arXiv:1701.03625v1 [math.PR] 13 Jan 2017

DERIVATIVE AND DIVERGENCE FORMULAE FOR DIFFUSION SEMIGROUPS

ANTON THALMAIER AND JAMES THOMPSON Mathematics Research Unit, FSTC, University of Luxembourg 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg, Grand Duchy of Luxembourg Abstract. For a semigroup Pt generated by an elliptic operator on a smooth manifold M, we use straightforward martingale arguments to derive probabilistic formulae for Pt (V( f )), not involving derivatives of f , where V is a vector field on M. For non-symmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, these formulae can be used to derive various shift-Harnack inequalities.

Introduction For a Banach space E, e ∈ E and a Markov operator P on Bb (E), it is known that certain estimates on P(∇e f ) are equivalent to corresponding shift-Harnack inequalities. This was proved by F.-Y. Wang in [18]. For example, for δe ∈ (0, 1) and βe ∈ C((δe , ∞) × E; [0, ∞)), he proved that the derivative-entropy estimate P(∇e f ) ≤ δ P( f log f ) − (P f ) log P f + βe (δ, ·)P f

holds for any δ ≥ δe and positive f ∈ Cb1 (E) if and only if the inequality ! ! Z 1 p−1 pr p p βe , · + sre ds (P f ) ≤ P( f (re + ·)) exp r + r(p − 1)s 0 1 + (p − 1)s

holds for any p ≥ 1/(1 − rδe), r ∈ (0, 1/δe ) and positive f ∈ Bb (E). Furthermore, he also proved that if C ≥ 0 is a constant then the L2 -derivative inequality 2 P(∇e f ) ≤ CP f 2

holds for any non-negative f ∈ Cb1 (E) if and only if the inequality q P f ≤ P f (αe + ·) + |α| CP f 2

holds for any α ∈ R and non-negative f ∈ Bb (E). The objective of this article is to find probabilistic formulae for PT (V( f )) from which such estimates can be derived, for the case in which PT is the Markov operator associated to a non-degenerate diffusion Xt on a smooth, finite-dimensional manifold M, and V a vector field. E-mail address: [email protected], [email protected] Date: January 16, 2017. 2010 Mathematics Subject Classification. 58J65, 60J60, 53C21. Key words and phrases. Diffusion semigroup, Heat kernel, Gradient estimate, Harnack inequality, Ricci curvature. 1

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DERIVATIVE AND DIVERGENCE FORMULAE

In Section 1 we suppose that M is a Riemannian manifold and that the generator of Xt is ∆ + Z, for some smooth vector field Z. Any non-degenerate diffusion on a smooth manifold induces a Riemannian metric with respect to which its generator takes this form. The basic strategy is then to use the relation V( f ) = div( f V) − f div V to reduce the problem to finding a suitable formula for PT (div( f V)). Such a formula was given in [3] for the case Z = 0, which we extend to the general case with Theorem 1.16. In doing so, we do not make any assumptions on the derivatives of the curvature tensor, as occurred in [2]. For an adapted process ht with paths in the Cameron-Martin space L1,2 ([0, T ]; R), with h0 = 0 and hT = 1 and under certain additional conditions, we obtain the formula PT (V( f )) (x) = − E f (XT (x)) (div V)(XT (x)) " Z T E# D 1 dB (div Z)(Xt (x))ht − h˙ t Θ−1 − E f (XT (x)) V(XT (x)), //T ΘT t t 2 0 where Θ is the Aut(T x M)-valued process defined by the pathwise differential equation d Θt = −//t−1 Ric♯ + (∇. Z)∗ − div Z //t Θt dt with Θ0 = idT x M . Here //t denotes the stochastic parallel transport associated to Xt (x), whose antidevelopment to T x M has martingale part B. In particular, B is a diffusion on Rn generated by the Laplacian; it is a standard Brownian motion sped up by 2, so that j dBit dBt = 2δi j dt. Choosing ht explicitly yields a formula from which estimates then can be deduced, as described in Subsection 1.5. The problem of finding a suitable formula for PT (V( f )) is dual to that of finding an analogous one for V(PT f ). A formula for the latter is called the Bismut formula [1] or the Bismut-Elworthy-Li formula, on account of [6]. We provide a brief proof of it in Subsection 1.3, since we would like to compare it to our formula for PT (V( f )). Our approach to these formulae is based on martingale arguments; integration by parts is done at the level of local martingales. Under conditions which assure that the local martingales are true martingales, the wanted formulae are then obtained by taking expectations. They allow for the choice of a finite energy process. Depending on the intended type, conditions are imposed either on the right endpoint, as in the formula for PT (V( f )), or the left endpoint, as in the formula for V(PT f ). The formula for PT (V( f )) requires non-explosivity; the formula for V(PT f ) does not. From the latter can be deduced Bismut’s formula for the logarithmic derivative in the backward variable x of the heat kernel pT (x, y) determined by Z f (y)pT (x, y) vol(dy), f ∈ Cb (M). (PT f )(x) = M

From our formula for PT (V( f )) can be deduced the following formula for the derivative in the forward variable y: " # Z T 1 −1 ˙ (div Z)(Xt (x))ht − ht Θt dBt XT (x) = y . (∇ log pT (x, ·))y = − E //T ΘT 2 0

In Section 2 we consider the general case in which M is a smooth manifold and Xt a non-degenerate diffusion solving a Stratonovich equation of the form dXt = A0 (Xt ) dt + A(Xt ) ◦ dBt . We denote by T Xt the derivative (in probability) of the solution flow. Using a similar approach to that of Section 1, and a variety of geometric objects naturally associated to the

DERIVATIVE AND DIVERGENCE FORMULAE

3

equation, we obtain, under certain conditions, the formula PT (V( f )) m X =− E f (XT ) Ai hV, Ai i(XT ) i=1

" Z T # 1 −1 A ˆ ˙ Ξt (trace ∇A0 )(Xt )ht − ht A(Xt ) dBt + 2ht A0 dt − E f (XT ) V(XT ), ΞT 2 0

with Ξt = T Xt − T Xt A0A =

m X i=1

Z

t

0

˘ 0 )∗ + ∇A ˘ 0 + trace ∇A ˆ 0 (Ξ s ) ds, T X s−1 (∇A

˘ 0 )∗ + ∇A ˘ 0 T˘ (·, Ai )∗ (Ai ) + A0 , T˘ (·, Ai )∗ (Ai ), (∇A

ˆ 0 , ∇A ˘ 0 and T˘ (·, Ai ) are given at each x ∈ M and v ∈ T x M by where the operators ∇A ∇ˆ v A0 = A(x) d(A∗ (·)A0 (·)) x (v) − (dA∗) x (v, A0 ) , ∇˘ v A0 = A(x)d A(·)∗ A0 (·) x (v), T˘ (v, Ai ) x = A(x)(dA∗ ) x (v, Ai ). This formula has the advantage of involving neither parallel transport nor Riemannian curvature, both typically difficult to calculate in terms of A. 1. Intrinsic Formulae 1.1. Preliminaries. Let M be a complete and connected n-dimensional Riemannian manifold, ∇ the Levi-Civita connection on M and π : O(M) → M the orthonormal frame bundle over M. Let E → M be an associated vector bundle with fibre V and structure group G = O(n). The induced covariant derivative ∇ : Γ(E) → Γ(T ∗ M ⊗ E)

determines the so-called connection Laplacian (or rough Laplacian) on Γ(E), a = trace∇2 a. P Note that ∇2 a ∈ Γ(T ∗ M ⊗ T ∗ M ⊗ E) and (a) x = i ∇2 a(vi , vi ) ∈ E x where vi runs through an orthonormal basis of T x M. For a, b ∈ Γ(E) of compact support it is immediate to check that ha, biL2 (E) = −h∇a, ∇biL2 (T ∗ M⊗E) . In this sense we have = −∇∗ ∇. Let H be the horizontal subbundle of the G-invariant splitting of T O(M) and ∼ H ֒→ T O(M) h : π∗ T M −→ the horizontal lift of the G-connection; fibrewise this bundle isomorphism reads as ∼ H , u ∈ O(M). h : T M −→ u

π(u)

u

In terms of the standard horizontal vector fields H1 , . . . , Hn on O(M), Hi (u) := hu (uei ),

Bochner’s horizontal Laplacian

∆hor ,

u ∈ O(M),

acting on smooth functions on O(M), is given as n X ∆hor = Hi2 . i=1

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DERIVATIVE AND DIVERGENCE FORMULAE

To formulate the relation between and ∆hor , it is convenient to write sections a ∈ Γ(E) as equivariant functions Fa : O(M) → V via Fa (u) = u−1 aπ(u) where we read u ∈ O(M) as an ∼ E . Equivariance means that isomorphism u : V −→ π(u) Fa (ug) = g−1 Fa (u),

u ∈ O(M), g ∈ G = O(n).

Lemma 1.1 (see [9], p. 115). For a ∈ Γ(E) and Fa the corresponding equivariant function on O(M), we have (Hi Fa )(u) = F∇uei a (u), u ∈ O(M). Hence ∆hor Fa = Fa , where as above ∇

∇

trace

: Γ(E) −→ Γ(T ∗ M ⊗ E) −→ Γ(T ∗ M ⊗ T ∗ M ⊗ E) −→ Γ(E). Proof. Fix u ∈ O(M) and choose a curve γ in M such that γ(0) = π(u) and γ˙ = uei . Let t 7→ u(t) be the horizontal lift of γ to O(M) such that u(0) = u. Note that u˙ (t) = hu(t) (γ˙ (t)), and in particular u˙ (0) = hu (uei ) = Hi (u). Hence, denoting the parallel transport along γ by //ε = u(ε)u(0)−1, we get F∇uei a (u) = u−1 ∇uei a = u−1 lim

π(u) −1 //ε aγ(ε) − aγ(0)

ε −1 γ(ε) − u(0) aγ(0) = lim ε↓0 ε Fa (u(ε)) − Fa(u(0)) = lim ε↓0 ε = (Hi )u Fa ε↓0

u(ε)−1 a

= (Hi Fa )(u).

