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Elect. Comm. in Probab. 2 (1997) 71{81

ELECTRONIC COMMUNICATIONS in PROBABILITY

MARTINGALE REPRESENTATION AND A SIMPLE PROOF OF LOGARITHMIC SOBOLEV INEQUALITIES ON PATH SPACES MIREILLE CAPITAINE Departement de Mathematiques, Laboratoire de Statistique et Probabilites associe au C.N.R.S., Universite Paul-Sabatier, 31062, Toulouse (France) ELTON P. HSU1 Department of Mathematics, Northwestern University, Evanston, Illinois 60208 (U.S.A.) MICHEL LEDOUX Departement de Mathematiques, Laboratoire de Statistique et Probabilites associe au C.N.R.S., Universite Paul-Sabatier, 31062, Toulouse (France) submitted March 24, 1997; revised November 11, 1997 AMS 1991 Subject classi cation: 60H07, 60D05, 60B15 Keywords and phrases: Martingale Representation, Logarithmic Sobolev Inequality, Hypercontractivity, Path Space. Abstract We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diusion measure as long as the torsion of the corresponding Riemannian connection satis es Driver's total antisymmetry condition.

1 Introduction

Let ! = (!t)t0 be a standard Brownian motion starting from the origin with values in IRn and denote by W0 (IRn ) the path space of continuous functions from [0; 1] to IRn starting from the origin. We denote further by IE expectation with respect to the law  (the Wiener measure) of ! on W0 (IRn). Gross [G] proved the following logarithmic Sobolev inequality holds: ?




(1) IE(F 2 log F 2) ? IE(F 2) logIE(F 2)  2 IE jDF j2IH where D : L2 () 7! L2(; IH) is the (Malliavin) gradient operator on W0 (IRn ) and IH the Cameron-Martin Hilbert space. Logarithmic Sobolev inequalities were introduced to study 1

Research supported in part by the NSF grant DMS 94-06888.

71

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Electronic Communications in Probability hypercontractivity properties of Markov semigroups and they imply spectral gap inequalities such as ?
?
(2) IE(F 2) ? IE(F ) 2  IE jDF j2IH in the present case. Thanks to the linear structure of W0 (IRn), the proof of the logarithmic Sobolev inequality (1) may be reduced to the case of nite dimensional Gaussian measures, for which rather elementary semigroup arguments may be used (cf. [Ba]). Such semigroup arguments can actually be formulated in terms of stochastic calculus on Brownian paths, which may then be shown to work easily in the in nite dimensional setting as well. To illustrate the purpose of this paper, let us rst describe a proof of (1) and (2) along these lines. This proof is known to a number of people although we were unable to trace it back in the literature2 . The starting point is the so-called Clark-Ocone-Haussmann formula (see e.g. [N]) which indicates that, for every functional F in the domain of the gradient operator D,

F ? IE(F ) =

(3)

Z

1 0

?




hIE (DF ):t jBt ; d!ti

where (Bt )t0 is the ltration of !. Taking the L2 -norm of both sides of (1) already yields the spectral gap inequality (2), since


?

IE F ? IE(F ) 2

= IE = IE

 IE

Z Z

1

0

?

1

? IE (

0

Z




hIE (DF ):t jBt ; d!ti

1

0




DF ):t j Bt 2dt : 2 t



2






?

DF ) dt = IE jDF j2IH :

(

Now, the same idea can be used to establish the logarithmic Sobolev inequality. Namely, what (3) tells us is that the martingale Mt = IE(F j Bt), 0  t  1, is such that
? dMt = hIE (DF ):t jBt ; d!ti: For simplicity we may assume that F is in the domain of D and F  " for some " > 0. The latter can be removed afterwards by letting " tend to 0. Applying It^o's formula to Mt log Mt , we get  Z 1 1 IE?(DF ): j B
2dt IE(M1 log M1 ) ? IE(M0 log M0 ) = 21 IE t t M 0

that is,

t

Z



1 1 IE?(DF ): j B
2 dt : IE(F log F ) ? IE(F ) log IE(F ) = 21 IE t t IE( F j Bt ) 0 Replace now F by F 2. By the Cauchy-Schwarz inequality,

? IE (

DF 2 ):t j Bt


2

?

= 4 IE F (DF ):t j Bt


2

?






