Δa ≡ − (d^{∗}d + dd^{∗}) a = 0) on M^{\}∂M, then

a(x) = 𝔼 [Q_{τ} //_{τ}^{−1}a(X_{τ} (x))] (6.4)

and for any v ∈ ΛT_{x}M the following formulas hold:

(6.5)  ⟨(da)_{x}, v⟩ = −𝔼 [⟨Q_{τ} //_{τ}^{−1}a(X_{τ} (x)), ∫_{0}^{τ} Q_{t}^{−1} (A_{dB_{t}}Q_{t} ℓ′_{t})⟩]
(6.6)  ⟨(d^{∗}a)_{x}, v⟩ = −𝔼 [⟨Q_{τ} //_{τ}^{−1}a(X_{τ} (x)), ∫_{0}^{τ} Q_{t}^{−1} (dB_{t} ∧ Q_{t} ℓ′_{t})⟩]

where {ℓ}_{s∈[0,∞[} is a bounded L^{1}-finite energy process taking values in ΛT_{x}M such

that ℓ_{0} = v, ℓ_{τ} = 0, and the property that (∫_{0}^{τ} |ℓ′_{s}|^{2}ds)^{1/2} ∈ L^{1}.

Proof. Note that a, da, d^{∗}a extend as bounded sections to M since M is compact. The proof is now essentially the same as the proof of Theorem 5.7 above with P_{t}a replaced by a and T by τ. The key point is that the local martingales of Proposition 3.2 and Theorem 3.7 are easily seen to be martingales up to the first exit time τ. Hence the optional sampling theorem applies to give stopped versions of Proposition 4.5, Corollary 5.3 and Corollary 5.5 from which Eqs. (6.4), (6.5) and (6.6) follow.

The method used in Theorem 6.3 can easily be adapted to other situations, for instance, if a is smooth on M^{\}∂M and extends only continuously to ∂M.

For technical details about how to construct finite energy processes ℓ satisfying the conditions in Theorems 6.1, 6.2 and 6.3 the reader is referred [58] and [60].

7. Higher Derivative Formulas

We can get higher derivative formulas by iterating our previous formulas following the ideas of Elworthy and Li in [24] and [26]. In order to carry this out, we will need a minor extension of the results in Section 4. Let {F_{t}} denote the filtration associated to the Brownian motion X_{t}(x) and 𝔼^{F_{t}} denote conditional expectation relative to the σ-field F_{t}. As usual we will assume that E, Ẽ, L, L̃, m satisfy Assumption 2 at the beginning of Section 4. For simplicity, let us assume that M is a compact manifold. The next theorem is the conditioned version of Theorem 3.7 and Corollaries 5.3 and 5.5.

Theorem 7.1. Let 0 ≤ τ < T, α be a bounded measurable section of E,

a_{t} := e^{(T −t)L/2}α,

{ℓ_{t}}_{τ≤t≤T} be an L^{1}-finite energy process with values in Ẽ_{x}^{∗} (see Definition 3.6), and N and Ñ be as in Eqs. (3.4) and (3.5). For 0 ≤ τ ≤ t ≤ T let

U_{τ,t}^{ℓ} := ∫_{τ}^{t} Q_{s}^{−1}m_{dB_{s}}^{tr} Q̃_{s}Q̃_{τ}^{−1}ℓ′_{s} + \frac{1}{2} ∫_{τ}^{t} Q_{s}^{−1}ρ_{//_{s}}^{tr} Q̃_{s}Q̃_{τ}^{−1}ℓ_{s}ds (7.1)

and

Z_{τ,t}^{ℓ} := ⟨Ñ_{t}, Q̃_{τ}^{−1}ℓ_{t}⟩ − ⟨N_{t}, U_{τ,t}^{ℓ}⟩. (7.2)

Then {Z_{τ,t}^{ℓ}}_{τ≤t≤T} is an {F_{t}}-martingale with

dZ_{τ,t}^{ℓ} = ⟨Q̃_{t}//_{t}^{−1}∇_{//_{t} dB_{t}}Da_{t}(X_{t}(x)), Q̃_{τ}^{−1}ℓ_{t}⟩

− ⟨Q_{t} //_{t}^{−1}∇_{//_{t}dB_{t}}a_{t}(X_{t}(x)), U_{τ,t}^{ℓ}⟩ − ⟨N_{t}, Q_{t}^{−1}m_{dB_{t}}^{tr} Q̃_{t} Q̃_{τ}^{−1}ℓ′_{t}⟩ (7.3)

and

⟨//_{τ}^{−1}Da_{τ} (X_{τ} (x)), ℓ_{τ} ⟩ = 𝔼^{F_{τ}} [⟨Q̃_{T} //_{T}^{−1}Da_{T} (X_{T} (x)), Q̃_{τ}^{−1}ℓ_{T} ⟩]
(7.4)  − 𝔼^{F_{τ}} [⟨Q_{T} //_{T}^{−1}a_{T} (X_{T} (x)), U_{τ,T}^{ℓ} ⟩].

Therefore, 1. if ℓ_{T} = 0 then

⟨//_{τ}^{−1}De^{(T −τ)L/2}α(X_{τ} (x)), ℓ_{τ} ⟩ = −𝔼^{F_{τ}} [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{τ,T}^{ℓ} ⟩], (7.5)

2. or if ℓ_{τ} = 0 then

𝔼^{F_{τ}} [⟨Q̃_{T} //_{T}^{−1}Dα(X_{T} (x)), Q̃_{τ}^{−1}ℓ_{T} ⟩] = 𝔼^{F_{τ}} [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), U_{τ,T}^{ℓ} ⟩]. (7.6)

Remark 7.2. The results in this theorem are direct analogues of Theorems 3.7, and Corollaries 5.3 and 5.5. In fact this theorem could be deduced using these results along with the strong Markov property of the Brownian motion X_{t}(x). (If H : C(ℝ_{+}; M) → ℝ_{+} is bounded measurable and τ a finite stopping time, then

𝔼^{F_{τ}} [H ∘ X_{τ+·}(x)] = 𝔼 [H ∘ X.(y)] |_{y=X_{τ} (x)} a.s.) (7.7)

We will sketch a proof here using the methods already developed in Section 3.

Proof. Notice that dU_{τ,t}^{ℓ} = Q_{t}^{−1}m_{//_{t} dB_{t}}^{tr} Q̃_{t}Q̃_{τ}^{−1}ℓ′_{t} + \frac{1}{2}Q_{t}^{−1}ρ_{//_{t}}^{tr} Q̃_{t}Q̃_{τ}^{−1}ℓ_{t} dt and hence by Theorem 3.4, U_{τ,t}^{ℓ} and Q̃_{τ}^{−1}ℓ_{t} for τ ≤ t ≤ T are a dual pair, i.e. {Z_{τ,t}^{ℓ}}_{τ≤t≤T} is an {F_{t}}-martingale. The same computations leading to Eq. (3.16) in Theorem 3.7 proves Eq. (7.3). (In fact Eq. (7.3) is Eq. (3.16) with ℓ_{t} replaced by Q̃_{τ}^{−1}ℓ_{t} and the lower limits in the integrals defining U_{t}^{Q̃_{τ}^{−1}ℓ} being changed from 0 to τ.)

Since Z_{τ,t}^{ℓ} is an {F_{t}}-martingale, Z_{τ,τ}^{ℓ} = 𝔼^{F_{τ}} [Z_{τ,T}^{ℓ}]. This identity is the same as Eq. (7.4) because

Z_{τ,τ}^{ℓ} = ⟨Ñ_{τ} , Q̃_{τ}^{−1}ℓ_{τ} ⟩ = ⟨Q̃_{τ} //_{τ}^{−1}Da_{τ} (X_{τ} (x)), Q̃_{τ}^{−1}ℓ_{τ} ⟩ = ⟨//_{τ}^{−1}Da_{τ} (X_{τ} (x)), ℓ_{τ} ⟩

and

𝔼^{F_{τ}} [Z_{τ,T}^{ℓ} ] = 𝔼^{F_{τ}} [⟨Ñ_{T} , Q̃_{τ}^{−1}ℓ_{T} ⟩] − 𝔼^{F_{τ}} [⟨N_{T} , U_{τ,T}^{ℓ} ⟩]

= 𝔼^{F_{τ}} [⟨Q̃_{T} //_{T}^{−1}Da_{T} (X_{T} (x)), Q̃_{τ}^{−1}ℓ_{T} ⟩] − 𝔼^{F_{τ}} [⟨Q_{T} //_{T}^{−1}a_{T} (X_{T} (x)), U_{τ,T}^{ℓ} ⟩].