Now consider diffusion processes Xt on M generated by the operator L = ∆+Z where Z ∈ Γ(T M) is a smooth vector field. Such diffusions on M may be constructed from the corresponding horizontal diffusions on O(M) generated by ∆hor + Z¯ where the vector field Z¯ is the horizontal lift of Z to O(M), i.e. Z¯u = hu (Zπ(u) ), u ∈ O(M). More precisely, we start from the Stratonovich stochastic differential equation on O(M), (1.1)

dUt =

n X i=1

¯ t ) dt, Hi (Ut ) ◦ dBit + Z(U

U0 = u ∈ O(M) j

where Bt is a Brownian motion on Rn sped up by 2, that is dBit dBt = 2δi j dt. Then for Xt = π(Ut ), the following equation holds: (1.2)

dXt =

n X i=1

Ut ei ◦ dBit + Z(Xt ) dt,

X0 = x := πu.

DERIVATIVE AND DIVERGENCE FORMULAE

The Brownian motion B is the martingale part of the anti-development denotes the canonical 1-form ϑ on O(M), i.e. ϑu (e) = u−1 eπ(u) ,

5

R

U

ϑ of X, where ϑ

e ∈ T u O(M).

In particular, for F ∈ C ∞ (O(M)), resp. f ∈ C ∞ (M), we have d(F ◦ Ut ) =

n X

¯ (Hi F)(Ut ) ◦ dBit + (ZF)(U t ) dt

=

n X

(Hi F)(Ut ) dBit + ∆hor + Z¯ (F)(Ut ) dt,

d( f ◦ Xt ) =

n X

(d f )(Ut ei ) ◦ dBit + (Z f )(Xt ) dt

=

n X

(d f )(Ut ei ) dBit + (∆ + Z) ( f )(Xt ) dt.

(1.3)

i=1

i=1

respectively

i=1

i=1

Typically, solutions to (1.2) are defined up to some maximal lifetime ζ(x) which may be finite. Then we have, almost surely, ζ(x) < ∞ ⊂ Xt → ∞ as t ↑ ζ(x)

where on the right-hand side, the symbol ∞ denotes the point at infinity in the one-point compactification of M. It can be shown that the maximal lifetime of solutions to equation (1.1) and to (1.2) coincide, see e.g. [12]. In case of a non-trivial lifetime the subsequent stochastic equations should be read for t < ζ(x). Proposition 1.2. Let //t : E X0 → E Xt be parallel transport in E along X, induced by the parallel transport on M, //t = Ut U0−1 : T X0 M → T Xt M. Then, for a ∈ Γ(E), we have n X d //t−1 a(Xt ) = //t−1 ∇Ut ei a ◦ dBit + //t−1 (∇Z a) (Xt ) dt, i=1

respectively in Itˆo form, n X d //t−1 a(Xt ) = //t−1 ∇Ut ei a dBit + //t−1 (a + ∇Z a) (Xt ) dt. i=1

More succinctly, the last two equations may be written as d //t−1a(Xt ) = //t−1 ∇◦dXt a,

respectively

d //t−1 a(Xt ) = //t−1 ∇dXt α + //t−1(a)(Xt ) dt.

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DERIVATIVE AND DIVERGENCE FORMULAE

¯ a = F ∇Z a . Proof. We have //t−1 a(Xt ) = U0 Ut−1 a(Xt ) = U0 Fa (Ut ). It is easily checked that ZF Thus, we obtain from equation (1.3) dFa (Ut ) =

n X ¯ a (Ut ) dt (Hi Fa )(Ut ) dBit + ∆hor Fa + ZF i=1

=

n X

=

n X

i=1

i=1

F∇Ut ei a (Ut ) dBit + Fa + F∇Z a (Ut ) dt

U −1 ∇Ut ei a (Xt ) dBit + Ut−1 (a + ∇Z a) (Xt ) dt.

Corollary 1.3. Fix T > 0 and let at ∈ Γ(E) solve the equation ∂ at = at + ∇Z at ∂t

on [0, T ] × M.

Then is a local martingale.

//t−1 aT −t (Xt ) ,

0 ≤ t < T ∧ ζ(x),

Proof. Indeed we have ! ∂ d(//t−1aT −t (Xt )) m = //t−1 aT −t + ∇Z at + aT −t (Xt ) dt = 0, ∂t | {z } =0

m

where = denotes equality modulo differentials of local martingales.

We are now going to look at operators L R on Γ(E) which differ from by a zero-order term, in other words, (1.4)

−LR = R

where R ∈ Γ(EndE).

Thus, by definition, the action R x : E x → E x is linear for each x ∈ M. Example 1.4. A typical example is E = Λ p T ∗ M and A p (M) = Γ(Λ p T ∗ M) with p ≥ 1. The de Rham-Hodge Laplacian ∆(p) = −(d∗ d + dd∗) : A p (M) → A p (M) then takes the form ∆(p) α = α − Rα

where R is given by the Weitzenb¨ock decomposition. In the special case p = 1, one obtains Rα = Ric(α♯ , ·) where Ric : T M ⊕ T M → R is the Ricci tensor. Definition 1.5. Fix x ∈ M and let Xt be a diffusion to L = ∆ + Z, starting at x. Let Qt be the Aut(E x )-valued process defined by the following linear pathwise differential equation d Qt = −Qt R//t , dt

Q0 = idEx ,

where R//t := //t−1 ◦ RXt ◦ //t ∈ End(E x )

and //t is parallel transport in E along X.

DERIVATIVE AND DIVERGENCE FORMULAE

7

Proposition 1.6. Let L R = − R be as in equation (1.4) and Xt be a diffusion to L = ∆ + Z, starting at x. Then, for any a ∈ Γ(E), n X d Qt //t−1 a(Xt ) = Qt //t−1 ∇Ut ei a dBit + Qt //t−1 (a + ∇Z a − Ra)(Xt ) dt. i=1

Proof. Let

nt := //t−1 a(Xt ).

Then

d(Qt nt ) = (dQt ) nt + Qt dnt = −Qt //t−1 RXt //t−1 nt dt + Qt dnt = −Qt //t−1 (Ra)(Xt ) dt + Qt dnt .

The claim thus follows from Proposition 1.2.

Corollary 1.7. Fix T > 0 and let Xt (x) be a diffusion to L = ∆ + Z, starting at x. Suppose that at solves ∂ at = ( − R + ∇Z ) at on [0, T ] × M, ∂t at |t=0 = a ∈ Γ(E). Then (1.5)

Nt := Qt //t−1 aT −t (Xt (x)) ,

0 ≤ t < T ∧ ζ(x),

is a local martingale, starting at aT (x). In particular, if ζ(x) = ∞ and if equation (1.5) is a true martingale on [0, T ], we arrive at the formula aT (x) = E QT //T−1 a(XT (x)) , a ∈ Γ(E). Proof. Indeed, we have

m

dNt =

Qt //t−1

! ∂ ( + ∇Z − R)aT −t + aT −t (Xt ) dt = 0. ∂t | {z }

=0

Remark 1.8. Note that

d Qt = −Qt R//t , dt implies the obvious estimate kQt kop ≤ exp −

with Q0 = idEx , Z

t

0

! R(X s (x))ds

where R(x) = inf {hR x v, wi : v, w ∈ E x , kvk ≤ 1 and kwk ≤ 1}. 1.2. Commutation formulae. In the sequel, we consider the special case E = T ∗ M. Thus Γ(E) is the space of differential 1-forms on M. The results of this section apply to vector fields as well, by identifying vector fields V ∈ Γ(T M) and 1-forms α ∈ Γ(T ∗ M) via the metric: V ←→ V ♭ , α ←→ α# .

Let Z ∈ Γ(T M) be a vector field on M. Then the divergence of Z, denoted by div Z ∈ C ∞ (M), is defined by div Z := trace(v 7→ ∇v Z). Therefore n X (div Z)(x) = h∇vi X, vi i i=1

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DERIVATIVE AND DIVERGENCE FORMULAE

for any orthonormal basis {vi }ni=1 for T x M. For compactly supported f we have hZ, ∇ f iL2 (T M) = −hdiv Z, f iL2 (M) .

The adjoint Z ∗ of Z is given by the relation

Z ∗ f = −Z f − (divZ) f,

f ∈ C ∞ (M).

If either f or h is compactly supported, this implies hZ f, hiL2 (M) = h f, Z ∗ hiL2 (M) .

Similarly, for α ∈ Γ(T ∗ M), we let

∇α # (divα)(x) = trace T x M −→ T x∗ M −→ T x M .

Thus div Y = div Y ♭ and div α = div α# . That is, if δ = d∗ denotes the usual codifferential then div α = −δα. Finally, we define RicZ (X, Y) := Ric(X, Y) − h∇X Z, Yi,

X, Y ∈ Γ(T M).

Notation 1.9. For the sake of convenience, we read bilinear forms on M, such as RicZ , likewise as sections of End(T ∗ M) or End(T M), e.g. RicZ (α) := RicZ (α♯ , ·), RicZ (v) := RicZ (v, ·)♯ ,

α ∈ T ∗ M, v ∈ T M.

If there is no risk of confusion, we do not distinguish in notation. In particular, depending on the context, (RicZ )//t may be a random section of End(T ∗ M) or of End(T M). Lemma 1.10 (Commutation rules). Let Z ∈ Γ(T M). (1) For the differential d, we have d ∆ + Z = − RicZ + ∇Z d;

(2) for the codifferential d∗ = − div, we have ∆ + Z ∗ d∗ = d∗ − Ric∗Z + ∇∗Z ,

where the formal adjoint of ∇Z (acting on 1-forms) is ∇∗Z α = −∇Z α − (divZ)α.