 4 IE(F 2 j Bt) IE (DF ):t 2 j Bt :

2 The third author learned it several years ago from B. Maurey. Recently, J. Neveu also mentioned this proof to him.

Logarithmic Sobolev Inequalities on Path Spaces Hence IE(F log F ) ? IE(F ) logIE(F )  2 IE 2

2

2

73 Z

2

0

1

: 2 t



?




DF ) dt = 2 IE jDF j2IH ;

(

which is the desired logarithmic Sobolev inequality (1). Recently, E. P. Hsu [H2], [H3] and S. Aida and K. D. Elworthy [A-K] established a logarithmic Sobolev inequality for the law of Brownian motion on a compact Riemannian manifold (or more generally for complete Riemannian manifolds with bounded Ricci curvature). The method of [H3] is based on logarithmic Sobolev inequalities for the heat kernel measures together with a Markovian tensorization and Bismut's formula to control the spatial derivative of the heat kernel. S. Aida and K. D. Elworthy embbed the manifold into an Euclidean space and then use the logarithmic Sobolev inequality (1) on at space. As a result, their logarithmic Sobolev constant also depends on the embedding rather than only on the bound on the Ricci curvature as in [H3]). This logarithmic Sobolev inequality improves upon the previous spectral gap inequality due to S. Fang [F1]. Now, S. Fang's beautiful proof is based on a version of the ClarkOcone-Haussmann formula for Brownian motion on a manifold. This representation formula is presented in a handy way in [H4] and simply involves an additional curvature term in (3). The aim of this note is then simply to observe that, together with this representation formula, the preceding simple proof of the logarithmic Sobolev inequality for Brownian motion in IRn yields in exactly the same way the logarithmic Sobolev inequality for the law of Brownian motion on a complete Riemannian manifold with bounded Ricci curvature. This result is presented in Section 2. In Section 3 we extend the argument by an appropriate integration by parts formula to general diusion processes with generators of the form 21
+ V over a complete Riemannian manifold. The manifold is assumed to be equipped with a connection compatible with the Riemannian metric, and its torsion satis es Driver's total antisymmetry condition. The resulting logarithmic Sobolev inequality extends simultaneously the recent results of F.-Y. Wang [W] and S. Fang [F2]. In the last section we show how the previous stochastic calculus argument may be applied to yield similarly the isoperimetric inequality on path spaces in [B-L].

2 Logarithmic Sobolev Inequality for Brownian Motion on a Manifold We rst recall Fang's martingale representation formula as presented in [H4]. We follow the exposition of [H4] and refer to it for further details. Let M be a complete and connected Riemannian manifold of dimension n equipped with the Levi-Civita connection r. Denote by
the Laplace-Beltrami operator on M . Fix a point x0 in M and let Wx0 (M ) be the space of (pinned) continuous paths from [0; 1] to M starting at x0. Let O(M ) be the bundle of orthonormal frames and let  : O(M ) 7! M be the canonical projection. Each frame u 2 O(M ) is a linear isometry u : IRn 7! T(u) (M ), the tangent space at (u). Let fHi ; 1  i  ng be the canonical horizontal vector elds on O(M ). Fix an orthonormal frame u0 at x0 and let U be the diusion process solution of the Stratonovich stochastic dierential equation

dUt =

n X i=1

Hi (Ut )  d!ti ; U0 = u0 ;

where ! = (!t )t0 is a P standard Brownian motion on IRn starting from the origin. The 1 diusion process U has 2 ni=1 Hi2 as its generator and is called horizontal Brownian motion.

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Electronic Communications in Probability The projection = (U ) is a Brownian motion on M starting from x0 whose law is the Wiener measure  on Wx0 (M ) with the generator 21
. The map J : W0 (IRn ) 7! Wx0 (M ) given by J! = is called the It^o map. Each h in the Cameron-Martin Hilbert space IH determines a vector eld Dh as follows. For a typical Brownian path , the map Ut : IRn 7! T t (M ) is an isometry. Hence Ut ht is a vector at t . The vector eld Dh is de ned by Dh ( t ) = Ut ht . Let F be a cylindrical function on Wx0 (M ) of the form F ( ) = ( t1 ; : : :; t` ) where 0  t1 <