Finally, Eqs. (7.5) and (7.6) are immediate consequences of Eq. (7.4).

The following Corollary is the conditioned analogue of Eq. (4.10) of Corollary 4.4.

Corollary 7.3. Let 0 ≤ τ < T, α be a bounded measurable section of E, {ℓ̄_{t}}_{τ≤t≤T} be an L^{1}-finite energy process with values in ℝ such that ℓ̄_{T} = 0. For 0 ≤ τ ≤ t ≤ T let

V ℓ̄ ; τ,t = ∫_{τ}^{t} Q̃_{τ}^{−1}Q̃_{s}(ℓ̄′_{s}m_{dB_{s}} + \frac{1}{2}ρ_{//_{s}} ℓ̄_{s} ds)Q_{s}^{−1}. (7.8)

Then

//_{τ}^{−1}De^{(T −τ)L/2}α(X_{τ} (x)) ℓ̄_{τ} = −𝔼^{F_{τ}} [V ℓ̄ ; τ,T Q_{T} //_{T}^{−1}α(X_{T} (x))] . (7.9)

Proof. Let ξ be a bounded F_{τ} -measurable random variable with values in Ẽ_{x}^{∗} and let ℓ_{t} = ℓ̄_{t}ξ. In this case

U_{τ,T}^{ℓ} = ∫_{τ}^{T} Q_{s}^{−1}m_{dB_{s}}^{tr} Q̃_{s}Q̃_{τ}^{−1}ℓ̄′_{s}ξ + \frac{1}{2} ∫_{τ}^{T} Q_{s}^{−1}ρ_{//_{s}}^{tr} Q̃_{s}Q̃_{τ}^{−1}ℓ̄_{s}ξ ds = (V ℓ̄ ; τ,T)^{tr}ξ

where V ℓ̄ ; τ,T is defined in Eq. (7.8). Hence by Eq. (7.5) above,

⟨//_{τ}^{−1}De^{(T −τ)L/2}α(X_{τ} (x)), ℓ̄_{τ} ξ⟩ = −𝔼^{F_{τ}} [⟨Q_{T} //_{T}^{−1}α(X_{T} (x)), (V ℓ̄ ; τ,T)^{tr}ξ⟩]

= −⟨𝔼^{F_{τ}} [V ℓ̄ ; τ,T Q_{T} //_{T}^{−1}α(X_{T} (x))], ξ⟩.

Since ξ is arbitrary, this proves Eq. (7.9).

7.1. Higher Derivative Formula for Dirac Operators. Let E = Ẽ = S → M be a spinor bundle over M, m = γ be the Clifford multiplication, D = D_{γ} the Dirac operator, and L = L̃ = −D^{2}. Recall that in this case ρ = 0, and R = R̃ = \frac{1}{4}scal.

Theorem 7.4. Suppose that 0 < T_{1} < T and ℓ is an L^{1}-finite energy process with values in ℝ such that ℓ_{0} = 2, ℓ_{T_{1}} = 1 and ℓ_{T} = 0, then

(D^{2}e^{−T D^{2}/2}α)(x) = 𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} γ_{∫_{0}^{T_{1}} ℓ′_{s_{1}} dB_{s_{1}}} γ_{∫_{T_{1}}^{T} ℓ′_{s_{2}} dB_{s_{2}}} //_{T}^{−1}α(X_{T} (x))] (7.10)

where α is a bounded measurable section of S. For example, if ℓ′_{s} = −(T_{1}^{−1}1_{[0,T_{1})} + (T − T_{1})^{−1}1_{[T_{1},T]}), Eq. (7.10) becomes

(D^{2}e^{−T D^{2}/2}α)(x)

= \frac{1}{T_{1}(T − T_{1})}𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} γ_{B_{T_{1}}} γ_{B_{T} −B_{T_{1}}} //_{T}^{−1} α(X_{T} (x))] . (7.11)

Proof. Since R = R̃ = \frac{1}{4}scal and ρ = 0,

Q_{t} = Q̃_{t} = e^{− \frac{1}{8} ∫_{0}^{t} scal(X_{s}(x))ds},

V ℓ̄ ; τ,t = e^{\frac{1}{8} ∫_{0}^{τ} scal(X_{s}(x))ds} ∫_{τ}^{t} ℓ̄′_{s}γ_{dB_{s}} = e^{\frac{1}{8} ∫_{0}^{τ} scal(X_{s}(x))ds}γ_{∫_{τ}^{t} ℓ̄′_{s}dB_{s}}

and hence by Corollary 7.3, for 0 ≤ τ < t,

//_{τ}^{−1}De^{−(t−τ)D^{2}/2}α(X_{τ} (x))ℓ̄_{τ} = −𝔼^{F_{τ}} [e^{− \frac{1}{8} ∫_{τ}^{t} scal(X_{s}(x))ds}γ_{∫_{τ}^{t} ℓ̄′_{s}dB_{s}}//_{t}^{−1}α(X_{t}(x))] (7.12)

where ℓ̄ is an ℝ-valued L^{1}-finite energy process such that ℓ̄_{t} = 0.

Let h := ℓ|_{[0,T_{1}]}−1 and k := ℓ|_{[T_{1},T]} so that h and k are L^{1}-finite energy processes on [0, T_{1}] and [T_{1}, T] respectively such that h_{0} = 1, h_{T_{1}} = 0, k_{T_{1}} = 1, and k_{0} = 0. Using the semigroup property of e^{−tD^{2}} and the fact that D commutes with e^{−tD^{2}},

D^{2}e^{−T D^{2}/2}α = De^{−T_{1} D^{2}/2}De^{−(T −T_{1})D^{2}/2}α.

By Eq. (7.12) with τ = 0 and t = T_{1},

(D^{2}e^{−T D^{2}/2}α)(x)
= −𝔼 [e^{− \frac{1}{8} ∫_{0}^{T_{1}} scal(X_{t}(x))dt} γ_{∫_{0}^{T_{1}} h′_{s}dB_{s}} //_{T_{1}}^{−1} De^{−(T −T_{1})D^{2}/2}α(X_{T_{1}}(x))]

and by Eq. (7.12) with τ = T_{1} and t = T

//_{T_{1}}^{−1} De^{−(T −T_{1})D^{2}/2}α(X_{T_{1}}(x))

= −𝔼^{F_{T_{1}}} [e^{− \frac{1}{8} ∫_{T_{1}}^{T} scal(X_{s}(x))ds}γ_{∫_{T_{1}}^{T} k′_{s}dB_{s}}//_{T}^{−1}α(X_{T} (x))] .

Combining these two equations implies that

(D^{2}e^{−T D^{2}/2}α) (x) = 𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} γ_{∫_{0}^{T_{1}} h′_{s_{1}} dB_{s_{1}}} γ_{∫_{T_{1}}^{T} k′_{s_{2}} dB_{s_{2}}} //_{T}^{−1}α(X_{T} (x))]

= 𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} γ_{∫_{0}^{T_{1}} ℓ′_{s_{1}} dB_{s_{1}}} γ_{∫_{T_{1}}^{T} ℓ′_{s_{2}} dB_{s_{2}}} //_{T}^{−1}α(X_{T} (x))] .

Remark 7.5. The method used in this proof already appears in the work of Elworthy and Li, see [24] in the context of 0-forms on a manifold and [26] for proving formulas for dd^{∗}P_{t}a, d^{∗}dP_{t}a, Δ^{k}P_{t}a where a is a differential form.