Proof. Indeed, for any smooth function f we have d ∆ + Z f = d − d∗ d f + (d f )Z

= ∆(1) d f + ∇Z d f + h∇. Z, ∇ f i = ( + ∇Z )(d f ) − RicZ (·, ∇ f ) = − RicZ + ∇Z (d f ).

The formula in (2) is then just dual to (1).

1.3. A formula for the differential. Now, let Xt (x) be a diffusion to ∆ + Z on M, starting R at X0 (x) = x, Ut a horizontal lift of X to O(M) and B = U0 U ϑ the martingale part of the anti-development of Xt (x) to T x M. Let Qt be the Aut(T x∗ M)-valued process defined by d Qt = −Qt (RicZ )//t dt with Q0 = idT x∗ M , let h i Pt f (x) = E 1{t 0 and let ℓt be an adapted process with paths in the Cameron-Martin space L1,2 ([0, T ]; T x M). By Corollary 1.7 (1.6)

Nt := Qt //t−1 (dPT −t f ),

t < T ∧ ζ(x),

is local martingale. Therefore Nt (ℓt ) −

Z

t

0

Q s //s−1 (dPT −s f )(ℓ˙s )ds

is a local martingale. By integration by parts Z t Z t 1 Q s //s−1 (dPT −s f )(ℓ˙s )ds − (PT −t f )(Xt (x)) hQtrs (ℓ˙s ), dB si 2 0 0 is also a local martingale and therefore (1.7)

1 Qt //t−1 (dPT −t f )(ℓt ) − (PT −t f )(Xt (x)) 2

Z

0

t

hQtrs ℓ˙s , dB si

is a local martingale, starting at (dPT f )(ℓ0 ). Choosing ℓt so that (1.7) is a true martingale on [0, T ] with ℓ0 = v and ℓT = 0, we obtain the formula # " Z T 1 hQtrs ℓ˙s , dB si . (1.8) (dPT f )(v) = − E 1{T 0 and h be an adapted process with paths in L1,2 ([0, T ]; R) such that h0 = 0 and hT = 1, and such that (1.11) is a true martingale. Then for all bounded smooth vector fields V on M, " Z T E# 1 D dB (div Z)(Xt (x))ht − h˙ t Θ−1 E (div V)(XT (x)) = − E V(XT (x)), //T ΘT t t 2 0 where Θ is the Aut(T x M)-valued process defined by the following pathwise differential equation: d Θt = −Ric//t Θt − (∇. Z)∗// Θt + (divZ)Θt t dt with Θ0 = idT x M .

DERIVATIVE AND DIVERGENCE FORMULAE

13

Corollary 1.17. Suppose f is a bounded smooth function and that V is a bounded smooth vector field with div V bounded. Then, under the assumptions of Theorem 1.16, by using the relation div( f V) = V f + f div V, we get PT V( f ) (x) = −E f (XT (x)) (div V)(XT (x)) " Z T E# D 1 dB (div Z)(Xt (x))ht − h˙ t Θ−1 − E f (XT (x)) V(XT (x)), //T ΘT t t 2 0 where the right-hand side does not contain any derivatives of f . Corollary 1.18. Under the assumptions of Theorem 1.16 we have " # Z T 1 ∇ log pT (x, ·) y = − E //T ΘT dB (div Z)(Xt (x))ht − h˙ t Θ−1 X (x) = y t T t 2 0

with Θ given as above.

Proof. By Theorem 1.16, for all smooth, compactly supported vector fields V we have PT (div V)(x) #E " Z T Z D 1 dB (div Z)(Xt (x))ht − h˙ t Θ−1 =− X (x) = y V(y), E //T ΘT pT (x, y) vol(dy), t T t 2 M 0

but on the other hand

Z

(div V)(y) pT (x, y) vol(dy) Z (d pT (x, ·))y V(y) vol(dy) =− ZM d log pT (x, ·) y V(y) pT (x, y) vol(dy) =−

PT (div V)(x) =

M

M

so the result follows.

1.5. Shift-Harnack Inequalities. Suppose RicZ is bounded below, that Ric + (∇. Z)∗ and div Z are bounded and that the following formula holds, for all t > 0, all f ∈ Cb1 (M) and all bounded vector fields V with div V bounded (see Corollary 1.17): Pt (V( f ))(x) = −E f (Xt (x)) (div V)(Xt (x)) " Z t E# D r 1 −1 1 (div Z)(Xr (x)) − Θr dBr . − E f (Xt (x)) V(Xt (x)), //t Θt 2 t t 0 Fix T > 0. Then, by Jensen’s inquality (see [13, Lemma 6.45]), there exist c,C1 (T ) > 0 such that ! C1 (T ) 2 (1.17) |Pt (V( f ))| ≤ δ Pt ( f log f ) − Pt f log Pt f + | div V|∞ + δc + |V|∞ Pt f δt | {z } =: α1 (δ,t,V)

for all δ > 0, t ∈ (0, T ] and positive f ∈ Cb1 (M). Alternatively, by the Cauchy-Schwarz inequality, there exists C2 (T ) > 0 such that !2 C2 (T ) 2 (1.18) |Pt (V( f ))| ≤ | div V|∞ + √ |V|∞ Pt f 2 t | {z } =: α2 (t,V)

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DERIVATIVE AND DIVERGENCE FORMULAE

for all t ∈ (0, T ] and f ∈ Cb1 (M). These estimates can be used to derive shift-Harnack inequalities, as shown by F.-Y. Wang for the case of a Markov operator on a Banach space (see [18, Proposition 2.3]). In particular, suppose {F s : s ∈ [0, 1]} is a C 1 family of diffeomorphisms of M with F0 = id M . For each s ∈ [0, 1] define a vector field V s on M by V s := (DF s )−1 F˙ s d ( f ◦ F s ) = ∇Vs ( f ◦ F s ). Fixing and assume V s and div V s are uniformly bounded. Note ds p ≥ 1 and setting β(s) = 1 + (p − 1)s, as in the first part of [18, Proposition 2.3], we deduce from inequality (1.17) that ! p/β(s) d β′ (s) p β(s) log Pt ( f ◦ F s) α1 , t, V s ≥− ds β(s) β(s)

for all s ∈ [0, 1], which when integrated gives the shift-Harnack inequality ! ! Z 1 β′ (s) p α1 , t, V s ds (Pt f ) p ≤ Pt ( f p ◦ F1 ) exp β(s) 0 β(s)

for each t ∈ [0, T ] and positive f ∈ Cb1 (M). Alternatively, from inequality (1.18) and following the calculation in the second part of [18, Proposition 2.3], we deduce !1/2 q Z 1 Pt f ≤ Pt ( f ◦ F 1 ) + α2 (t, V s )ds Pt f 2 0

for each t ∈ [0, T ] and positive f ∈ Cb1 (M). The shift F1 could be given by the exponential of a well-behaved vector field; the shifts considered in [18] are of the form x 7→ x + v, for some v belonging to the Banach space. 2. Extrinsic Formulae Suppose now that M is an n-dimensional smooth manifold. Suppose A0 is a smooth vector field and A : M × Rm → T M, (x, e) 7→ A(x)e, a smooth bundle map over M. This means A(·)e is a vector field on M for each e ∈ Rm , and A(x) : Rm → T x M is linear for each x ∈ M For an Rm -valued Brownian motion Bt , sped up by 2 so that d[B, B]t = 2 idRm dt, defined on a filtered probability space (Ω, F , P; (Ft )t∈R+ ), satisfying the usual completeness conditions, consider the Stratonovich stochastic differential equation (2.1)

dXt = A0 (Xt ) dt + A(Xt ) ◦ dBt .

m i Given an orthonormal basis {ei }m i=1 of R set Ai (·) := A(·)ei and Bt := hBt , ei i. Then the previous equation can be equivalently written m X dXt = A0 (Xt ) dt + Ai (Xt ) ◦ dBit . i=1

There is a partial flow Xt (·), ζ(·) associated to (2.1) (see [10] for details) such that for each x ∈ M the process Xt (x), 0 ≤ t < ζ(x) is the maximal strong solution to (2.1) with starting point X0 (x) = x, defined up to the explosion time ζ(x); moreover using the notation Xt (x, ω) = Xt (x)(ω) and ζ(x, ω) = ζ(x)(ω), if Mt (ω) = {x ∈ M : t < ζ(x, ω)}

then there exists Ω0 ⊂ Ω of full measure such that for all ω ∈ Ω0 : i) Mt (ω) is open in M for each t ≥ 0, i.e. ζ(·, ω) is lower semicontinuous on M;

DERIVATIVE AND DIVERGENCE FORMULAE

15

ii) Xt (·, ω) : Mt (ω) → M is a diffeomorphism onto an open subset of M; iii) The map s 7→ X s (·, ω) is continuous from [0, t] into C ∞ (Mt (ω), M) with its C ∞ topology, for each t > 0. The solution processes X = X(x) to (2.1) are diffusions on M with generator L := A0 +

m X

A2i

i=1

We will assume that the equation is non-degenerate, which is to say that A(x) : Rm → T x M is surjective for all x ∈ M. Then A induces a Riemannian metric on M, the quotient metric, with respect to which A(x)∗ = (A(x)|ker A(x)⊥ )−1 and whose inner product h·, ·i on a tangent space T x M is given by hv, ui = hA(x)∗ v, A(x)∗ uiRm .