< t`  1 and  is a smooth real-valued function on M ` . For such an F ,

Dh F ( ) =

` X i=1

hr(i)F ( ); Uti hti iT ti ;

where r(i)F is the (usual) gradient on M of  with respect to the i-th variable. The directional derivative operator Dh is closable in L2 ( ), and so is the gradient operator D : L2 ( ) 7! L2 ( ; IH) de ned by hDF; hiIH = Dh F . We denote by Dom(D) its domain in L2( ). We now present Fang's version Clark-Ocone-Haussmann formula for the Brownian motion in M , following [H4]. We denote by Ric the Ricci tensor on M and write more precisely Ricu , u 2 O(M ), for the linear symmetric transformation v 7! Ricu (v) from T(u) (M )  = IRn into itself. We assume throughout this section that M has bounded curvature, that is  sup kRicuk ; u 2 O(M ) = K < 1; where k
k is the operator norm on IRn . Let (At )0t1 be the matrix-valued process (Ricci

ow) satisfying the dierential equation dAt ? 1 A Ric = 0; A = I: 0 dt 2 t Ut Then, for every cylindrical function F in the domain of D, Z 1 Z 1D   E E (DF ):t ? 12 At (As )?1 RicUs (DF ):s ds Bt ; d!t : (4) F ? IE(F ) = t 0 The identity (4), which extends the at case (3), relies on the Bismut-Driver integration by parts formula (see [Bi], [D], [H4] and the references therein) to the eect that for any h 2 IH, the adjoint Dh of Dh with respect to  is given by (5)

Dh = ?Dh +

Z

1 0

hh_ t + 21 RicUt ht; d!ti:

Provided with this formula, the representation (4) is established from this formula through rather standard arguments (cf. [H4]). We can now prove the logarithmic Sobolev inequality for following almost exactly the previous proof in the at case. Let F be a cylindrical function in Dom(D) such that F  " for some " > 0. By (4), the martingale Mt = IE(F jBt ), 0  t  1, satis es dMt = hHt; d!ti where  Z 1  1 : : ? 1   Ht = E (DF )t ? 2 At (As ) RicUs (DF )s ds Bt : t

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75

Applying It^o's formula to Mt log Mt , we get

Z

IE(M1 log M1 ) ? IE(M0 log M0 ) = 21 IE In other words, IE(F log F ) ? IE(F ) logIE(F ) = 21 IE

Z

1 0



1 H 2 dt : Mt t 

1 H 2 dt : t 0 IE(F j Bt ) Replace now F by F 2 and set for simplicity jt = j(DF 2):t j. We have,  Z 1 

  ?1 1

jHtj  IE jt + 2 At (As ) kRicUs k js ds Bt : t R s Since for every s  t, A?s 1 At = ? 12 t RicUr A?r 1 At dr, by Gronwall's lemmawe have kA?s 1At k  eK (s?t)=2. Thus,  Z 1  1 K (s?t)=2 js ds Bt : jHtj  IE jt + 2 K e 1

t

Since jt = 2jF jj(DF ) j, it follows from the Cauchy-Schwarz inequality that,  Z 1 2 
? 2 1 : : 2 K (s?t)=2 jHtj  4 IE F j Bt IE (DF )t + 2 K e (DF )s ds Bt : t Therefore,  Z 1 Z 1  Z 1 2 2 1 1 : : K (s?t)=2 Ht dt  4 IE j(DF )t j + 2 K e (DF )s ds dt: IE 2 0 t 0 IE(F j Bt ) Now we have : t

Z

t

1

K (s?t)=2

e

j(DF ) j ds : s

Z

where



Z

t

1

K (s?t)

e

ds

Z

1 0



DF ):s 2 ds

(

  = K1 eK (1?t) ? 1 jDF j2IH:

It then follows easily that IE

2

1 0

1

2 t



?




2 IE(F j Bt) H dt  4 c(K ) IE jDF jIH ; 2

?
p c(K ) = 1 + 14 eK ? 1 ? K + eK ? 1 ? K  eK : We thus have established a logarithmic Sobolev inequality for Brownian motion on Wx0 (M ) in the form ?
(6) IE(F 2 log F 2) ? IE(F 2 ) logIE(F 2 )  2 eK IE jDF j2IH where we recall that K is the uniform bound on the Ricci curvature of M . The extension to all functions F in the domain of D follows in a standard way. In particular, when K = 0 (M = IRn or IRn=ZZn for example), we recover (1).