Remark 7.6. Let 0 < T_{1} < T_{2} < ⋯ < T_{n−1} < T_{n} = T. The previous Theorem may easily be extended to give

(D^{n}e^{−T D^{2}/2}α)(x)

= 𝔼 [e^{− \frac{1}{8} ∫_{0}^{T} scal(X_{t}(x))dt} ∫_{J} ℓ′_{s_{1}} ℓ′_{s_{2}} … ℓ′_{s_{n}}γ_{dB_{s_{1}}} γ_{dB_{s_{2}}} … γ_{dB_{sn}} //_{T}^{−1} a(X_{T} (x))] ,

where ℓ is an ℝ-valued L^{1}-finite energy process such that ℓ_{0} = n, ℓ_{T_{1}} = n − 1, …, ℓ_{T_{n−1}} = 1 and ℓ_{T_{n}} = 0 and

J = {(s_{1}, s_{2}, …, s_{n}) : 0 ≤ s_{1} ≤ T_{1} ≤ s_{2} ≤ T_{2} ≤ ⋯ ≤ T_{n−1} ≤ s_{n} ≤ T_{n}}.

7.2. A Hessian Formula. In this section we will work out a formula for

Hess(e^{T Δ/2}f) := ∇de^{T Δ/2}f

where f is a bounded measurable function on M.

Similar formulas hold for ∇de^{T Δ/2}α, where α is a differential form on M. Related Hessian formulas may also be found in Norris [44], Elworthy and Li [24, 26], Stroock and Turetsky [54, 55] and Hsu [35, 34].

Theorem 7.7. Let M be a compact Riemannian manifold, ξ ∈ T_{x}M ⊗ T_{x}M for some x ∈ M, further f : M → ℝ be a bounded measurable function on M. Also for 0 < T_{1} < T, let {ℓ_{s}}_{0≤s≤T_{1}} be an L^{1}-finite energy process with values in T_{x}M ⊗T_{x}M such that ℓ_{0} = ξ and ℓ_{T_{1}} = 0, and let {h_{s}}_{T_{1}≤s≤T} be an ℝ-valued L^{1}-finite energy process such that h_{T_{1}} = 1 and h_{T} = 0. Then

⟨Hess(e^{T Δ/2}f)(x), ξ⟩ = 𝔼 [f(X_{T} (x)) ∫_{T_{1}}^{T} h′_{s}(Q_{s}U_{T_{1}}^{ℓ}, dB_{s})] , (7.13)

where Q solves

(7.14) Q′_{s} = −\frac{1}{2} Ric_{//_{s}} Q_{s} with Q_{0} = id_{T_{x}M}

and U_{T_{1}}^{ℓ} is given in Eq. (5.33) of Proposition 5.20. (The process U_{T_{1}}^{ℓ} depends on the curvature tensor and its first derivatives.)

Proof. Since dΔ = Δd,

∇de^{T Δ/2}f = ∇de^{T_{1}Δ/2}e^{(T −T_{1})Δ/2}f = ∇e^{T_{1}Δ/2}de^{(T −T_{1})Δ/2}f.

Consequently by Proposition 5.20,

⟨∇de^{T Δ/2}f, ξ⟩ = −𝔼 [⟨Q_{T_{1}}//_{T_{1}}^{−1}de^{(T −T_{1})Δ/2}f(X_{T_{1}}(x)), U_{T_{1}}^{ℓ}⟩]
(7.15)  = −𝔼 [⟨//_{T_{1}}^{−1}de^{(T −T_{1})Δ/2}f(X_{T_{1}}(x)), Q_{T_{1}}U_{T_{1}}^{ℓ}⟩] .

Suppose now that {k_{s}}_{T_{1}≤s≤T} is an T_{x}M-valued L^{1}-finite energy process such that k_{T} = 0 and k_{T_{1}} = Q_{T_{1}}U_{T_{1}}^{ℓ}. By Eq. (7.5) of Theorem 7.1 above applied to the case where E = M × ℝ, Ẽ = T^{∗}M, L = Δ^{(0)}, L̃ = Δ^{(1)}, R = 0, R̃ = Ric^{tr} and m = C (as in item 1. of Example 2.11) gives

⟨//_{T_{1}}^{−1}de^{(T −T_{1})Δ/2}f(X_{T_{1}}(x)), Q_{T_{1}}U_{T_{1}}^{ℓ}⟩ = −𝔼^{F_{T_{1}}} [f(X_{T} (x))U_{T_{1},T}^{k} ] (7.16) where

U_{T_{1},T}^{k} = ∫_{T_{1}}^{T} C_{dB_{s}}^{tr}Q_{s}Q_{T_{1}}^{−1}k′_{s} = ∫_{T_{1}}^{T} (Q_{s}Q_{T_{1}}^{−1}k′_{s}, dB_{s}). (7.17)

(Notice that when applying Theorem 7.1, Q = id and Q̃ is the Q defined by Eq. (7.14) above.) Plugging Eq. (7.16) into Eq. (7.15) shows that

⟨∇de^{T Δ/2}f, ξ⟩ = 𝔼 [f(X_{T} (x)) ∫_{T_{1}}^{T} (Q_{s}Q_{T_{1}}^{−1}k′_{s}, dB_{s})] .

Taking k_{s} := h_{s}Q_{T_{1}}U_{T_{1}}^{ℓ} in this formula implies Eq. (7.13) which proves the theorem.

Remark 7.8. In a more general setting, it is possible to develop higher derivative formulas in the following situation. Let D_{1} : Γ(E) → Γ(E′) be a “Dirac type” operator such that D_{1}L−L′D_{1} = 0, assuming conditions to ensure that D_{1}e^{−tL/2} = e^{−tL′/2}D_{1}. Let D_{2} : Γ(E′) → Γ(E′′) be another Dirac type operator such that D_{2}L′−L′′D_{2} = ρ. Then using the ideas described above, one can derive a stochastic representation formula for D_{2}D_{1}e^{−tL/2}.

8. Appendix I: Differential Geometric Notation and Identities

8.1. Conventions on Differential Forms. Let V and W be finite dimensional vector spaces and let ΛV^{∗} := ⊕_{k}Λ^{k}V^{∗} be the exterior algebra over V^{∗}. As is usual, we will identify elements of Λ^{k}V^{∗} ⊗ W and alternating k-forms on V with values in W. Our convention for doing this is to define, when α = α_{1} ∧ α_{2} ∧ ⋯ ∧ α_{k} ⊗ w,

α (v_{1}, v_{2}, …, v_{k}) := det [{α_{i}(v_{j})}_{i,j=1}^{k}] w (8.1)

for all {v_{i}}_{i=1}^{k} in V. Eq. (8.1) gives rise to the pairing ⟨·, ·⟩ : Λ^{k}V^{∗} × Λ^{k}V → ℝ determined by

⟨α, v_{1} ∧ v_{2} ∧ ⋯ ∧ v_{k}⟩ = α (v_{1}, v_{2}, …, v_{k}) . (8.2)

This pairing allows us to identify (Λ^{k}V)^{∗} with Λ^{k}V^{∗}.

Suppose V and W are equipped with inner products (·, ·)_{V} and (·, ·)_{W} respectively. Given this data, we may define the inner product of α ∈ Λ^{k}V^{∗} ⊗ W and β ∈ Λ^{ℓ}V^{∗} ⊗ W by

(α, β) = δ_{k,ℓ} \frac{1}{k!} ∑_{v_{1},v_{2},...,v_{k}∈Γ}(α (v_{1}, v_{2}, …, v_{k}) , β (v_{1}, v_{2}, …, v_{k}))_{W}

= δ_{k,ℓ} ∑_{1≤i_{1}<i_{2}<···<i_{k}≤n}(α (e_{i_{1}} ∧ e_{i_{2}} ∧ ⋯ ∧ e_{i_{k}}) , β (e_{i_{1}} ∧ e_{i_{2}} ∧ ⋯ ∧ e_{i_{k}}))_{W} (8.3)

where Γ = {e_{1}, e_{2}, …, e_{n}} is any orthonormal basis of V. It may be checked that (·, ·) on Λ^{k}V^{∗} ⊗ W is the unique inner product with the property that

(α_{1} ∧ ⋯ ∧ α_{k} ⊗ w, β_{1} ∧ ⋯ ∧ β_{ℓ} ⊗ w′) = δ_{k,ℓ} det [{(α_{i}, β_{j})_{V^{∗}}}_{i,j=1}^{k}] (w, w′)_{W}

for all α_{i}, β_{j} ∈ V^{∗} and w, w′ ∈ W. The dual inner product on ΛV relative to the pairing in Eq. (8.2) is determined by

(ξ_{1} ∧ ξ_{2} ∧ ⋯ ∧ ξ_{k}, η_{1} ∧ η_{2} ∧ ⋯ ∧ η_{ℓ}) = δ_{k,ℓ} det [{(ξ_{i}, η_{j})_{V} }_{i,j=1}^{k}]

as is easily checked because, by Eq. (8.3),

{e_{i_{1}} ∧ e_{i_{2}} ∧ ⋯ ∧ e_{i_{k}} : 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n}

is an orthonormal basis for Λ^{k}V.