2.1. A formula for the differential. Denote by i h Pt f (x) := E 1{t 0, by Itˆo’s formula we have d(αT −t (Ξt (x))) = (2.15)

m X i=1

ˆ A α (Ξt (x)) dt + ∂t α (Ξt (x)) dt ∇ˆ Ai αT −t (Ξt (x)) dBit + ∇ 0 T −t T −t

ˆ + trace ∇ˆ 2 αT −t (Ξt (x)) dt + αT −t (d∇ Ξt (x)) m X = ((∇ˆ Ai αT −t )· + αT −t (∇˘ . Ai ))(Ξt (x)) dBit . i=1

DERIVATIVE AND DIVERGENCE FORMULAE

21

It follows that αT −t (Ξt (x)) is a local martingale, starting at αT . Furthermore, according to equation (26) in [5], for the derivative process T Xt (x) we have ˆ

˘ T Xt (x) A0 dt + ˘ d∇ T Xt (x) = −Ric(T Xt (x)) dt + ∇

m X i=1

∇˘ T Xt (x) Ai dBit

and therefore, by the variation of constants formula, we have Z t ˘ 0 )∗ + ∇A ˘ 0 + trace ∇A ˆ 0 (Ξ s (x)) ds. T X s (x)−1 (∇A Ξt (x) = T Xt (x) − T Xt (x) 0

Thus it is possible to calculate Ξt (x) without using the parallel transport implicit in the original equation. Moreover, if the vector field A0 vanishes then Ξt (x) is given precisely by the derivative process T Xt (x). Proposition 2.7. Suppose ht is an adapted process with paths in L1,2 ([0, T ]; R). Then Z t ˆδα ht − h s αT −s (A0A) ds T −t 0 ! (2.16) Z t 1 −1 ˆ ˙ + αT −t Ξt (x) h s − (trace ∇A0 )(X s (x))h s Ξ s (x) A(X s (x)) dB s 2 0 ˆ h0 , where the vector field AA is given by (2.14). is a local martingale, starting at δα T 0 Proof. Set At := exp

Z

0

t

! ˆ 0 )(X s (x)) ds (trace ∇A

A−1 t ht .

and define ℓt := By equation (2.15), integration by parts and formula (2.11), we have, suppressing the summation over i, that Z ! 1 t ˙ −1 A s ℓ s Ξ s (x) A(X s (x)) dB s d αT −t Ξt (x) 2 0 j m 1 (∇ˆ Ai αT −t ) . + αT −t (∇˘ . Ai ) (Ξt (x)) dBit At ℓ˙t Ξt (x)−1 A j (Xt (x)) dBt = (2.17) 2 = (∇ˆ Ai αT −t )Ai + αT −t (∇˘ Ai Ai ) At ℓ˙t dt = (∇ˆ Ai αT −t )Ai At ℓ˙t dt ˆ = −(δα )At ℓ˙t dt T −t

m

where = denotes equality modulo the differential of a local martingale. By Proposition 2.5 and Itˆo’s formula we have m A ˆ ˆ ˆ ˆ d(At δα T −t ) = At δ∂t αT −t dt + At (L + trace ∇A0 )δαT −t dt = At αT −t (A0 ) dt

which implies ˆ nt := At δα T −t −

Z

0

t

A s αT −s (A0A)ds

ˆ . This implies is a local martingale, starting at δα T d(nt ℓt ) m = nt ℓ˙t dt ˙ ˙ ˆ = (δα T −t )At ℓt dt − ℓt

Z

t 0

A s αT −s (A0A ) dsdt.

22

DERIVATIVE AND DIVERGENCE FORMULAE

Substituting the definition of nt into the left-hand side and performing integration by parts to the second term on the right-hand side implies Z t Z t ˙ ˆ ˆ ℓ ds − h s αT −s (A0A ) ds (2.18) δα h − ( δα )A s s T −t t T −s 0

0

is another local martingale. Since ˆ 0 )(Xt (x))ht , ℓ˙t = A−1 h˙ t − (trace ∇A t

substituting formula (2.17) into the second term in (2.18) completes the proof.

Theorem 2.8. Suppose ht is any adapted process with paths in L1,2 ([0, ∞); R) such that h0 = 0 and hT = 1 and that α is a bounded smooth 1-form. Suppose (2.1) is complete and that the local martingales αT −t (Ξt ) and (2.16) are true martingales. Then " Z T !# 1 A −1 ˆ ˙ ˆ (trace ∇A0 )(Xt )ht − ht A(Xt ) dBt + 2ht A0 dt . Ξt PT (δα) = E α ΞT 2 0 Proof. By (2.15) we have αT −t (Ξt ) = α(ΞT ) −

Z

t

T

ˆ A α . + α (∇˘ . Ai ) (Ξt ) dBit ∇ i T −s T −s

and therefore E

"Z

T

0

# " Z A αT −t (Ξt ht Ξ−1 A ) dt = E α Ξ T t 0

0

T

A ht Ξ−1 t A0 dt

!#

since αT −t (Ξt ) is assumed to be a martingale. The result now follows from Proposition 2.7, by taking expectations. In analogue to Lemma 1.14, an integrability assumption on h plus suitable bounds on ˘ 0 , trace ∇A ˆ 0 , AA and δα ˆ and on the moments of T Xt and T Xt−1 would be sufficient to ∇A 0 guarantee that αT −t (Ξt ) and (2.16) are true martingales. Corollary 2.9. Suppose f is a bounded smooth function. Suppose V is a bounded smooth P vector field with m i=1 Ai hV, Ai i bounded. Then, under the assumptions of Theorem 2.8 with ♭ α = f V , we have PT (V( f )) m X E f (XT ) Ai hV, Ai i(XT ) =− i=1

" Z T # 1 A ˆ ˙ − E f (XT ) V(XT ), ΞT A(X ) dB + 2h A dt (trace ∇A )(X )h − h Ξ−1 t t t 0 0 t t t t 2 0

with Ξt = T Xt − T Xt

Z

0

t

˘ 0 )∗ + ∇A ˘ 0 + trace ∇A ˆ 0 (Ξ s ) ds, T X s−1 (∇A

m X ˘ 0 )∗ + ∇A ˘ 0 T˘ (·, Ai )∗ (Ai ) + A0 , T˘ (·, Ai )∗ (Ai ), (∇A A0A = i=1

DERIVATIVE AND DIVERGENCE FORMULAE

23

ˆ 0 , ∇A ˘ 0 and T˘ (·, Ai ) are given at each x ∈ M and v ∈ T x M by where the operators ∇A ∇ˆ v A0 = A(x) d A∗ (·)A0 (·) x (v) − (dA∗) x (v, A0 ) , ∇˘ v A0 = A(x)d A(·)∗ A0 (·) x (v), T˘ (v, Ai ) x = A(x)(dA∗) x (v, Ai ). Proof. This follows from Theorem 2.8. In particular, Lemma 2.6 implies ˆ ♭ ) − δ( ˆ f V ♭) V( f ) = f δ(V

while formula (2.12), the Le Jan-Watanabe property and the adaptedness of ∇˘ imply m m X X ˆ ♭ ) = − h∇˘ Ai V, Ai i = − δ(V Ai hV, Ai i. i=1

i=1

Note that if (2.1) is a gradient system then L = ∆ + A0 and A0A vanishes and m X i=1

Ai hV, Ai i = div V.

ˆ 0 = div A0 , Corollary 2.9 yields the unfiltered version of CorolIn this case, since trace ∇A lary 1.17. Corollary 2.10. Under the assumptions of Corollary 2.9 we have d log pT (x, ·) y (v) m E D X T˘ (·, Ai )∗ (Ai )(y) = − v, i=1

+

1D

2

" Z v, E ΞT

0

T

Ξ−1 t

#E A ˆ ˙ (trace ∇A0 )(Xt )ht − ht A(Xt ) dBt + 2ht A0 dt XT (x) = y

for all v ∈ T y M where the various terms appearing in the right-hand side can be calculated as in Corollary 2.9. Proof. Since Corollary 2.9 holds for all smooth functions f and vector fields V of compact support, and since by Lemma 2.6 ˆ ♭ ) − δ( ˆ f V ♭ ) = V( f ) = f δ(V ♭ ) − δ( f V ♭ ), f δ(V the result follows from equation (2.13), Lemma 2.4 and Corollary 2.9.

Example 2.11. Consider the special case M = Rn . Denote by qT (x, y) the smooth density of XT (x) with respect to the standard n-dimensional Lebesgue measure. Recall that pT (x, y) denotes the density with respect to the induced Riemannian measure. It follows that qT (x, y) = pT (x, y) ρ1/2 (y) where ρ(y) denotes the absolute value of the determinant of the matrix ∗ hA ∂i , A∗ ∂ j iRm (y) ni, j=1

in which {∂i }ni=1 denotes the standard basis of vector fields on Rn . Consequently d log qT (x, ·) y (v) = d log pT (x, ·) y (v) + d log ρ1/2 (·) y (v)

with the first term on the right-hand side given, in terms of the induced metric, by Corollary 2.10.