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3 Diusion Processes and Connections with Torsion In this section, we extend the preceding logarithmic Sobolev inequality to the law of a general diusion process with generator L = 21
+ V on the path space over a Riemannian manifold M equipped with a connection compatible with the Riemannian metric but not necessarily torsionfree. We refer to [D] and [H1] for the general setting, notation and geometric consideration. Let M be a complete, connected Riemannian manifold equipped now with a connection r compatible with the Riemannian metric. We assume that the torsion  = fu; u 2 O(M )g satis es Driver's total antisymmetry condition i.e., hu (x; y); z iIRn is alternating in all three variables (x; y; z ). Let V be some smooth vector eld on M . The main step of the proof is to get an integration by parts formula for the connection r and the diusion measure generated by L. To this aim, one may either redo parts of [H1], or use the integration by parts formula of Driver [D] for 21
to derive that of L = 12
+ V via Girsanov's theorem. We follow here the second route. We rst recall Driver's integration by parts formula for the Wiener measure  on M [D]. Namely, the adjoint Dh of Dh with respect to  is given by (cf. [D], [H1] and the notation therein) Dh = ?Dh + `h where Z 1 (7) `h = hh_ t + 21 b Ut ht + 12 RicUt ht ; d!ti: 0 Here, the map b u : IRn 7! IRn is given by b u (v) =

n X i=1

Hi u (ei ; v); v 2 IRn ;

where fei ; 1  i  ng are the coordinate unit vectors in IRn and fHi ; 1  i  ng are the canonical horizontal vector elds on O(M ). Let  be the diusion measure on Wx0 (M ) for L = 21
+ V . For u 2 O(M ) set V (u) = u?1V ((u)) 2 IRn and rV (u) = u?1 rV ((u)) 2 IRn  IRn; namely, V and rV are the scalarization of the tensors V and rV , respectively. Set nally dbt = d!t ? V (Ut )dt: The process ! is not a Brownian motion under , but b is. We have the following integration by parts formula. Proposition 1 For any h in IH, the adjoint Deh of Dh with respect to  is given by

De h = ?Dh +

Z 0

1

hh_ t + 12 b Ut ht + 12 RicUt ht ? Ut (V (Ut ); ht) ? rV (Ut )ht ; dbti:

We outline a proof of this proposition using (7) and the techniques of [H1]. It is enough to show that De h 1 is given by the second term on the right-hand side of the above formula. First of all it is easy to check that  d ?1  d  De h 1 = d : Dh d

Logarithmic Sobolev Inequalities on Path Spaces

77

By Girsanov's theorem,

d ( ) = log d

Z

1

0

hV (Ut ); d!ti ? 12

Z

1

0

jV (Ut )j2dt:

We thus need to compute

Dh

Z

1 0

hV (Ut ); d!ti and Dh

Z

1



0



V (Ut ) 2 dt:

Now, by Theorem 2.1 of [H1], (Dh !)t = ht ? where K = Kh is de ned as

Z t

Us (d!s ; hs) ?

0

Kt =

Z t 0

Z t 0

Ks  d!s

Us (d!s; hs );

being the curvature tensor (cf. [H1]). Furthermore (cf. [H1, (2.4) and (2.8)]),

Dh V = rV h ? KV : Hence,

Dh

Z 0

1

hV (Ut ); d!ti =

Z 0

1

hrV (Ut )ht ? Kt V (Ut ); d!ti +

Z 0

1

hV (Ut ); h_ t dt ? Ut (d!t; ht) ? Kt  d!ti:

Since K is antisymmetric, we have hV ; KV i = 0. Therefore, 1 D Z 1 V (U ) 2dt = Z 1 hV (U ); rV (U )h idt: t t t t 2 h 0 0 Putting everything together, and recalling (7), we get   De h 1 = ?Dh log d d + `h Z 1 = ? hrV (Ut )ht ? Kt V (Ut ); d!ti 0

? +

Z Z

1

hV (Ut ); h_ tdt ? Ut (d!t; ht) ? Kt  d!ti

1

hV (Ut ); rV (Ut )htidt +

0

0

Z 0

1

hh_ t + 12 b Ut ht + 12 RicUt ht; d!ti:

