Integration by parts in loop spaces

Math. Ann. 309, 331–339 (1997)

Mathematische Annalen c Springer-Verlag 1997

Integration by parts in loop spaces Elton P. Hsu Department of Mathematics, Northwestern University, Evanston, IL 60208, USA (e-mail: elton@@math.nwu.edu) Received: 28 April 1996 / Revised version: 6 September 1996

Mathematics Subject Classification (1991): 60D58, 28D05

1. Introduction We assume throughout this paper that M is a n-dimensional compact Riemannian manifold and O(M ) its orthonormal frame bundle. We use H to denote the Rd valued Cameron-Martin space over the interval [0, 1] with zero initial values and H0 the subspace of H with zero values at 1. We fix a point o ∈ M and a frame uo ∈ O(M ) over o. We use Wo (M ) to denote the set of M -valued paths (of time length 1) starting from o and Lo (M ) the set of loops at o, i.e., the set of paths γ in Wo (M ) such that γ(1) = o. The Levi-Civita connection determines a Laplace-Beltrami operator ∆ on M . We use ν to denote the Wiener measure on Wo (M ) generated by ∆/2. The measure νo defined intuitively by νo (·) = ν(·|ω(1) = o) is a measure on the loop space Lo (M ), which we call the Wiener measure on Lo (M ). For a smooth or a typical Brownian γ ∈ Wo (M ) or Lo (M ), let U (γ) be the horizontal lift of γ such that U (γ)o = uo . Fix an h ∈ H (or H0 ) , the “vector field” Dh on Wo (M ) (or Lo (M )) is defined by (1)

Dh (γ)s = U (γ)s hs .

There is a complete theory of integration by parts for Dh on Wo (M ), developed by Driver[1] and supplemented by Hsu[5]. See also Enchev and Stroock[3] for The research was supported in part by the NSF grant 9406888-DMS

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another approach. In the case of the loop space Lo (M ), Driver[2] proved an integration by parts formula for vector fields Dh with lipschitzian h and the complete result for all Cameron-Martin vector fields was proved in Enchev and Stroock[4]. The purpose of this paper is to give an alternative approach to integration by parts in loop spaces. Armed with an upper estimate on the ∇2 log p(s, x , y) due to Sheu[8] (see (7) below), we prove an integration by parts formula in the loop space Lo (M ) through the corresponding formula for the path space Wo (M ). Such an approach avoids the quasi-invariance of the Wiener measure in the loop spaces thus providing a more direct route to the result.

2. Integration by parts in path spaces Let µ be the usual Wiener measure on the path Wo (Rn ). Let {Us } be the solution of the stochastic differential equation on O(M ) (2)

dUs = HUs ◦ d ωs ,

U0 = uo .

Here H = {Hi , i = 1, . . . , d } are the canonical horizontal vector fields on O(M ) and {ωs } is the coordinate process on Wo (Rn ). Let γs = π(Us ) be the projection of U in Wo (M ). The Itˆo map J : Wo (Rn ) → Wo (M ) is defined by J ω = γ. It is well known that the law of γ is ν, the Wiener measure on Wo (M ), i.e., J carries the Wiener measure µ on Wo (Rd ) to the Wiener measure ν on Wo (M ). The inverse J −1 : Wo (M ) → Wo (Rd ) is the stochastic development map. A function F : Wo (M ) → R1 is called cylindrical if there is a positive integer l , a set of l points 0 ≤ s1 < · · · < sl ≤ 1 and a smooth function f : M × · · · × M → R1 such that
(3) F (γ) = f γs1 , · · · , γsl . The set of cylindrical functions on Wo (M ) is denoted by C . We will use L2 (ν) to denote the Hilbert space of measurable functions F on Wo (M ) such that Z kF k2L2 (ν) = |F (γ)|2E ν(d γ) < ∞. Wo (M )

The inner product on L2 (ν) is denoted by (·, ·)L2 (ν) or simply (·, ·). Let F ∈ C be given by (3). From the defintion of the vector field Dh in (1) it is natural to define (4)

Dh F (γ) =

l X

h∇(p) F (γ), U (γ)sp hsp i,

p=1

where ∇(p) F denotes the gradient of f with respect to the pth variable. Let h ∈ H, define