24

DERIVATIVE AND DIVERGENCE FORMULAE

Acknowledgment. This work has been supported by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746). References 1. Jean-Michel Bismut, Large deviations and the Malliavin calculus, Progress in Mathematics, vol. 45, Birkh¨auser Boston, Inc., Boston, MA, 1984. MR 755001 2. Ana Bela Cruzeiro and Xicheng Zhang, Bismut type formulae for diffusion semigroups on Riemannian manifolds, Potential Anal. 25 (2006), no. 2, 121–130. MR 2238939 3. Bruce K. Driver and Anton Thalmaier, Heat equation derivative formulas for vector bundles, J. Funct. Anal. 183 (2001), no. 1, 42–108. MR 1837533 4. K. D. Elworthy, Y. Le Jan, and Xue-Mei Li, On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics, vol. 1720, Springer-Verlag, Berlin, 1999. MR 1735806 5. K. D. Elworthy, Yves Le Jan, and Xue-Mei Li, Concerning the geometry of stochastic differential equations and stochastic flows, New trends in stochastic analysis (Charingworth, 1994), World Sci. Publ., River Edge, NJ, 1997, pp. 107–131. MR 1654511 6. K. D. Elworthy and X.-M. Li, Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1994), no. 1, 252–286. MR 1297021 7. K. David Elworthy, Yves Le Jan, and Xue-Mei Li, The geometry of filtering, Frontiers in Mathematics, Birkh¨auser Verlag, Basel, 2010. MR 2731662 8. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989 9. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR 0152974 10. Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990. MR 1070361 11. Mikio Nakahara, Geometry, topology and physics, second ed., Graduate Student Series in Physics, Institute of Physics, Bristol, 2003. MR 2001829 12. Ichiro Shigekawa, On stochastic horizontal lifts, Z. Wahrsch. Verw. Gebiete 59 (1982), no. 2, 211–221. MR 650613 13. Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, Mathematical Surveys and Monographs, vol. 74, American Mathematical Society, Providence, RI, 2000. MR 1715265 14. Anton Thalmaier, On the differentiation of heat semigroups and Poisson integrals, Stochastics Stochastics Rep. 61 (1997), no. 3-4, 297–321. MR 1488139 15. Anton Thalmaier and Feng-Yu Wang, Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal. 155 (1998), no. 1, 109–124. MR 1622800 , A stochastic approach to a priori estimates and Liouville theorems for harmonic maps, Bull. Sci. 16. Math. 135 (2011), no. 6-7, 816–843. MR 2838103 17. Feng-Yu Wang, Analysis for diffusion processes on Riemannian manifolds, Advanced Series on Statistical Science & Applied Probability, vol. 18, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014. MR 3154951 , Integration by parts formula and shift Harnack inequality for stochastic equations, Ann. Probab. 42 18. (2014), no. 3, 994–1019. MR 3189064

DERIVATIVE AND DIVERGENCE FORMULAE FOR DIFFUSION SEMIGROUPS

ANTON THALMAIER AND JAMES THOMPSON Mathematics Research Unit, FSTC, University of Luxembourg 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg, Grand Duchy of Luxembourg Abstract. For a semigroup Pt generated by an elliptic operator on a smooth manifold M, we use straightforward martingale arguments to derive probabilistic formulae for Pt (V( f )), not involving derivatives of f , where V is a vector field on M. For non-symmetric generators, such formulae correspond to the derivative of the heat kernel in the forward variable. As an application, these formulae can be used to derive various shift-Harnack inequalities.

Introduction For a Banach space E, e ∈ E and a Markov operator P on Bb (E), it is known that certain estimates on P(∇e f ) are equivalent to corresponding shift-Harnack inequalities. This was proved by F.-Y. Wang in [18]. For example, for δe ∈ (0, 1) and βe ∈ C((δe , ∞) × E; [0, ∞)), he proved that the derivative-entropy estimate P(∇e f ) ≤ δ P( f log f ) − (P f ) log P f + βe (δ, ·)P f

holds for any δ ≥ δe and positive f ∈ Cb1 (E) if and only if the inequality ! ! Z 1 p−1 pr p p βe , · + sre ds (P f ) ≤ P( f (re + ·)) exp r + r(p − 1)s 0 1 + (p − 1)s

holds for any p ≥ 1/(1 − rδe), r ∈ (0, 1/δe ) and positive f ∈ Bb (E). Furthermore, he also proved that if C ≥ 0 is a constant then the L2 -derivative inequality 2 P(∇e f ) ≤ CP f 2

holds for any non-negative f ∈ Cb1 (E) if and only if the inequality q P f ≤ P f (αe + ·) + |α| CP f 2

holds for any α ∈ R and non-negative f ∈ Bb (E). The objective of this article is to find probabilistic formulae for PT (V( f )) from which such estimates can be derived, for the case in which PT is the Markov operator associated to a non-degenerate diffusion Xt on a smooth, finite-dimensional manifold M, and V a vector field. E-mail address: [email protected], [email protected] Date: January 16, 2017. 2010 Mathematics Subject Classification. 58J65, 60J60, 53C21. Key words and phrases. Diffusion semigroup, Heat kernel, Gradient estimate, Harnack inequality, Ricci curvature. 1

2

DERIVATIVE AND DIVERGENCE FORMULAE

In Section 1 we suppose that M is a Riemannian manifold and that the generator of Xt is ∆ + Z, for some smooth vector field Z. Any non-degenerate diffusion on a smooth manifold induces a Riemannian metric with respect to which its generator takes this form. The basic strategy is then to use the relation V( f ) = div( f V) − f div V to reduce the problem to finding a suitable formula for PT (div( f V)). Such a formula was given in [3] for the case Z = 0, which we extend to the general case with Theorem 1.16. In doing so, we do not make any assumptions on the derivatives of the curvature tensor, as occurred in [2]. For an adapted process ht with paths in the Cameron-Martin space L1,2 ([0, T ]; R), with h0 = 0 and hT = 1 and under certain additional conditions, we obtain the formula PT (V( f )) (x) = − E f (XT (x)) (div V)(XT (x)) " Z T E# D 1 dB (div Z)(Xt (x))ht − h˙ t Θ−1 − E f (XT (x)) V(XT (x)), //T ΘT t t 2 0 where Θ is the Aut(T x M)-valued process defined by the pathwise differential equation d Θt = −//t−1 Ric♯ + (∇. Z)∗ − div Z //t Θt dt with Θ0 = idT x M . Here //t denotes the stochastic parallel transport associated to Xt (x), whose antidevelopment to T x M has martingale part B. In particular, B is a diffusion on Rn generated by the Laplacian; it is a standard Brownian motion sped up by 2, so that j dBit dBt = 2δi j dt. Choosing ht explicitly yields a formula from which estimates then can be deduced, as described in Subsection 1.5. The problem of finding a suitable formula for PT (V( f )) is dual to that of finding an analogous one for V(PT f ). A formula for the latter is called the Bismut formula [1] or the Bismut-Elworthy-Li formula, on account of [6]. We provide a brief proof of it in Subsection 1.3, since we would like to compare it to our formula for PT (V( f )). Our approach to these formulae is based on martingale arguments; integration by parts is done at the level of local martingales. Under conditions which assure that the local martingales are true martingales, the wanted formulae are then obtained by taking expectations. They allow for the choice of a finite energy process. Depending on the intended type, conditions are imposed either on the right endpoint, as in the formula for PT (V( f )), or the left endpoint, as in the formula for V(PT f ). The formula for PT (V( f )) requires non-explosivity; the formula for V(PT f ) does not. From the latter can be deduced Bismut’s formula for the logarithmic derivative in the backward variable x of the heat kernel pT (x, y) determined by Z f (y)pT (x, y) vol(dy), f ∈ Cb (M). (PT f )(x) = M

From our formula for PT (V( f )) can be deduced the following formula for the derivative in the forward variable y: " # Z T 1 −1 ˙ (div Z)(Xt (x))ht − ht Θt dBt XT (x) = y . (∇ log pT (x, ·))y = − E //T ΘT 2 0

In Section 2 we consider the general case in which M is a smooth manifold and Xt a non-degenerate diffusion solving a Stratonovich equation of the form dXt = A0 (Xt ) dt + A(Xt ) ◦ dBt . We denote by T Xt the derivative (in probability) of the solution flow. Using a similar approach to that of Section 1, and a variety of geometric objects naturally associated to the

DERIVATIVE AND DIVERGENCE FORMULAE

3

equation, we obtain, under certain conditions, the formula PT (V( f )) m X =− E f (XT ) Ai hV, Ai i(XT ) i=1

" Z T # 1 −1 A ˆ ˙ Ξt (trace ∇A0 )(Xt )ht − ht A(Xt ) dBt + 2ht A0 dt − E f (XT ) V(XT ), ΞT 2 0

with Ξt = T Xt − T Xt A0A =

m X i=1

Z

t

0

˘ 0 )∗ + ∇A ˘ 0 + trace ∇A ˆ 0 (Ξ s ) ds, T X s−1 (∇A

˘ 0 )∗ + ∇A ˘ 0 T˘ (·, Ai )∗ (Ai ) + A0 , T˘ (·, Ai )∗ (Ai ), (∇A

ˆ 0 , ∇A ˘ 0 and T˘ (·, Ai ) are given at each x ∈ M and v ∈ T x M by where the operators ∇A ∇ˆ v A0 = A(x) d(A∗ (·)A0 (·)) x (v) − (dA∗) x (v, A0 ) , ∇˘ v A0 = A(x)d A(·)∗ A0 (·) x (v), T˘ (v, Ai ) x = A(x)(dA∗ ) x (v, Ai ). This formula has the advantage of involving neither parallel transport nor Riemannian curvature, both typically difficult to calculate in terms of A. 1. Intrinsic Formulae 1.1. Preliminaries. Let M be a complete and connected n-dimensional Riemannian manifold, ∇ the Levi-Civita connection on M and π : O(M) → M the orthonormal frame bundle over M. Let E → M be an associated vector bundle with fibre V and structure group G = O(n). The induced covariant derivative ∇ : Γ(E) → Γ(T ∗ M ⊗ E)

determines the so-called connection Laplacian (or rough Laplacian) on Γ(E), a = trace∇2 a. P Note that ∇2 a ∈ Γ(T ∗ M ⊗ T ∗ M ⊗ E) and (a) x = i ∇2 a(vi , vi ) ∈ E x where vi runs through an orthonormal basis of T x M. For a, b ∈ Γ(E) of compact support it is immediate to check that ha, biL2 (E) = −h∇a, ∇biL2 (T ∗ M⊗E) . In this sense we have = −∇∗ ∇. Let H be the horizontal subbundle of the G-invariant splitting of T O(M) and ∼ H ֒→ T O(M) h : π∗ T M −→ the horizontal lift of the G-connection; fibrewise this bundle isomorphism reads as ∼ H , u ∈ O(M). h : T M −→ u

π(u)

u

In terms of the standard horizontal vector fields H1 , . . . , Hn on O(M), Hi (u) := hu (uei ),

Bochner’s horizontal Laplacian

∆hor ,

u ∈ O(M),

acting on smooth functions on O(M), is given as n X ∆hor = Hi2 . i=1

4

DERIVATIVE AND DIVERGENCE FORMULAE

To formulate the relation between and ∆hor , it is convenient to write sections a ∈ Γ(E) as equivariant functions Fa : O(M) → V via Fa (u) = u−1 aπ(u) where we read u ∈ O(M) as an ∼ E . Equivariance means that isomorphism u : V −→ π(u) Fa (ug) = g−1 Fa (u),

u ∈ O(M), g ∈ G = O(n).