We now convert Stratonovich integrals into It^o's integrals using the relations Ut (d!t; ht) = Ut (d!t; ht) + 21 b Ut ht dt

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Kt  d!t = Kt d!t ? 12 RicUt ht dt;

which follow from the de nitions of b and K . Driver's condition on the torsion form  implies that hV ; (d!; h)i = ?h(V ; h); d!i and h(V ; h); V i = 0. Hence we easily get Z 1
? De h 1 = hh_ t + 12 b Ut ht + 21 RicUt ht ? Ut V (Ut ); ht ? rV (Ut )ht; d!ti 0 Z 1 ?
? hh_ t + 12 b Ut ht + 21 RicUt ht ? Ut V (Ut ); ht ? rV (Ut )ht ; V (Ut )dti: 0 Recalling that dbt = d!t ? V (Ut )dt, we obtain the desired result immediately. Given the preceding integration by parts formula, the proof of the logarithmic Sobolev inequality for the diusion measure  on a manifold with torsion  is entirely similar to the proof in Section 2. As was indicated there, one deduces from the proposition with standard arguments (following e.g. [H4]) a representation formula for a smooth cylindrical functional F which takes the form (with IE for integration with respect to ) Z 1 Z 1D   E F ? IE(F ) = E (DF ):t ? 21 At (As )?1 Ms (DF ):s ds Bt ; dbt t 0 where the ow (At )0t1 now satis es dAt ? M A = 0; A = I; t t 0 dt and ?
Mt = 12 b Ut + 21 RicUt ? Ut V (Ut );
? rV (Ut ): Therefore, following the arguments of Section 2, if


 K = sup b u + Ricu ? 2u(V (u);
? 2rV (u) ; u 2 O(M ) < 1; then, for any F in the domain of D, we have ?
IE(F 2 log F 2) ? IE(F 2) logIE(F 2)  2 eK IE jDF j2IH : We summarize our conclusions in the following theorem, which covers, with its simple proof, both Wang's result on the case  = 0, V 6= 0 [W] and Fang's result on the case  6= 0, V = 0 [F2]. Theorem 1 Consider a diusion process with generator L = 21
+ V on a complete Riemannian manifold with a connection compatible with the Riemannian metric whose torsion form

 satis es Driver's total antisymmetry condition. Then, if


K = sup b u + Ricu ? 2u(V (u);
? 2rV (u) ; u 2 O(M ) < 1; for any functional F in the domain of D, ?
IE(F 2 log F 2) ? IE(F 2) logIE(F 2)  2 eK IE jDF j2IH :

Logarithmic Sobolev Inequalities on Path Spaces

79

4 Isoperimetric Inequalities on Path Spaces In this last section, we show how the preceding stochastic calculus approach may be used similarly to recover and extend the isoperimetric inequality on path spaces proved in [B-L]. For the sake of comparison, let us rst recall a functional form of the isoperimetric inequality proved in [B-L]: on W0 (IRn ), for any smooth functional F with values in [0; 1], ?




U IE(F )  IE

(9)

q



U 2 (F ) + jDF j2IH ; R

x '(t)dt, x 2 IR. We refer to [B-L] where U = '  ?1 , '(x) = (2)?1=2e?x2 =2 , (x) = ?1 for further details and comments on this inequality and on its isoperimetric content. Note in particular that it always implies the corresponding logarithmic Sobolev inequalities (cf. Proposition 3.2 in [B-L]). The isoperimetric inequality for the law of Brownian motion on a manifold with bounded Ricci curvature was established in [B-L] using a Markovian tensorization property together with the same argument as the one developed in [H3] for the logarithmic Sobolev inequality. We apply here the simple martingale representation techniques of the preceding sections to extend this result to the generality of Theorem 1. That is, we have Theorem 2 Under the notation and hypotheses of Theorem 1, for any F in the domain of D and with values in [0; 1],

?