Integration by parts in loop spaces

333

Z

1

lh (γ) = 0

1 hh˙ s + RicUs hs , d ωs i, 2

where ω = J −1 γ, U = U (γ) is the horizontal lift of γ to O(M ), and Ricu : Rn → Rn is the Ricci transform at u ∈ O(M ). Theorem 2.1. (Integration by parts in path space) Let F , G be two cylindrical functions on Wo (M ). Then
(5) (Dh F , G) = F , Dh∗ G , where Dh∗ = −Dh + lh . The assumption that h ∈ H implies that there exists a constant c > 0 such 2 that Eν e c|lh | < ∞. By a standard functional analysis argument, the integration by parts formula implies that Dh is closable and the adjoint Dh∗ is densely defined (the in Dom closability of Dh requires only lh ∈ L2 (ν)). There are plenty of functions S (Dh∗ ). More precisely, we have the following result. Let L2+ (ν) = p>2 Lp (ν). Theorem 2.2. Let h ∈ H. Then Dh : C → L2 (ν) is closable in L2 (ν) and has a densely defined adjoint Dh∗ . Furtheremore,
Dom (Dh ) ∩ L2+ (ν) ⊂ Dom Dh∗ and for all G ∈ Dom (Dh ) ∩ L2+ (ν) we have Dh∗ G = −Dh G + lh G.

3. Some preliminary results In this section we collect some results which will be used in the proof of integration by parts formula on the loop space in the next section. We denote by p(s, x , y) the heat kernel of the half Laplacian ∆/2 on M . Proposition 3.1. There exists a constant depending only on M such that for all (s, x , y) ∈ (0, 1) × M × M ,   1 d (x , y) +√ (6) |∇ log p(s, x , y)| ≤ C s s  (7)

|∇ log p(s, x , y)| ≤ C 2

d (x , y)2 1 + s2 s

 .

Proof. As far as we know, these results are due to Sheu[8]. See also Hsu[7] and Stroock and Trubetsky[9] for further discussions. t u

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Lemma 3.2. For each positive integer N there is a constant CN depending only on N and M such that n o Eνo d (γs , o)N ≤ CN min s N /2 , (1 − s)N /2 . Proof. This inequality is intuitively clear and can be proved based on the estimate (6). See Driver[2] or Hsu[6] for details. t u Lemma 3.3. (Hardy’s inequality) Let h ∈ H0 , then Z 1 Z 1 hs 2 ds ≤ 4 |h˙ s |2 ds. 1 − s 0 0 Proof. We have for any t ∈ (0, 1),   Z t Z t hs 2 1 2 ds = |h | d s 1−s 0 1−s 0 Z t ˙ |ht |2 hs · hs = 2 ds + 1−t 0 1−s 2 Z t Z t 2 hs 1 ds + 2 ˙ s |2 ds + |ht | . ≤ | h 2 0 1 − s 1−t 0 In the last step we have used inequality 2ab ≤ Therefore

1 2 a + 2b 2 . 2

Z t Z t 2 hs 2 ds ≤ 4 ˙ s |2 ds + 2|ht | . | h 1−t 0 1−s 0

The desired inequality follows by letting t → 1 in the above inequality because Z 2 Z 1 1 1 ˙ |ht |2 hs ds ≤ = |h˙ s |2 ds → 0. t u 1−t 1−t t t 4. Integration by parts on loop spaces Recall that in path space Wo (M ) the adjoint of Dh is given by (8)

Dh∗ = −Dh + lh ,

where lh : Wo (M ) → Rn is defined by Z 1 1 lh (γ) = hh˙ s + RicUs hs , d ωs i. 2 0 Here U is the horizontal lift of γ and ω = J −1 γ is the stochastic development of γ. On the loop space Lo (M ), we define Dh F for a cylindrical function by

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the same formula (4) as in the path space. The next proposition shows that lh is well defined under the measure νo . This step is necessary because ν and νo are mutually singular. Let {lh,s } be the ν-martingale Z

s

hh˙ τ +

lh,s = 0

1 RicUτ hτ , d ωτ i. 2

Let {Bs , 0 ≤ s ≤ 1} be the standard filtration of σ-fields on Wo (M ). Then the measures νo and ν are mutually absolutely continuous on Bs for all s < 1. Hence the process {lh,s , 0 ≤ s < 1} is well defined under the measure νo . The next lemma concerns the limit of lh,s as s → 1 under the measure νo . Proposition 4.1. The limit lh,s → lh exists in L1 (νo ) as s → 1. Furthermore lh ∈ L2 (νo ). Proof. Under the measure ν, the stochastic development ω = J −1 γ is a Brownian motion. Under the measure νo , it is a local semimartingale before time 1 and its martingale part {bs } is a Brownian motion. The measure νo is characterized by the fact that Z s Uτ−1 ∇ log p(1 − τ, γτ , o)d τ. ωs = bs + 0

Let Qs = Us−1 ∇ log p(1 − s, γs , o), Z

1 Fs = hs − 2

1

RicUτ hτ d τ s

for simplicity. We have for s < 1 Z lh,s

Z

s

Z

hF˙ τ , Qτ d τ i Z

0

0

s

s

hF˙ τ , dbτ i −

= 0

=

s

hF˙ τ , dbτ i +

=

hFτ , dQτ i + hQs , Fs i − hQ0 , F0 i 0

I1,s − I2,s + I3,s − hQ0 , F0 i.