Lemma 1.1 (see [9], p. 115). For a ∈ Γ(E) and Fa the corresponding equivariant function on O(M), we have (Hi Fa )(u) = F∇uei a (u), u ∈ O(M). Hence ∆hor Fa = Fa , where as above ∇

∇

trace

: Γ(E) −→ Γ(T ∗ M ⊗ E) −→ Γ(T ∗ M ⊗ T ∗ M ⊗ E) −→ Γ(E). Proof. Fix u ∈ O(M) and choose a curve γ in M such that γ(0) = π(u) and γ˙ = uei . Let t 7→ u(t) be the horizontal lift of γ to O(M) such that u(0) = u. Note that u˙ (t) = hu(t) (γ˙ (t)), and in particular u˙ (0) = hu (uei ) = Hi (u). Hence, denoting the parallel transport along γ by //ε = u(ε)u(0)−1, we get F∇uei a (u) = u−1 ∇uei a = u−1 lim

π(u) −1 //ε aγ(ε) − aγ(0)

ε −1 γ(ε) − u(0) aγ(0) = lim ε↓0 ε Fa (u(ε)) − Fa(u(0)) = lim ε↓0 ε = (Hi )u Fa ε↓0

u(ε)−1 a

= (Hi Fa )(u).

Now consider diffusion processes Xt on M generated by the operator L = ∆+Z where Z ∈ Γ(T M) is a smooth vector field. Such diffusions on M may be constructed from the corresponding horizontal diffusions on O(M) generated by ∆hor + Z¯ where the vector field Z¯ is the horizontal lift of Z to O(M), i.e. Z¯u = hu (Zπ(u) ), u ∈ O(M). More precisely, we start from the Stratonovich stochastic differential equation on O(M), (1.1)

dUt =

n X i=1

¯ t ) dt, Hi (Ut ) ◦ dBit + Z(U

U0 = u ∈ O(M) j

where Bt is a Brownian motion on Rn sped up by 2, that is dBit dBt = 2δi j dt. Then for Xt = π(Ut ), the following equation holds: (1.2)

dXt =

n X i=1

Ut ei ◦ dBit + Z(Xt ) dt,

X0 = x := πu.

DERIVATIVE AND DIVERGENCE FORMULAE

The Brownian motion B is the martingale part of the anti-development denotes the canonical 1-form ϑ on O(M), i.e. ϑu (e) = u−1 eπ(u) ,

5

R

U

ϑ of X, where ϑ

e ∈ T u O(M).

In particular, for F ∈ C ∞ (O(M)), resp. f ∈ C ∞ (M), we have d(F ◦ Ut ) =

n X

¯ (Hi F)(Ut ) ◦ dBit + (ZF)(U t ) dt

=

n X

(Hi F)(Ut ) dBit + ∆hor + Z¯ (F)(Ut ) dt,

d( f ◦ Xt ) =

n X

(d f )(Ut ei ) ◦ dBit + (Z f )(Xt ) dt

=

n X

(d f )(Ut ei ) dBit + (∆ + Z) ( f )(Xt ) dt.

(1.3)

i=1

i=1

respectively

i=1

i=1

Typically, solutions to (1.2) are defined up to some maximal lifetime ζ(x) which may be finite. Then we have, almost surely, ζ(x) < ∞ ⊂ Xt → ∞ as t ↑ ζ(x)

where on the right-hand side, the symbol ∞ denotes the point at infinity in the one-point compactification of M. It can be shown that the maximal lifetime of solutions to equation (1.1) and to (1.2) coincide, see e.g. [12]. In case of a non-trivial lifetime the subsequent stochastic equations should be read for t < ζ(x). Proposition 1.2. Let //t : E X0 → E Xt be parallel transport in E along X, induced by the parallel transport on M, //t = Ut U0−1 : T X0 M → T Xt M. Then, for a ∈ Γ(E), we have n X d //t−1 a(Xt ) = //t−1 ∇Ut ei a ◦ dBit + //t−1 (∇Z a) (Xt ) dt, i=1

respectively in Itˆo form, n X d //t−1 a(Xt ) = //t−1 ∇Ut ei a dBit + //t−1 (a + ∇Z a) (Xt ) dt. i=1

More succinctly, the last two equations may be written as d //t−1a(Xt ) = //t−1 ∇◦dXt a,

respectively

d //t−1 a(Xt ) = //t−1 ∇dXt α + //t−1(a)(Xt ) dt.

6

DERIVATIVE AND DIVERGENCE FORMULAE

¯ a = F ∇Z a . Proof. We have //t−1 a(Xt ) = U0 Ut−1 a(Xt ) = U0 Fa (Ut ). It is easily checked that ZF Thus, we obtain from equation (1.3) dFa (Ut ) =

n X ¯ a (Ut ) dt (Hi Fa )(Ut ) dBit + ∆hor Fa + ZF i=1

=

n X

=

n X

i=1

i=1

F∇Ut ei a (Ut ) dBit + Fa + F∇Z a (Ut ) dt

U −1 ∇Ut ei a (Xt ) dBit + Ut−1 (a + ∇Z a) (Xt ) dt.

Corollary 1.3. Fix T > 0 and let at ∈ Γ(E) solve the equation ∂ at = at + ∇Z at ∂t

on [0, T ] × M.

Then is a local martingale.

//t−1 aT −t (Xt ) ,

0 ≤ t < T ∧ ζ(x),

Proof. Indeed we have ! ∂ d(//t−1aT −t (Xt )) m = //t−1 aT −t + ∇Z at + aT −t (Xt ) dt = 0, ∂t | {z } =0

m

where = denotes equality modulo differentials of local martingales.

We are now going to look at operators L R on Γ(E) which differ from by a zero-order term, in other words, (1.4)

−LR = R

where R ∈ Γ(EndE).

Thus, by definition, the action R x : E x → E x is linear for each x ∈ M. Example 1.4. A typical example is E = Λ p T ∗ M and A p (M) = Γ(Λ p T ∗ M) with p ≥ 1. The de Rham-Hodge Laplacian ∆(p) = −(d∗ d + dd∗) : A p (M) → A p (M) then takes the form ∆(p) α = α − Rα

where R is given by the Weitzenb¨ock decomposition. In the special case p = 1, one obtains Rα = Ric(α♯ , ·) where Ric : T M ⊕ T M → R is the Ricci tensor. Definition 1.5. Fix x ∈ M and let Xt be a diffusion to L = ∆ + Z, starting at x. Let Qt be the Aut(E x )-valued process defined by the following linear pathwise differential equation d Qt = −Qt R//t , dt

Q0 = idEx ,

where R//t := //t−1 ◦ RXt ◦ //t ∈ End(E x )

and //t is parallel transport in E along X.

DERIVATIVE AND DIVERGENCE FORMULAE

7

Proposition 1.6. Let L R = − R be as in equation (1.4) and Xt be a diffusion to L = ∆ + Z, starting at x. Then, for any a ∈ Γ(E), n X d Qt //t−1 a(Xt ) = Qt //t−1 ∇Ut ei a dBit + Qt //t−1 (a + ∇Z a − Ra)(Xt ) dt. i=1

Proof. Let

nt := //t−1 a(Xt ).

Then

d(Qt nt ) = (dQt ) nt + Qt dnt = −Qt //t−1 RXt //t−1 nt dt + Qt dnt = −Qt //t−1 (Ra)(Xt ) dt + Qt dnt .

The claim thus follows from Proposition 1.2.

Corollary 1.7. Fix T > 0 and let Xt (x) be a diffusion to L = ∆ + Z, starting at x. Suppose that at solves ∂ at = ( − R + ∇Z ) at on [0, T ] × M, ∂t at |t=0 = a ∈ Γ(E). Then (1.5)

Nt := Qt //t−1 aT −t (Xt (x)) ,

0 ≤ t < T ∧ ζ(x),

is a local martingale, starting at aT (x). In particular, if ζ(x) = ∞ and if equation (1.5) is a true martingale on [0, T ], we arrive at the formula aT (x) = E QT //T−1 a(XT (x)) , a ∈ Γ(E). Proof. Indeed, we have

m

dNt =

Qt //t−1

! ∂ ( + ∇Z − R)aT −t + aT −t (Xt ) dt = 0. ∂t | {z }

=0

Remark 1.8. Note that

d Qt = −Qt R//t , dt implies the obvious estimate kQt kop ≤ exp −

with Q0 = idEx , Z

t

0

! R(X s (x))ds

where R(x) = inf {hR x v, wi : v, w ∈ E x , kvk ≤ 1 and kwk ≤ 1}. 1.2. Commutation formulae. In the sequel, we consider the special case E = T ∗ M. Thus Γ(E) is the space of differential 1-forms on M. The results of this section apply to vector fields as well, by identifying vector fields V ∈ Γ(T M) and 1-forms α ∈ Γ(T ∗ M) via the metric: V ←→ V ♭ , α ←→ α# .

Let Z ∈ Γ(T M) be a vector field on M. Then the divergence of Z, denoted by div Z ∈ C ∞ (M), is defined by div Z := trace(v 7→ ∇v Z). Therefore n X (div Z)(x) = h∇vi X, vi i i=1

8

DERIVATIVE AND DIVERGENCE FORMULAE

for any orthonormal basis {vi }ni=1 for T x M. For compactly supported f we have hZ, ∇ f iL2 (T M) = −hdiv Z, f iL2 (M) .

The adjoint Z ∗ of Z is given by the relation

Z ∗ f = −Z f − (divZ) f,

f ∈ C ∞ (M).

If either f or h is compactly supported, this implies hZ f, hiL2 (M) = h f, Z ∗ hiL2 (M) .