Definition 8.1. For v ∈ V, the creation (exterior product) operator C_{v} is the linear operator on ΛV ⊕ ΛV^{∗} given by C_{v}ξ = v ∧ ξ and C_{v}α := (v, ·)_{V} ∧ α for ξ ∈ ΛV and α ∈ ΛV^{∗}. The annihilation (interior product) operator A_{v} is the linear operator on ΛV ⊕ΛV^{∗} given by the adjoint of C_{v}. Notice that C_{v} (Λ^{k}V ⊕ Λ^{k}V^{∗}) = Λ^{k+1}V ⊕ Λ^{k+1}V^{∗}and A_{v} (Λ^{k}V ⊕ Λ^{k}V^{∗}) = Λ^{k−1}V ⊕ Λ^{k−1}V^{∗}.

We have the following well known (and easily checked) facts about C and A.

Lemma 8.2. Let A and C be as in Definition 8.1, then

1. (Formula for A_{v}) For v, v_{1}, v_{2}, … v_{k} ∈ V and α ∈ ΛV^{∗}, A_{v}α := α(v, ·, …, ·) and

A_{v}(v_{1} ∧ v_{2} ∧ ⋯ ∧ v_{k}) = ∑_{i=1}^{n}(−1)^{i+1}(v, v_{i}) v_{1} ∧ v_{2} ∧ ⋯ ∧ v̂_{i} ⋯ ∧ v_{k}.

2. (Derivation Property) For each v ∈ V,

A_{v}(α ∧ β) = A_{v}α ∧ β + (−1)^{k}α ∧ A_{v}β

where α ∈ Λ^{k}V^{∗} and β ∈ ΛV^{∗} or α ∈ Λ^{k}V and β ∈ ΛV.

3. (Multiplicative Property) For each v ∈ V,

C_{v}(α ∧ β) = (−1)^{k}α ∧ C_{v}β

for α ∈ Λ^{k}V^{∗} and β ∈ ΛV^{∗} or α ∈ Λ^{k}V and β ∈ ΛV.

4. (Transposes) The pairing in Eq. (8.2) allows us to identify ΛV^{∗} with (ΛV)^{∗} and hence (ΛV^{∗})^{∗} with (ΛV)^{∗∗} ≅ ΛV. Under these identifications, C_{v}^{tr} = A_{v}.

5. (Commutation Relations) For all v, w ∈ V,

C_{v}C_{w} + C_{w}C_{v} = 0,
A_{v}A_{w} + A_{w}A_{v} = 0, and
A_{v}C_{w} + C_{w}A_{v} = (v, w) id_{(ΛV ⊕ΛV^{∗})} .

Let γ_{v} := C_{v} − A_{v}, an operator on E := Λ^{k}V^{∗} for all v ∈ V. Then

γ_{v}^{∗} = C_{v}^{∗} − A_{v}^{∗} = A_{v} − C_{v} = −γ_{v}

and by property 5. above,

γ_{v}γ_{w} + γ_{w}γ_{v} = −2(v, w)_{V} id . (8.4)

Notation 8.3. If W is another vector space, we will abuse notation by using A_{v} and C_{v} to denote the operators A_{v} ⊗ id_{W} and C_{v} ⊗ id_{W} on (ΛV ⊕ ΛV^{∗}) ⊗ W.

Definition 8.4 (Clifford Multiplication). Let V and E be inner product spaces, a multiplication map γ : V^{∗} ⊗ E → E is called a Clifford multiplication provided γ_{v}^{∗} = −γ_{v} for all v ∈ V and

γ_{v}γ_{w} + γ_{w}γ_{v} = −2(v, w)_{V} id_{E}

holds for all v, w ∈ V.

8.2. Curvature. Suppose that E → M is a vector bundle and that T M and E are equipped with covariant derivatives ∇^{E} and ∇^{T M} respectively. As in Section 2.2, these covariant derivatives induce a covariant derivative on T^{∗}M ⊗ E as well. For a ∈ Γ(E) and v, w ∈ T_{x}M, let ∇_{v⊗w}^{2}a = (∇^{T^{∗}M⊗E}∇^{E}a) (v, w). Using this notation, we have the following useful formula for the curvature of ∇^{E}.

Lemma 8.5. For v, w ∈ T_{x}M,

∇_{v⊗w}^{2}a − ∇_{w⊗v}^{2}a = R^{E}(v, w)a − ∇_{T (v,w)}^{E}a,

where R^{E} is the curvature tensor of ∇^{E} and T is the torsion tensor of ∇^{T M}. Hence if ∇^{T M} is torsion free, as is the Levi-Civita connection, then

∇_{v⊗w}^{2}a − ∇_{w⊗v}^{2}a = R^{E}(v, w)a.

Proof. Let X, Y ∈ Γ(T M), then

∇_{X⊗Y}^{2} a = ∇_{X}^{E}∇_{Y}^{E}a − ∇_{∇_{X}^{T M} Y}^{E} a

and hence

∇_{X⊗Y}^{2} a − ∇_{Y ⊗X}^{2}a = ∇_{X}^{E}∇_{Y}^{E}a − ∇_{∇_{X}^{T M} Y}^{E} a − ∇_{Y}^{E}∇_{X}^{E}a + ∇_{∇_{Y}^{T M} X}^{E}a
= ∇_{X}^{E}∇_{Y}^{E}a − ∇_{Y}^{E}∇_{X}^{E}a − ∇_{[X,Y]}^{E}a
−∇_{(∇_{X}^{T M} Y −∇_{Y}^{T M} X−[X,Y])}^{E}a
= R^{E}(X, Y)a − ∇_{T (X,Y)}^{E}a.

8.3. Weitzenböck Formulas for Generalized Dirac Operators. Suppose that E → M is a Riemannian vector bundle with metric compatible covariant derivative. Suppose further that γ : T^{∗}M ⊗ E → E is a Clifford multiplication map, i.e. γ_{v}γ_{w} + γ_{w}γ_{v} = −2(v, w) for all v, w ∈ T_{x}M and x ∈ M. We also assume that γ is compatible with the covariant derivative on E and that γ_{v} acts as a skew adjoint operator on E_{x} for all x ∈ M. This implies by Lemma 2.16 that the formal adjoint of D_{γ} is D_{γ}. Let L := −D_{γ}^{2}, then the following Weitzenböck formula holds:

L = □ − \frac{1}{2} ∑_{i,j} γ_{e_{i}}γ_{e_{j}}R^{E}(e_{i}, e_{j}), (8.5)

where {e_{i}}_{i=1}^{n} is any local orthonormal frame of T M. Fixing x and choosing {e_{i}}_{i=1}^{n} such that (∇e_{i})_{x} = 0 for all i, this is verified by the following computation:

D_{γ}^{2} = ∑_{i,j} γ_{e_{i}}∇_{e_{i}}γ∇ = ∑_{i,j} γ_{e_{i}}γ∇_{e_{i}}∇

= ∑_{i,j} γ_{e_{i}}γ_{e_{j}}∇_{e_{i}⊗e_{j}}^{2} = \frac{1}{2} ∑_{i,j} (γ_{e_{i}}γ_{e_{j}}∇_{e_{i}⊗e_{j}}^{2} + γ_{e_{j}}γ_{e_{i}}∇_{e_{j}⊗e_{i}}^{2})

= \frac{1}{2} ∑_{i,j} (−γ_{e_{j}}γ_{e_{i}}∇_{e_{i}⊗e_{j}}^{2} − 2δ_{i,j}∇_{e_{i}⊗e_{j}}^{2} + γ_{e_{j}}γ_{e_{i}}∇_{e_{j}⊗e_{i}}^{2})

= −□ + \frac{1}{2} ∑_{i,j} γ_{e_{j}}γ_{e_{i}} (∇_{e_{j}⊗e_{i}}^{2} − ∇_{e_{i}⊗e_{j}}^{2})

= −□ + \frac{1}{2} ∑_{i,j} γ_{e_{j}}γ_{e_{i}}R(e_{j}, e_{i}),

where the last equality is a consequence of Lemma 8.5. Clearly we also have that D_{γ}L = LD_{γ}.