Now Ricu hτ is uniformly bounded, and h˙ ∈ L2 [0, 1]. These facts imply that the limit I1,s → I1 exists in L2 (νo ) and Z

1

hF˙ s , dbs i.

I1 = 0

For I3,s we have |Fs | ≤ C {|hs | + (1 − s)} and using (6) and Lemma 3.2 we have

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Eνo |hFs , Qs i|

≤ ≤ ≤ →

C {|hs | + (1 − s)} Eνo |∇ log p(1 − s, γs , o)|   1 Eνo d (γs , o) +√ C1 {|hs | + (1 − s)} 1−s 1−s   s Z 1  √ 1 C2 1−s |h˙ τ |2 d τ + (1 − s) √  1−s  s 0.

This shows that I3,s → 0 in L1 (νo ). For I2,s we use Itˆo’s formula on the

Rn -valued function

Qs = Us−1 ∇ log p(1 − s, γs , 0) = ∇H log P (1 − s, Us ) of (s, Us ) ∈ (0, 1) × O(M ), where P (s, u) = p(s, πu, o). Using the stochastic differential equation (2) for Us we have for the i th component dHi log P (1 − s, Us )

(9) =

1 hHi ∇H log P (1 − s, Us ), dbs i + hRicUs ei , ∇H log P (1 − s, Us )ids 2   1 +Hi H log P (1 − s, Us ) + |∇H log P (1 − s, Us )|2 ds, 2

where

H = 21 ∆H + ∂s∂ ,

Pn and ∆H = j =1 Hj2 is Bochner’s horizontal Laplacian. Note that in the above computation we need to use the second structural equation [Hi , Hj ] = Ω(Hi , Hj )∗ to exchange Hi and Hj (Ω ∗ is the canonical vertical vector field corresponding to Ω ∈ o(n)). The last term in (9) vanishes because p(t, x , y) satisfies the heat equation. Hence we have Z s hFτ , Uτ−1 ∇2 log p(1 − τ, γτ , o), dbτ i I2,s = 0 Z 1 s hRicUτ Fτ , Uτ−1 ∇ log p(1 − τ, γτ , o)d τ i. + 2 0 To show that the limit I2,s → I2 exists in L2 (νo ) it is enough to show that Z 1 Eν o (10) |Fs |2 · |∇2 log p(1 − s, γs , o)|2 ds < ∞ 0

and (11)

Z

1

|Fs |2 · |∇ log p(1 − s, γs , o)|2 ds < ∞.

Eν o 0

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From the definition of Fs there exists a constant C such that |Fs | ≤ C {|hs | + (1 − s)} .

(12)

Using the estimate (7) and Lemma 3.2 we see that there exists a constant C such that C . Eνo |∇2 log p(1 − s, γs , o)|2 ≤ (1 − s)2 It follows from Lemma 3.3 that Z 1 Eνo |Fs |2 · |∇2 log p(1 − s, γs , o)|2 ds 0 1

2 |hs | + (1 − s) ds (1 − s) 0 (Z ) 1 2 8C |h˙ s | ds + 1 . Z

≤

C

≤

0

This proves (10). From (6) and Lemma 3.2 there is a constant C such that Eνo |∇ log p(1 − s, γs , o)|2 ≤

C . 1−s

Using this inequality and (12) we have Z 1 Eν o |Fs |2 |∇ log p(1 − s, γs , o)|2 ds Z

0 1

2

{|hs | + (1 − s)} ds 1−s 0 (Z ) 1 2 ˙ 8C |hs | ds + 1 .

≤

C

≤

0

This proves (11). It follows that the limit I2,s → I2 exists in L2 (νo ) and Z 1 I2 = hFs , Us−1 ∇2 log p(1 − s, γs , o), dbs i 0

+

1 2

Z

1

hRicUs Fs , Us−1 ∇ log p(1 − s, γs , o)dsi.