Similarly, for α ∈ Γ(T ∗ M), we let

∇α # (divα)(x) = trace T x M −→ T x∗ M −→ T x M .

Thus div Y = div Y ♭ and div α = div α# . That is, if δ = d∗ denotes the usual codifferential then div α = −δα. Finally, we define RicZ (X, Y) := Ric(X, Y) − h∇X Z, Yi,

X, Y ∈ Γ(T M).

Notation 1.9. For the sake of convenience, we read bilinear forms on M, such as RicZ , likewise as sections of End(T ∗ M) or End(T M), e.g. RicZ (α) := RicZ (α♯ , ·), RicZ (v) := RicZ (v, ·)♯ ,

α ∈ T ∗ M, v ∈ T M.

If there is no risk of confusion, we do not distinguish in notation. In particular, depending on the context, (RicZ )//t may be a random section of End(T ∗ M) or of End(T M). Lemma 1.10 (Commutation rules). Let Z ∈ Γ(T M). (1) For the differential d, we have d ∆ + Z = − RicZ + ∇Z d;

(2) for the codifferential d∗ = − div, we have ∆ + Z ∗ d∗ = d∗ − Ric∗Z + ∇∗Z ,

where the formal adjoint of ∇Z (acting on 1-forms) is ∇∗Z α = −∇Z α − (divZ)α.

Proof. Indeed, for any smooth function f we have d ∆ + Z f = d − d∗ d f + (d f )Z

= ∆(1) d f + ∇Z d f + h∇. Z, ∇ f i = ( + ∇Z )(d f ) − RicZ (·, ∇ f ) = − RicZ + ∇Z (d f ).

The formula in (2) is then just dual to (1).

1.3. A formula for the differential. Now, let Xt (x) be a diffusion to ∆ + Z on M, starting R at X0 (x) = x, Ut a horizontal lift of X to O(M) and B = U0 U ϑ the martingale part of the anti-development of Xt (x) to T x M. Let Qt be the Aut(T x∗ M)-valued process defined by d Qt = −Qt (RicZ )//t dt with Q0 = idT x∗ M , let h i Pt f (x) = E 1{t 0 and let ℓt be an adapted process with paths in the Cameron-Martin space L1,2 ([0, T ]; T x M). By Corollary 1.7 (1.6)

Nt := Qt //t−1 (dPT −t f ),

t < T ∧ ζ(x),

is local martingale. Therefore Nt (ℓt ) −

Z

t

0

Q s //s−1 (dPT −s f )(ℓ˙s )ds

is a local martingale. By integration by parts Z t Z t 1 Q s //s−1 (dPT −s f )(ℓ˙s )ds − (PT −t f )(Xt (x)) hQtrs (ℓ˙s ), dB si 2 0 0 is also a local martingale and therefore (1.7)

1 Qt //t−1 (dPT −t f )(ℓt ) − (PT −t f )(Xt (x)) 2

Z

0

t

hQtrs ℓ˙s , dB si

is a local martingale, starting at (dPT f )(ℓ0 ). Choosing ℓt so that (1.7) is a true martingale on [0, T ] with ℓ0 = v and ℓT = 0, we obtain the formula # " Z T 1 hQtrs ℓ˙s , dB si . (1.8) (dPT f )(v) = − E 1{T 0 and h be an adapted process with paths in L1,2 ([0, T ]; R) such that h0 = 0 and hT = 1, and such that (1.11) is a true martingale. Then for all bounded smooth vector fields V on M, " Z T E# 1 D dB (div Z)(Xt (x))ht − h˙ t Θ−1 E (div V)(XT (x)) = − E V(XT (x)), //T ΘT t t 2 0 where Θ is the Aut(T x M)-valued process defined by the following pathwise differential equation: d Θt = −Ric//t Θt − (∇. Z)∗// Θt + (divZ)Θt t dt with Θ0 = idT x M .

DERIVATIVE AND DIVERGENCE FORMULAE

13

Corollary 1.17. Suppose f is a bounded smooth function and that V is a bounded smooth vector field with div V bounded. Then, under the assumptions of Theorem 1.16, by using the relation div( f V) = V f + f div V, we get PT V( f ) (x) = −E f (XT (x)) (div V)(XT (x)) " Z T E# D 1 dB (div Z)(Xt (x))ht − h˙ t Θ−1 − E f (XT (x)) V(XT (x)), //T ΘT t t 2 0 where the right-hand side does not contain any derivatives of f . Corollary 1.18. Under the assumptions of Theorem 1.16 we have " # Z T 1 ∇ log pT (x, ·) y = − E //T ΘT dB (div Z)(Xt (x))ht − h˙ t Θ−1 X (x) = y t T t 2 0

with Θ given as above.

Proof. By Theorem 1.16, for all smooth, compactly supported vector fields V we have PT (div V)(x) #E " Z T Z D 1 dB (div Z)(Xt (x))ht − h˙ t Θ−1 =− X (x) = y V(y), E //T ΘT pT (x, y) vol(dy), t T t 2 M 0

but on the other hand

Z

(div V)(y) pT (x, y) vol(dy) Z (d pT (x, ·))y V(y) vol(dy) =− ZM d log pT (x, ·) y V(y) pT (x, y) vol(dy) =−

PT (div V)(x) =

M

M

so the result follows.

1.5. Shift-Harnack Inequalities. Suppose RicZ is bounded below, that Ric + (∇. Z)∗ and div Z are bounded and that the following formula holds, for all t > 0, all f ∈ Cb1 (M) and all bounded vector fields V with div V bounded (see Corollary 1.17): Pt (V( f ))(x) = −E f (Xt (x)) (div V)(Xt (x)) " Z t E# D r 1 −1 1 (div Z)(Xr (x)) − Θr dBr . − E f (Xt (x)) V(Xt (x)), //t Θt 2 t t 0 Fix T > 0. Then, by Jensen’s inquality (see [13, Lemma 6.45]), there exist c,C1 (T ) > 0 such that ! C1 (T ) 2 (1.17) |Pt (V( f ))| ≤ δ Pt ( f log f ) − Pt f log Pt f + | div V|∞ + δc + |V|∞ Pt f δt | {z } =: α1 (δ,t,V)

for all δ > 0, t ∈ (0, T ] and positive f ∈ Cb1 (M). Alternatively, by the Cauchy-Schwarz inequality, there exists C2 (T ) > 0 such that !2 C2 (T ) 2 (1.18) |Pt (V( f ))| ≤ | div V|∞ + √ |V|∞ Pt f 2 t | {z } =: α2 (t,V)

14

DERIVATIVE AND DIVERGENCE FORMULAE

for all t ∈ (0, T ] and f ∈ Cb1 (M). These estimates can be used to derive shift-Harnack inequalities, as shown by F.-Y. Wang for the case of a Markov operator on a Banach space (see [18, Proposition 2.3]). In particular, suppose {F s : s ∈ [0, 1]} is a C 1 family of diffeomorphisms of M with F0 = id M . For each s ∈ [0, 1] define a vector field V s on M by V s := (DF s )−1 F˙ s d ( f ◦ F s ) = ∇Vs ( f ◦ F s ). Fixing and assume V s and div V s are uniformly bounded. Note ds p ≥ 1 and setting β(s) = 1 + (p − 1)s, as in the first part of [18, Proposition 2.3], we deduce from inequality (1.17) that ! p/β(s) d β′ (s) p β(s) log Pt ( f ◦ F s) α1 , t, V s ≥− ds β(s) β(s)

for all s ∈ [0, 1], which when integrated gives the shift-Harnack inequality ! ! Z 1 β′ (s) p α1 , t, V s ds (Pt f ) p ≤ Pt ( f p ◦ F1 ) exp β(s) 0 β(s)

for each t ∈ [0, T ] and positive f ∈ Cb1 (M). Alternatively, from inequality (1.18) and following the calculation in the second part of [18, Proposition 2.3], we deduce !1/2 q Z 1 Pt f ≤ Pt ( f ◦ F 1 ) + α2 (t, V s )ds Pt f 2 0

for each t ∈ [0, T ] and positive f ∈ Cb1 (M). The shift F1 could be given by the exponential of a well-behaved vector field; the shifts considered in [18] are of the form x 7→ x + v, for some v belonging to the Banach space. 2. Extrinsic Formulae Suppose now that M is an n-dimensional smooth manifold. Suppose A0 is a smooth vector field and A : M × Rm → T M, (x, e) 7→ A(x)e, a smooth bundle map over M. This means A(·)e is a vector field on M for each e ∈ Rm , and A(x) : Rm → T x M is linear for each x ∈ M For an Rm -valued Brownian motion Bt , sped up by 2 so that d[B, B]t = 2 idRm dt, defined on a filtered probability space (Ω, F , P; (Ft )t∈R+ ), satisfying the usual completeness conditions, consider the Stratonovich stochastic differential equation (2.1)

dXt = A0 (Xt ) dt + A(Xt ) ◦ dBt .

m i Given an orthonormal basis {ei }m i=1 of R set Ai (·) := A(·)ei and Bt := hBt , ei i. Then the previous equation can be equivalently written m X dXt = A0 (Xt ) dt + Ai (Xt ) ◦ dBit . i=1

There is a partial flow Xt (·), ζ(·) associated to (2.1) (see [10] for details) such that for each x ∈ M the process Xt (x), 0 ≤ t < ζ(x) is the maximal strong solution to (2.1) with starting point X0 (x) = x, defined up to the explosion time ζ(x); moreover using the notation Xt (x, ω) = Xt (x)(ω) and ζ(x, ω) = ζ(x)(ω), if Mt (ω) = {x ∈ M : t < ζ(x, ω)}

then there exists Ω0 ⊂ Ω of full measure such that for all ω ∈ Ω0 : i) Mt (ω) is open in M for each t ≥ 0, i.e. ζ(·, ω) is lower semicontinuous on M;

DERIVATIVE AND DIVERGENCE FORMULAE

15

ii) Xt (·, ω) : Mt (ω) → M is a diffeomorphism onto an open subset of M; iii) The map s 7→ X s (·, ω) is continuous from [0, t] into C ∞ (Mt (ω), M) with its C ∞ topology, for each t > 0. The solution processes X = X(x) to (2.1) are diffusions on M with generator L := A0 +

m X

A2i

i=1

We will assume that the equation is non-degenerate, which is to say that A(x) : Rm → T x M is surjective for all x ∈ M. Then A induces a Riemannian metric on M, the quotient metric, with respect to which A(x)∗ = (A(x)|ker A(x)⊥ )−1 and whose inner product h·, ·i on a tangent space T x M is given by hv, ui = hA(x)∗ v, A(x)∗ uiRm .