8.3.1. Dirac Operator on a Spinor Bundle. If M is a spin manifold, E = S is a spinor bundle over M, and ∇^{S} is the spin connection on S, then the previous formula reduces to D_{γ}^{2} = −□ + \frac{1}{4}scal, where scal is the scalar curvature of M, see p. 126 in [4].

8.3.2. Vector-Valued Differential Forms. Let M be a Riemannian manifold with Levi-Civita connection and E → M be a Riemannian vector bundle over M, endowed with a Riemannian linear connection. Let A^{p}(E) = Γ(Λ^{p}T^{∗}M ⊗ E) be the space of p-forms on M with values in the vector bundle E and let

A(E) = Γ(ΛT^{∗}M ⊗ E) = ⨁_{p≥0} A^{p}(E).

The same symbol ∇ will be used to denote various covariant derivations induced naturally from the Riemannian metric on M and the metric connection in E. Let C : T^{∗}M ⊗(ΛT^{∗}M ⊗ E) → ΛT^{∗}M ⊗E be the creation multiplication operator and define d_{∇} : A(E) → A(E) by d_{∇} := D_{C} = C∇. The explicit formula for d_{∇} is

d_{∇}a(v_{1}, …, v_{p+1}) = ∑_{i=1}^{p+1}(−1)^{i+1}(∇_{v_{i}}a)(v_{1}, …, v̂_{i}, …, v_{p+1}),

where a ∈ A^{p}(E) and v_{1}, …, v_{p+1} ∈ T_{x}M for some x ∈ M.

Since

∇_{X} (C(α ⊗ a)) = ∇_{X} (α ∧ a) = ∇_{X}α ∧ a + α ∧ ∇_{X}a
(8.6)  = C (∇_{X}α ⊗ a + α ⊗ ∇_{X}a)

the exterior multiplication C is compatible with ∇. The following properties of d_{∇} (see [20], [21]) follow from the product rule for ∇, Eq. (8.6), and basic properties of C:

1. d_{∇}a = ∇a for a ∈ A^{0}(E) = Γ(E).

2. d_{∇}(α ∧ a) = dα ∧ a + (−1)^{deg(α)}α ∧ d_{∇}a for all homogeneous differential forms α on M and a ∈ A(E).

3. d_{∇}^{2}(α ∧ a) = α ∧ d_{∇}^{2}a for all homogeneous differential forms α on M and a ∈ A(E).

4. If a ∈ A^{0}(E) := Γ(E), then d_{∇}^{2}a = R^{E}a because

d_{∇}^{2}a = C∇C∇a
= ∑_{i,j=1}^{n} C_{e_{i}}C_{e_{j}}∇_{e_{i}⊗e_{j}}^{2}a

= \frac{1}{2} ∑_{i,j=1}^{n} [C_{e_{i}}C_{e_{j}} (∇_{e_{i}⊗e_{j}}^{2}a − ∇_{e_{j}⊗e_{i}}^{2}a)]

= \frac{1}{2} ∑_{i,j=1}^{n} C_{e_{i}}C_{e_{j}}R(e_{i}, e_{j})a = R^{E}a,

wherein we have used Lemma 8.5.

5. Property 4 may be extended to all a ∈ A(E) to read d_{∇}^{2}a = R^{E}∧a, where R^{E}∧ is the linear operator on Λ(T^{∗}M) ⊗ E determined by R^{E} ∧ (α ⊗ b) = α ∧ R^{E}b for α ∈ Λ(T^{∗}M) and b ∈ A^{0}(E). Alternatively,

d_{∇}^{2}a = \frac{1}{2} ∑_{i,j=1}^{n} C_{e_{i}}C_{e_{j}}R(e_{i}, e_{j})a, (8.7)

where now R is the curvature tensor for the induced connection on Λ(T^{∗}M)⊗ E.

Remark 8.6. If E = M × ℝ is the trivial vector bundle so that R^{E} = 0 and hence d_{∇}^{2} = 0, then d_{∇} is precisely the exterior differential d on differential forms on M.

Relative to the Riemannian structures on E and T M, the co-differential operator d_{∇}^{∗}: A^{p}(E) → A^{p−1}(E) is characterized as the adjoint of d_{∇} via

∫_{M}(d_{∇}a(x), b(x))_{Λ^{p}(T^{∗}M)⊗E} vol(dx) = ∫_{M}(a(x), d_{∇}^{∗}b(x))_{Λ^{p−1}(T^{∗}M)⊗E} vol(dx) (8.8)

for a ∈ A^{p−1}(E) of compact support and b ∈ A^{p}(E). By Lemma 2.16 and the fact that C_{v}^{∗} = A_{v,}

d_{∇}^{∗}a = − ∑_{i} A_{e_{i}}∇_{e_{i}}a for a ∈ A^{p}(E) (8.9)

see [21], p. 8. Since d_{∇}^{2} is a zero order operator, see Eq. (8.7), it follows that d_{∇}^{∗2} is the zero order operator:

d_{∇}^{∗2}b = \frac{1}{2} ∑_{i,j=1}^{n} R^{E}(e_{i}, e_{j})A_{e_{i}}A_{e_{j}}b = ∑_{1≤i<j≤n} R^{E}(e_{i}, e_{j})A_{e_{i}}A_{e_{j}}b, (8.10)

because

(R^{E} ∧ a, b) = \frac{1}{2} ∑_{i,j=1}^{n} (C_{e_{i}}C_{e_{j}}R^{E}(e_{i}, e_{j})a, b) = \frac{1}{2} ∑_{i,j=1}^{n} (R^{E}(e_{i}, e_{j})a, A_{e_{j}}A_{e_{i}}b)

= −\frac{1}{2} ∑_{i,j=1}^{n} (a, R^{E}(e_{i}, e_{j})A_{e_{j}}A_{e_{i}}b) = \frac{1}{2} ∑_{i,j=1}^{n} (a, R^{E}(e_{i}, e_{j})A_{e_{i}}A_{e_{j}}b) .

There are now different Laplacians on E-valued differential forms, e.g.,

Δ = −(d_{∇}d_{∇}^{∗} + d_{∇}^{∗}d_{∇}) and Δ′ = −(d_{∇} + d_{∇}^{∗})^{2} = −D_{γ}^{2}, (8.11)

where γ is the Clifford multiplication defined by γ_{v} = C_{v} − A_{v}. The Laplacians Δ and

Δ′ = Δ − d_{∇}^{2} − d_{∇}^{∗2} = Δ − ∑_{1≤i<j≤n} R^{E}(e_{i}, e_{j}) (A_{e_{i}}A_{e_{j}} + C_{e_{i}}C_{e_{j}}) (8.12)

do not coincide on A.(E), except in the case of flat bundles. Both Δ and Δ′ are elliptic, negative, and essentially self-adjoint on complete manifolds. The operator Δ is the most popular choice since it has the advantage of being homogeneous, i.e. Δ : A^{p}(E) →A^{p}(E). For our purposes however, Δ′ seems to be more natural.