0

To summarize, we have lh,s = I1,s − I2,s + I3,s − hQ0 , F0 i; I1,s → I1 , I2,s → I2 , both in L2 (νo ), and I3,s → 0 in L1 (νo ). It follows that the stochastic integral Z 1 1 hh˙ s + RicUs hs , d ωs i lh = 2 0 (ω = J −1 γ) exists as the L1 (νo )-limit of lh,s as s → 1 and lh ∈ L2 (νo ).

t u

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We now prove the main theorem. Theorem 4.2. (Integration by parts formula in loop space) Let F , G be two cylindrical functions on Lo (M ). Then
(Dh F , G)L2 (νo ) = F , Dh∗ G L2 (νo ) , where

Dh∗ = −Dh + lh

and lh ∈ L2 (νo ) is defined by Z

1

lh (γ) = 0

1 hh˙ s + RicUs hs , d ωs i. 2

Here ω = J −1 γ is the stochastic development of γ and U is the horizontal lift of γ. Proof. Suppose that F and G depend on the path up to time s ∗ < 1. Then we have for all s ∈ (s ∗ , 1), (Dh F , G)L2 (νo ) = Co (Dh F , Gp(1 − s, γs , o))L2 (ν) , where Co = p(1, o, o)−1 . By the integration by parts formula (5) for the path space, we have (Dh F , G)L2 (νo )




=

Co F , Dh∗ {Gp(1 − s, γs , o)}

=

−Co (F , Dh Gp(1 − s, γs , o))L2 (ν)

L2 (ν)

−Co (F , GDh p(1 − s, γs , o))L2 (ν) +Co (F , lh Gp(1 − s, γs , o))L2 (ν) =

− (F , Dh G)L2 (νo ) − (F , GDh log p(1 − s, γs , o))L2 (νo )
+ F , lh,s G L2 (ν ) . o

Since F and G are uniformly bounded, by Proposition 4.1 we have
F , lh,s G L2 (ν ) → (F , lh G)L2 (νo ) . o It is therefore enough to show (F , GDh log p(1 − s, γs , o))L2 (νo ) → 0. This is implied by (13)

Eνo |Dh log p(1 − s, γs , o)| → 0.

We have Dh log p(1 − s, γs , o) = hhs , Us−1 ∇ log p(1 − s, γs , o)i. By (6) and Lemma 3.2 we have

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Eνo |Dh log p(1 − s, γs , o)|

≤

|hs | · Eνo |∇p(1 − s, γs , o)|   1 Eνo d (γs , o) +√ C |hs | 1−s 1−s C1 |hs | √ 1−s Z C1 1 ˙ √ hτ d τ 1−s s s Z 1 |h˙ τ |2 d τ C1

→

0.

= ≤ ≤ =

s

This shows (13) and the theorem is proved.

t u

As a consequence of the above integration by parts formula and the fact that lh ∈ L2 (νo ), we have the following result parallel to Theorem 2.2. Let B (νo ) be the space of νo -essentially bounded measurable functions on Lo (M ). Theorem 4.3. Let h ∈ H0 . Then Dh : C → L2 (νo ) is closable in L2 (νo ) and has a densely defined adjoint Dh∗ . Furthermore Dom(Dh ) ∩ B (νo ) ⊂ Dom(Dh∗ ) and for all G ∈ Dom(Dh ) ∩ B (νo ) we have (14)

Dh∗ G = −Dh G + lh G.

Acknowledgement. I want to thank Professor M. Cranston for his generous help throughout the work.

References 1. Driver, B.: A Cameron-Martin type of quasiinvarance for the Brownian motion on a compact manifold. J. Funct. Anal., 110 (1992), 237–376. 2. Driver, B.: A Cameron-Martin type of quasiinvarance theorem for pinned Brownian motion on a compact manifold. TAMS, 342, No. 1 (1994), 375-395. 3. Enchev, O., Stroock, D. W.: Towards a Riemannian geometry on the path space over a Riemannian manifold. J. Funct. Anal., 134 (1995). 4. Enchev, O., Stroock, D. W.: Integration by parts for pinned Brownian motion. To appear in Advances in Mathematics. 5. Hsu, E. P.: Quasiinvariance of the Wiener measure and integration by parts in the path space over a compact Riemannian manifold. J. Funct. Anal., 134 (1995), 417–450. 6. Hsu, E. P.: Stochastic Gauss-Bonnet-Chern formula. J. Theoret. Probability (1995). 7. Hsu, E. P.: Estimates of the derivatives of the heat kernel, preprint (1996). 8. Sheu, S.-Y.: Some estimates of the transition density function of a nondegenerate diffusion Markov process. Annals of Probability, 19, No. 2 (1991), 538–561. 9. Stroock, D. W., Trubetsky, J.: Upper bounds for the derivatives of the logarithm of the heat kernel, preprint (1996)