2.1. A formula for the differential. Denote by i h Pt f (x) := E 1{t 0, by Itˆo’s formula we have d(αT −t (Ξt (x))) = (2.15)

m X i=1

ˆ A α (Ξt (x)) dt + ∂t α (Ξt (x)) dt ∇ˆ Ai αT −t (Ξt (x)) dBit + ∇ 0 T −t T −t

ˆ + trace ∇ˆ 2 αT −t (Ξt (x)) dt + αT −t (d∇ Ξt (x)) m X = ((∇ˆ Ai αT −t )· + αT −t (∇˘ . Ai ))(Ξt (x)) dBit . i=1

DERIVATIVE AND DIVERGENCE FORMULAE

21

It follows that αT −t (Ξt (x)) is a local martingale, starting at αT . Furthermore, according to equation (26) in [5], for the derivative process T Xt (x) we have ˆ

˘ T Xt (x) A0 dt + ˘ d∇ T Xt (x) = −Ric(T Xt (x)) dt + ∇

m X i=1

∇˘ T Xt (x) Ai dBit

and therefore, by the variation of constants formula, we have Z t ˘ 0 )∗ + ∇A ˘ 0 + trace ∇A ˆ 0 (Ξ s (x)) ds. T X s (x)−1 (∇A Ξt (x) = T Xt (x) − T Xt (x) 0

Thus it is possible to calculate Ξt (x) without using the parallel transport implicit in the original equation. Moreover, if the vector field A0 vanishes then Ξt (x) is given precisely by the derivative process T Xt (x). Proposition 2.7. Suppose ht is an adapted process with paths in L1,2 ([0, T ]; R). Then Z t ˆδα ht − h s αT −s (A0A) ds T −t 0 ! (2.16) Z t 1 −1 ˆ ˙ + αT −t Ξt (x) h s − (trace ∇A0 )(X s (x))h s Ξ s (x) A(X s (x)) dB s 2 0 ˆ h0 , where the vector field AA is given by (2.14). is a local martingale, starting at δα T 0 Proof. Set At := exp

Z

0

t

! ˆ 0 )(X s (x)) ds (trace ∇A

A−1 t ht .

and define ℓt := By equation (2.15), integration by parts and formula (2.11), we have, suppressing the summation over i, that Z ! 1 t ˙ −1 A s ℓ s Ξ s (x) A(X s (x)) dB s d αT −t Ξt (x) 2 0 j m 1 (∇ˆ Ai αT −t ) . + αT −t (∇˘ . Ai ) (Ξt (x)) dBit At ℓ˙t Ξt (x)−1 A j (Xt (x)) dBt = (2.17) 2 = (∇ˆ Ai αT −t )Ai + αT −t (∇˘ Ai Ai ) At ℓ˙t dt = (∇ˆ Ai αT −t )Ai At ℓ˙t dt ˆ = −(δα )At ℓ˙t dt T −t

m

where = denotes equality modulo the differential of a local martingale. By Proposition 2.5 and Itˆo’s formula we have m A ˆ ˆ ˆ ˆ d(At δα T −t ) = At δ∂t αT −t dt + At (L + trace ∇A0 )δαT −t dt = At αT −t (A0 ) dt

which implies ˆ nt := At δα T −t −

Z

0

t

A s αT −s (A0A)ds

ˆ . This implies is a local martingale, starting at δα T d(nt ℓt ) m = nt ℓ˙t dt ˙ ˙ ˆ = (δα T −t )At ℓt dt − ℓt

Z

t 0

A s αT −s (A0A ) dsdt.

22

DERIVATIVE AND DIVERGENCE FORMULAE

Substituting the definition of nt into the left-hand side and performing integration by parts to the second term on the right-hand side implies Z t Z t ˙ ˆ ˆ ℓ ds − h s αT −s (A0A ) ds (2.18) δα h − ( δα )A s s T −t t T −s 0

0

is another local martingale. Since ˆ 0 )(Xt (x))ht , ℓ˙t = A−1 h˙ t − (trace ∇A t

substituting formula (2.17) into the second term in (2.18) completes the proof.

Theorem 2.8. Suppose ht is any adapted process with paths in L1,2 ([0, ∞); R) such that h0 = 0 and hT = 1 and that α is a bounded smooth 1-form. Suppose (2.1) is complete and that the local martingales αT −t (Ξt ) and (2.16) are true martingales. Then " Z T !# 1 A −1 ˆ ˙ ˆ (trace ∇A0 )(Xt )ht − ht A(Xt ) dBt + 2ht A0 dt . Ξt PT (δα) = E α ΞT 2 0 Proof. By (2.15) we have αT −t (Ξt ) = α(ΞT ) −

Z

t

T

ˆ A α . + α (∇˘ . Ai ) (Ξt ) dBit ∇ i T −s T −s

and therefore E

"Z

T

0

# " Z A αT −t (Ξt ht Ξ−1 A ) dt = E α Ξ T t 0

0

T

A ht Ξ−1 t A0 dt

!#

since αT −t (Ξt ) is assumed to be a martingale. The result now follows from Proposition 2.7, by taking expectations. In analogue to Lemma 1.14, an integrability assumption on h plus suitable bounds on ˘ 0 , trace ∇A ˆ 0 , AA and δα ˆ and on the moments of T Xt and T Xt−1 would be sufficient to ∇A 0 guarantee that αT −t (Ξt ) and (2.16) are true martingales. Corollary 2.9. Suppose f is a bounded smooth function. Suppose V is a bounded smooth P vector field with m i=1 Ai hV, Ai i bounded. Then, under the assumptions of Theorem 2.8 with ♭ α = f V , we have PT (V( f )) m X E f (XT ) Ai hV, Ai i(XT ) =− i=1

" Z T # 1 A ˆ ˙ − E f (XT ) V(XT ), ΞT A(X ) dB + 2h A dt (trace ∇A )(X )h − h Ξ−1 t t t 0 0 t t t t 2 0

with Ξt = T Xt − T Xt

Z

0

t

˘ 0 )∗ + ∇A ˘ 0 + trace ∇A ˆ 0 (Ξ s ) ds, T X s−1 (∇A

m X ˘ 0 )∗ + ∇A ˘ 0 T˘ (·, Ai )∗ (Ai ) + A0 , T˘ (·, Ai )∗ (Ai ), (∇A A0A = i=1

DERIVATIVE AND DIVERGENCE FORMULAE

23

ˆ 0 , ∇A ˘ 0 and T˘ (·, Ai ) are given at each x ∈ M and v ∈ T x M by where the operators ∇A ∇ˆ v A0 = A(x) d A∗ (·)A0 (·) x (v) − (dA∗) x (v, A0 ) , ∇˘ v A0 = A(x)d A(·)∗ A0 (·) x (v), T˘ (v, Ai ) x = A(x)(dA∗) x (v, Ai ). Proof. This follows from Theorem 2.8. In particular, Lemma 2.6 implies ˆ ♭ ) − δ( ˆ f V ♭) V( f ) = f δ(V

while formula (2.12), the Le Jan-Watanabe property and the adaptedness of ∇˘ imply m m X X ˆ ♭ ) = − h∇˘ Ai V, Ai i = − δ(V Ai hV, Ai i. i=1

i=1

Note that if (2.1) is a gradient system then L = ∆ + A0 and A0A vanishes and m X i=1

Ai hV, Ai i = div V.

ˆ 0 = div A0 , Corollary 2.9 yields the unfiltered version of CorolIn this case, since trace ∇A lary 1.17. Corollary 2.10. Under the assumptions of Corollary 2.9 we have d log pT (x, ·) y (v) m E D X T˘ (·, Ai )∗ (Ai )(y) = − v, i=1

+

1D

2

" Z v, E ΞT

0

T

Ξ−1 t

#E A ˆ ˙ (trace ∇A0 )(Xt )ht − ht A(Xt ) dBt + 2ht A0 dt XT (x) = y

for all v ∈ T y M where the various terms appearing in the right-hand side can be calculated as in Corollary 2.9. Proof. Since Corollary 2.9 holds for all smooth functions f and vector fields V of compact support, and since by Lemma 2.6 ˆ ♭ ) − δ( ˆ f V ♭ ) = V( f ) = f δ(V ♭ ) − δ( f V ♭ ), f δ(V the result follows from equation (2.13), Lemma 2.4 and Corollary 2.9.

Example 2.11. Consider the special case M = Rn . Denote by qT (x, y) the smooth density of XT (x) with respect to the standard n-dimensional Lebesgue measure. Recall that pT (x, y) denotes the density with respect to the induced Riemannian measure. It follows that qT (x, y) = pT (x, y) ρ1/2 (y) where ρ(y) denotes the absolute value of the determinant of the matrix ∗ hA ∂i , A∗ ∂ j iRm (y) ni, j=1

in which {∂i }ni=1 denotes the standard basis of vector fields on Rn . Consequently d log qT (x, ·) y (v) = d log pT (x, ·) y (v) + d log ρ1/2 (·) y (v)

with the first term on the right-hand side given, in terms of the induced metric, by Corollary 2.10.

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DERIVATIVE AND DIVERGENCE FORMULAE

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