Proposition 8.7 (Weitzenböck’s Formula). The relation between Δ and □ is Δ = □ − R, where

R = − ∑_{i,j=1}^{n} C_{e_{i}}A_{e_{j}}R(e_{i}, e_{j}). (8.13)

Alternatively, R may be described by requiring

(Ra) (v_{1}, …, v_{p}) = ∑_{k=1}^{p} ∑_{j=1}^{n}(−1)^{k}(R(v_{k}, e_{j})a) (e_{j}, v_{1}, …, v̂_{k}, …, v_{p})

for all a ∈ A^{p}(E), v_{1}, …, v_{p} ∈ T_{x}M, x ∈ M and p ≥ 1 and Ra = 0 if a ∈ A^{0}(E). Similarly (using Eq. (8.12)), Δ′ = □ − R′, where

R′ = R + ∑_{1≤i<j≤n} R^{E}(e_{i}, e_{j}) (A_{e_{i}}A_{e_{j}} + C_{e_{i}}C_{e_{j}}) . (8.14)

Proof. By the definition of Δ,

Δ = C∇A∇ + A∇C∇
= C_{e_{i}}∇_{e_{i}}A∇ + A_{e_{i}}∇_{e_{i}}C∇
= C_{e_{i}}A∇_{e_{i}}∇ + A_{e_{i}}C∇_{e_{i}}∇
= C_{e_{i}}A_{e_{j}}∇_{e_{i}⊗e_{j}}^{2} + A_{e_{j}}C_{e_{i}}∇_{e_{j}⊗e_{i}}^{2}
= C_{e_{i}}A_{e_{j}}∇_{e_{i}⊗e_{j}}^{2} + (δ_{ij} − C_{e_{i}}A_{e_{j}}) ∇_{e_{j}⊗e_{i}}^{2}
= □ + C_{e_{i}}A_{e_{j}} (∇_{e_{i}⊗e_{j}}^{2} − ∇_{e_{j}⊗e_{i}}^{2})
= □ + C_{e_{i}}A_{e_{j}}R(e_{i}, e_{j}),

where we are summing on repeated indices. This proves Δ = □ − R where R is given as in Eq. (8.13). For v_{1}, …, v_{p} ∈ T_{x}M,

−(Ra)(v_{1}, …, v_{p}) = ((e_{i}, ·) ∧ A_{e_{j}}R(e_{i}, e_{j})a) (v_{1}, …, v_{p})

= ∑_{k=1}^{p}(−1)^{k+1}(e_{i}, v_{k})(A_{e_{j}}R(e_{i}, e_{j})a) (v_{1}, …, v̂_{k}, …, v_{p})

= ∑_{k=1}^{p}(−1)^{k+1}(R(v_{k}, e_{j})a) (e_{j}, v_{1}, …, v̂_{k}, …, v_{p})

which proves Eq. (8.14).

Remark 8.8. Because Δ, Δ′, and □ are asymmetric operators on compactly supported smooth sections, it follows that R and R′ are fiberwise symmetric operators as well. In particular it follows that

R = R^{∗} = − ∑_{i,j=1}^{n} (R(e_{i}, e_{j}))^{∗} A_{e_{j}}^{∗}C_{e_{i}}^{∗}

= ∑_{i,j=1}^{n} R(e_{i}, e_{j})C_{e_{j}}A_{e_{i}} = − ∑_{i,j=1}^{n} R(e_{j}, e_{i})C_{e_{j}}A_{e_{i}}. (8.15)

The derivative formulas that appear in the body of this article often involve the operator R^{tr} rather than R.

Lemma 8.9. The transpose R^{tr} of R ∈ Γ(End(Λ(T^{∗}M) ⊗ E)) in Eq. (8.13) is formally given by the same formula as R,

R^{tr} = − ∑_{i,j=1}^{n} C_{e_{i}}A_{e_{j}}R(e_{i}, e_{j}) = − ∑_{i,j=1}^{n} R(e_{j}, e_{i})C_{e_{j}}A_{e_{i}}, (8.16)

but now acting on Λ(T M)⊗E^{∗}. In particular when E = M ×ℝ (so Λ(T^{∗}M)⊗E = Λ(T^{∗}M)), if v ∈ T M = Λ^{1}T M ⊂ ΛT M, then

R^{tr}v = ∑_{i,j=1}^{n} R(e_{i}, e_{j})e_{j}(v, e_{i}) = Ric v. (8.17)

Proof. Since R is the curvature tensor induced by the covariant derivatives on E and T M, it follows that R(v, w)^{tr}|(Λ(T M)⊗E^{∗}) = −R(v, w)|(Λ(T^{∗}M)⊗E). Hence starting with Eq. (8.13) and using Lemma 8.2 we find,

R^{tr} = − ∑_{i,j=1}^{n} R(e_{i}, e_{j})^{tr}A_{e_{j}}^{tr}C_{e_{i}}^{tr}

= ∑_{i,j=1}^{n} R(e_{i}, e_{j})C_{e_{j}}A_{e_{i}} = − ∑_{i,j=1}^{n} R(e_{j}, e_{i})C_{e_{j}}A_{e_{i}}.

An analogous computation starting with Eq. (8.15) shows

R^{tr} = − ∑_{i,j=1}^{n} C_{e_{i}}A_{e_{j}}R(e_{i}, e_{j}).

Proposition 8.10. Let a ∈ A^{0}(E) = Γ(E) and v ∈ T M, then

[□, ∇]_{v} a = ∇_{Ric v}a + (∇ · R^{E})_{v}a − 2R^{E}(v, e_{i})∇_{e_{i}}a (8.18)

where Ric v ≡ ∑_{i=1}^{n} R^{T M}(v, e_{i})e_{i} and (∇ · R^{E})_{v} ≡ ∑_{i=1}^{n}(∇_{e_{i}}R^{E})(e_{i}, v).

Proof. We start with the relation,

Δd_{∇} − d_{∇}Δ = d_{∇}(d_{∇}^{∗}d_{∇} + d_{∇}d_{∇}^{∗}) − (d_{∇}^{∗}d_{∇} + d_{∇}d_{∇}^{∗})d_{∇}
= d_{∇}^{2}d_{∇}^{∗} − d_{∇}^{∗}d_{∇}^{2} = R ∧ d_{∇}^{∗} − d_{∇}^{∗} R ∧ .

Applying this relation to a ∈ A_{0}(E) gives

(□ − R)∇a − ∇□a = Δd_{∇}a − d_{∇}Δa = −d_{∇}^{∗} (R ∧ a)
= ∑_{i=1}^{n} A_{e_{i}}∇_{e_{i}} (Ra) = ∑_{i=1}^{n} A_{e_{i}} (∇_{e_{i}}R a + R∇_{e_{i}}a)

= ∑_{i=1}^{n}{(∇_{e_{i}}R) (e_{i}, ·) a + R(e_{i}, ·)∇_{e_{i}}a}
= (∇ · R) a − R · ∇a,

where R is defined in Proposition 8.7 above. Adding this equation to the formula for R∇a,

(R∇a) (v) = − ∑_{i=1}^{n} (R(v, e_{i})∇a) (e_{i})

= − ∑_{i=1}^{n} R^{E}(v, e_{i})∇_{e_{i}}a + ∑_{i=1}^{n} ∇_{R(v,e_{i})e_{i}}a
= −R · ∇a + ∇_{Ric v}a,

completes the proof.

9. Appendix II: Semigroup Results

9.1. Some Spectral Theory for Vector-Valued Schrödinger Operators. Let M be a Riemannian manifold and π: E → M a Riemannian vector bundle over M, endowed with a Riemannian connection. Recall that Γ(E) denotes the smooth sections, Γ_{c}(E) the compactly supported smooth sections, and L^{2}-Γ(E) the smooth square-integrable sections of E. Finally, L^{2}(E) is the Hilbert space of squareintegrable sections of E with the inner product

(a, b)_{L^{2}(E)} = ∫_{M}(a(x), b(x))_{E_{x}} vol(dx). (9.1)

Lemma 9.1. The operator □ = tr ∇^{2} is non-positive and formally self-adjoint. More precisely,

(□a, b)_{L^{2}(E)} = −(∇a, ∇b)_{L^{2}(T^{∗}M⊗E)} (9.2)

for a, b ∈ Γ(E) with a or b of compact support.