U IE(F )  IE

q



U 2 (F ) + eK jDF j2IH :

For simplicity we only give the proof in the at case (9). The arguments presented in the preceding sections for logarithmic Sobolev inequality show clearly that the general result will follow almost identically. Fix F , say smooth and cylindrical, with values in [0; 1]. Consider as before Mt = IE(F jBt ), 0  t  1, and besides the IH-valued martingale Nt = IE(DF jBt ), 0  t  1. For an element h in IH, and 0  t  1, de ne ht 2 IH by hts = ht^s . We will apply It^o's formula to the (Hilbert space valued) semimartingale (Mt ; Nt ; t) between t = 0 and t = 1 with q

(x; y; t) = U 2 (x) + jyt j2IH ; x 2 [0; 1]; y 2 IH; t 2 [0; 1]:: Note that, using the basic relation UU 00 = ?1, we have @x2 = 13 U 0 2 jyt j2IH ? 1 ; h? i
@y2 = 13 U 2 + jyt j2 h(
)t; (
)tiIH ? yt yt (
;
) (2 IH IH ); 2 @xy = ? 13 UU 0yt (2 IH ) 1 y_ (t) 2 : @t = 2

In the notation of the present proof, the Clark-Ocone-Haussman formula can be written as dhM it = jN_ t(t)j2 dt. Applying It^o's formula we obtain (with the obvious notational simpli cations), q

IE

U 2 (F ) + jDF j2IH



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Electronic Communications in Probability 

Z

n   1 1 X (Nt )t 2 (S i )t 2 ? h(Nt )t ; S i i2 dt = U IE(F ) + 12 IE t t IH IH IH 3 0 i=1 Z 1 n    1 X (Nt )t 2 (Ri )2 U 0 2 ? 2Ri h(Nt )t ; S i i UU 0 + (S i )t 2 U 2 dt ; + 21 IE t t t IH t IH 3 IH 0 i=1

?




where we denote for simplicity

dhM it = and

n X i=1

n

X (Rit)2 dt (= jN_ t (t)j2dt); dhN it = Sti Sti dt i=1

dhM; N it =





n X i=1

RitSti dt:

Since (Nt )t 2IH (Sti )t 2IH ? h(Nt )t ; Sti i2IH  0 and ( t )t 2 ( i )2 t IH









N R U 0 2 ? 2Rith(Nt )t ; Sti iIH UU 0 + (Sti )t 2IH U 2 = (Nt )tRit U 0 ? (Sti )t U 2IH  0; the conclusion immediately follows. Acknowlegdement. We are grateful to B. Driver for his help at the early stage of this work and for his suggestion of further developing this approach in the case of a connection with torsion.

References [A-E] S. Aida, K. D. Elworthy. Dierential calculus on path and loop spaces I. Logarithmic Sobolev inequalities on path spaces. C. R. Acad. Sci. Paris 321, (1995) 97{102. [Ba] D. Bakry. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New Trends in Stochastic Analysis (edited by K. D. Elworthy, Shigeo Kusuoka, and Ichiro Shigekawa), World Scienti c, 43{75 (1997).| [B-L] D. Bakry, M. Ledoux. Levy-Gromov's isoperimetric inequality for an in nite dimensional diusion generator. Invent. Math. 123, (1996) 259{281. [Bi] M. Bismut. Large deviations and the Malliavin calculus. Birkhauser 1984| [D] B. Driver. A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold. J. Funct. Anal. 110, (1992) 272{376. [F1] S. Fang. Inegalite du type de Poincare sur l'espace des chemins riemanniens. C. R. Acad. Sci. Paris, 318, (1994) 257{260. [F2] S. Fang. Weitzenbock formula relative to a connection with torsion. Preprint (1996) [G] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97, (1975) 1061{1083. [H1] E. P. Hsu. Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold. J. Funct. Anal. 134, (1995) 417{450.

Logarithmic Sobolev Inequalities on Path Spaces [H2] E. P. Hsu. Inegalites de Sobolev logarithmiques sur un espace de chemins. C. R. Acad. Sci. Paris, 320, (1995) 1009{1012. [H3] E. P. Hsu. Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. Commun. in Math. Phys. (to appear) [H4] E. P. Hsu. Analysis on Path and Loop Spaces (1996). To appear in IAS/Park City Mathematics Series, Vol. 5, edited by E. P. Hsu and S. R. S. Varadhan, American Mathematical Society and Institute for Advanced Study (1997) [N] D. Nualart. The Malliavin calculus and related topics. Springer Verlag (1995). [W] F.-Y. Wang. Logarithmic Sobolev inequalities for diusion processes with application to path space. Chinese J. Appl. Probab. Stat.12:3, (1996) 255{264.

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