Proof. Let m be the identity multiplication map from E to Ẽ := T^{∗}M ⊗ E. Then D_{m} = ∇^{E}, m_{v}^{∗}S = S(v) for S ∈ Γ(Ẽ) ≅ Γ(Hom(T M, E)) and □ = D_{m^{∗}}D_{m} and hence by Lemma 2.16,

(□a, b)_{L^{2}(E)} = (D_{m^{∗}}D_{m}a, b)_{L^{2}(E)}
= −(D_{m}a, D_{m}b)_{L^{2}(T^{∗}M⊗E)}
= −(∇a, ∇b)_{L^{2}(T^{∗}M⊗E)}.

9.1.1. Self-adjoint Extensions and Elliptic Regularity. Let L = □ − R where R ∈ Γ(End E) is assumed to be symmetric, i.e. R_{x} is a symmetric linear transformation of E_{x} for each x ∈ M. Suppose that (□ − R)|Γ_{c}(E) is bounded above, i.e.,

λ_{0}(R) := sup \{\frac{((□ − R)a, a)_{L^{2}}}{(a, a)_{L^{2}}} : 0 ≠ a ∈ Γ_{c}(E)\} < ∞. (9.3)

In this situation, there is a canonical self-adjoint extension of □ − R, the so-called Friedrichs extension, cf. [45]. We briefly sketch the construction: Defining E(a, b) := −(∇a, ∇b)_{L^{2}} − (Ra, b)_{L^{2}} for a, b ∈ D(E) := Γ_{c}(E), then for any c > λ_{0}(R),

q(a, b) := −E(a, b) + c (a, b)_{L^{2}} (9.4) is a positive quadratic form on D(E). On completing D(E) in the q-norm to D(E) and extending E by continuity to a closed quadratic form E on D(E), we get

E(a, b) = ((□ − R)ˆ a, b)_{L^{2}} (9.5)

for some self-adjoint operator (□ − R)ˆ with domain D(E) ⊂ L^{2}(E). The operator (□ − R)ˆ is called the Friedrichs extension of (□ − R)|Γ_{c}(E).

Remark 9.2. If the manifold M is complete, then (□ − R)|Γ_{c}(E) is essentially self-adjoint and (□ − R)ˆ = (□ − R)^{−} where (□ − R)^{−} denotes the closure of (□ − R)|Γ_{c}(E). See for example Strichartz [51] and Davies [11].

In the sequel, we are going to deal with the L^{2} semigroup

P_{t}a = e^{t(□−R)ˆ /2}a, a ∈ L^{2}(E), (9.6)

defined by the spectral theorem. First, we note some consequences from standard elliptic theory.

Remark 9.3 (Elliptic Regularity). For a ∈ L^{2}(E) the following properties hold:

(i) If a ∈ ker(□ − R)ˆ then a ∈ L^{2}-Γ(E). (ii) The map (t, x) ↦ P_{t}a(x) is smooth on ]0, ∞[ × M, for a ∈ L^{2}-Γ(E) even

on [0, ∞[ × M. In addition, there exists a kernel (t, x, y) ↦ p(t, x, y) ∈ Hom(E_{y}, E_{x}) which is smooth on ]0, ∞[ × M × M, such that

P_{t}a(x) = ∫_{M} p(t, x, y)a(y) vol(dy) (9.7)

for the C^{∞} version of P_{t}a, see [11].

9.1.2. Semigroup Domination and Feynman-Kac Identities. Given ρ: M → ℝ continuous and a measurable function f on M, let

P_{t}^{ρ}f(x) = 𝔼 [exp (−\frac{1}{2} ∫_{0}^{t} ρ(X_{s}(x)) ds) f(X_{t}(x)) 1_{\{t<ζ(x)\}}] (9.8)

when the right-hand side is well-defined.

Consider again L = □ − R where R ∈ Γ(End E) is assumed to be symmetric, and let

R(x) = min {(R_{x}v, v) : v ∈ E_{x}, |v| = 1} . (9.9)

By uniform continuity, R is a continuous function on M. If (Q_{t}) is defined by

\frac{d}{dt}Q_{t} = −\frac{1}{2} Q_{t}R_{//_{t}} with Q_{0} = id_{E_{x}}, (9.10)

then

|Q_{t}|_{op} ≤ exp (−\frac{1}{2} ∫_{0}^{t} R(X_{s}(x)) ds) . (9.11)

which can be seen, for instance, by representing the solution to (9.10) in terms of a product integral ([13], p. 28) or by Gronwall’s inequality.

Theorem 9.4. Let □ − R be as above where R ∈ Γ(End E) is a symmetric field of endomorphisms. Suppose that (□−R)|Γ_{c}(E) is bounded from above. For a ∈ L^{2}(E) let

P_{t}a = e^{t/2 (□−R)}ˆ a (9.12)

be the C^{∞} version of the L^{2} semigroup. Then the formula

P_{t}a(x) = 𝔼 [Q_{t} //_{t}^{−1}a(X_{t}(x)) 1_{{t<ζ(x)}}] (9.13)

holds for all a ∈ L^{2}(E) with P_{t}^{R}|a|(x) < ∞.

Proof. Since P_{t}a has an integral kernel, using a monotone class argument, it is sufficient to check (9.13) for a ∈ L^{2}(E)∩Γ_{c}(E). Further, note that any (connected) manifold M can be exhausted by a sequence of relatively compact open domains D_{n} with smooth boundary. For instance, let (φ_{ℓ})_{ℓ∈ℕ} be a partition of unity such that 0 ≤ φ_{ℓ} ∈ C_{c}^{∞}(M). Consider ϕ_{n} := ∑_{ℓ=1}^{n} φ_{ℓ}. Choosing numbers ε_{n} ↘ 0 such that {ϕ_{n} = ε_{n}} are smooth submanifolds of M (which is possible by Sard’s theorem), then

D′_{n} := {ϕ_{n} > ε_{n}} ↗ M

gives a smooth exhaustion of M, since ϕ_{n} ↗ 1 pointwise. Finally, by fixing a point x_{0} in M and defining D_{n} as the component of D′_{n} containing x_{0}, we get a sequence of connected sets D_{n} with the desired properties.

Now, fix an exhausting sequence D_{n} ↗ M as above, and let L̂_{n} denote the Friedrichs extensions of (□−R)|Γ_{c}(E/D_{n}). Then, by monotone convergence of the corresponding quadratic forms, see e.g. [36], Theorem VIII-3.11, we get

P_{t}^{(n)}a := e^{tL̂_{n}/2}a → P_{t}a in L^{2}. (9.14)

We use the following two properties of P_{t}^{(n)}a:

1. the map (t, x) ↦ P_{t}^{(n)}a(x) is smooth (in particular bounded) on [0, T] × D_{n},

2. the semigroup P_{t}^{(n)}a vanishes on ∂D_{n}.

Recall that for each t > 0,

N_{s}^{(n)} := Q_{s} //_{s}^{−1}P_{t−s}^{(n)}a(X_{s}(x)) (9.15)

is a local martingale with lifetime t ∧ τ_{n}(x) where τ_{n}(x) denotes the first exit time of X(x) from D_{n}. Since N_{s}^{(n)} in Eq. (9.15) is bounded, we may conclude 𝔼[N_{0}^{(n)}] = 𝔼[N_{t∧τ_{n}(x)}^{(n)}]. But note that

𝔼[N_{t∧τ_{n}(x)}^{(n)}] = 𝔼[N_{t}^{(n)} 1_{{t<τ_{n}(x)}}] + 𝔼[N_{τ_{n}(x)}^{(n)} 1_{{t≥τ_{n}(x)}}]
= 𝔼[N_{t}^{(n)} 1_{{t<τ_{n}(x)}}] = 𝔼[Q_{t} //_{t}^{−1} a(X_{s}(x)) 1_{{t<τ_{n}(x)}}]

and

|Q_{t} //_{t}^{−1} a(X_{s}(x)) 1_{\{t<τ_{n}(x)\}}| ≤ exp (−\frac{1}{2} ∫_{0}^{t} R(X_{s}(x)) ds) |a|(X_{t}(x)) 1_{\{t<ζ(x)\}}.

By assumption, the right-hand side is in L^{1}, thus by dominated convergence,

P_{t}^{(n)}a(x) = 𝔼[N_{t∧τ_{n}(x)}^{(n)}] → 𝔼[Q_{t} //_{t}^{−1}a(X_{t}(x)) 1_{{t<ζ(x)}}]

which combined with (9.14) gives the claim.

Note that the above proof shows in particular semigroup domination, see Theorem 4.3 of Donnelly and Li [14]:

|P_{t}a|(x) ≤ P_{t}^{R}|a|(x), (9.16)

see [50], [3] for a general account on this. Before discussing this point, let us specialize Theorem 9.4 to a Feynman-Kac identity on functions. The following Corollary is well-known at least in the case ρ = 0, e.g. [12].

Corollary 9.5. Let M be a Riemannian manifold, Δ_{M} its Laplace-Beltrami operator, and Δ_{M}^{ρ} := Δ_{M} −ρ where ρ: M → ℝ is continuous. Suppose that Δ_{M}^{ρ}|C_{c}^{∞}(M) is bounded from above, i.e.,

λ_{0}(ρ) := sup \{\frac{(Δ_{M}^{ρ}φ, φ)}{(φ, φ)} : 0 ≠ φ ∈ C_{c}^{∞}(M)\} < ∞. (9.17)

Then P_{t}^{ρ}|f|(x) < ∞ for any f ∈ L^{2}(M) and x ↦ P_{t}^{ρ}f(x) is continuous for t > 0. Let Δ̂_{M}^{ρ} be the Friedrichs extension of Δ_{M}^{ρ}|C_{c}^{∞}(M). Then,

e^{tΔ̂_{M}^{ρ} /2}f = P_{t}^{ρ}f (9.18)

for the L^{2} semigroup given by the spectral theorem.

Proof. Take the trivial bundle E = M × ℝ, then Γ(E) = C^{∞}(M) and □f = Δ_{M}f. Theorem 9.4 gives the claim at least for smooth ρ, a restriction which can easily be removed.

Theorem 9.6 (Semigroup Domination). For a field R ∈ Γ(End E) of symmetric

endomorphisms, let L^{R} = □ − R and Δ_{M}^{R} = Δ_{M} − R. Then

λ_{0}(R) ≤ λ_{0}(R). (9.19)

In particular, if Δ_{M}^{R}|C_{c}^{∞}(M) is bounded from above, then L^{R}|Γ_{c}(E) is also bounded from above, and moreover the following estimate holds:

|P_{t}a| ≤ P_{t}^{R}|a|. (9.20)

Proof. We may assume that λ_{0}(R) < ∞. If λ_{0}(R) < ∞, then the Friedrichs extension L̂^{R} of L^{R}|Γ_{c}(E) is well-defined, and by Eq. (9.16),

e^{λ_{0}(R)/2} = sup_{a∈Γ_{c}(E)\\{0\}}\{\frac{(a, e^{L̂^{R}/2}a)}{(a, a)}\} ≤ sup_{a∈Γ_{c}(E)\\{0\}}\{\frac{(|a|, e^{Δ̂_{M}^{R} /2}|a|)}{(|a|, |a|)}\} ≤ e^{λ_{0}(R)/2}.

To see that λ_{0}(R) < ∞ implies λ_{0}(R) < ∞, we note that the above argument can be applied first to give λ_{0}(R|D_{n}) ≤ λ_{0}(R) for each D_{n} of a sequence of smoothly bounded, relatively compact open domains D_{n} ↗ M. From this the claim follows obviously.

9.2. A Commutativity Result. In this subsection, we investigate conditions under which a Dirac type operator D commutes with the L^{2} semigroup generated by the Friedrichs extension of D^{2}. In particular, we shall recover the fact that d and d^{∗} commute with e^{tΔ̄} when M is complete. The precise statement is given in Remark 9.9 below. Similar discussions may be found in Brüning and Lesch [7], Xue-Mei Li [41, 40] and in Bueler [8].

Theorem 9.7. Let D be a closable densely defined operator on a Hilbert space H. Then D^{∗}D̄ and D̄D^{∗} are densely defined, self-adjoint and ≥ 0. Furthermore

D^{∗}e^{−tD̄D^{∗}} |D(D^{∗}) = e^{−tD^{∗}D̄}D^{∗} and D̄e^{−tD^{∗}D̄}|D(D̄) = e^{−tD̄D^{∗}} D̄. (9.21)

Proof. (By ”Nelson’s trick”, as in [57], section 5.2). The operator

Q := ( \begin{matrix}0 & D^{∗} \\ D̄ & 0 \end{matrix} )

on H ⊕ H is self-adjoint (see [57], Lemma 5.3). Thus, by the spectral theorem,

Q^{2} = ( \begin{matrix}D^{∗}D̄ & 0 \\ 0 & D̄D^{∗} \end{matrix} )

is densely defined, self-adjoint and non-negative and hence are its components D^{∗}D̄ respectively D̄D^{∗}. By the spectral theorem, Qe^{−tQ^{2}}|D(Q) = e^{−tQ^{2}}Q. It makes no difference applying the spectral theorem to Q^{2} as a whole or to its components. Hence

( \begin{matrix}0 & D^{∗}e^{−tD̄D^{∗}} \\ D̄e^{−tD^{∗}D̄} & 0 \end{matrix} ) = ( \begin{matrix}0 & D^{∗} \\ D̄ & 0 \end{matrix} ) ( \begin{matrix}e^{−tD^{∗}D̄} & 0 \\ 0 & e^{−tD̄D^{∗}} \end{matrix} )

= ( \begin{matrix}e^{−tD^{∗}D̄} & 0 \\ 0 & e^{−tD̄D^{∗}} \end{matrix} ) ( \begin{matrix}0 & D^{∗} \\ D̄ & 0 \end{matrix} )

= ( \begin{matrix}0 & e^{−tD^{∗}D̄}D^{∗} \\ e^{−tD̄D^{∗}} D̄ & 0 \end{matrix} ) .

Remark 9.8. In Theorem 9.7, the operator D^{∗}D̄ is the Friedrichs extension of D^{2}, e.g. Reed-Simon [45]. In particular, if D is essentially self-adjoint, i.e. D̄ = D^{∗}, then D̄ commutes (on the domain of D̄) with the semigroup generated by the Friedrichs extension of −D^{2}.

Remark 9.9. Let E = ΛT^{∗}M and D = d + δ on compactly supported smooth sections of E, where d and δ denote the exterior differential and its formal adjoint, both restricted to smooth sections of compact support. Under the assumption that M is complete, D = d + δ and all its powers are known to be essentially self-adjoint on Γ_{c}(E) ⊂ L^{2}(E), see [9], also [51], Theorem 2.4. An immediate consequence is

\overline{d + δ} = d + δ = δ^{∗} + d^{∗}. (9.22)

Hence, the Hodge-de Rham Laplacian L = −D^{2} is essentially self-adjoint on Γ_{c}(E) ⊂ L^{2}(E). Thus, on a complete manifold,

dP_{t} = P_{t}d on the domain of d, and
d^{∗}P_{t} = P_{t}d^{∗} on the domain of d^{∗},

where P_{t} is the semigroup generated by L̄ = L̂ = −D^{∗}D̄ = −D̄D^{∗}. Indeed, this follows from Theorem 9.7, together with Eq. (9.22), by taking into account that P_{t}, d, d^{∗} are homogeneous respectively of degree 0, 1, −1.

Remark 9.10. For non-complete manifolds M there are in general several different self-adjoint extensions of −D^{2} = −(d + δ)^{2}|Γ_{c}(E), see [32], [31], [10] for details.

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Department of Mathematics-0112 University of California at San Diego La Jolla, CA 92093-0112 USA

E-mail address: driver@math.ucsd.edu

Institut für Angewandte Mathematik Universität Bonn, Wegelerstr. 6 53115 Bonn, Germany

E-mail address: anton@wiener.iam.uni-bonn.de