Potential Analysis (2006) 25: 77–102 DOI: 10.1007/s11118-006-9009-1

#

Springer 2006

Logarithmic Sobolev Inequalities of Diffusions for the L2 Metric MATHIEU GOURCY1 and LIMING WU1,2 1 Laboratoire de Mathe´matiques Applique´es, CNRS-UMR 6620, Universite´ Blaise Pascal, 63177 Aubie´re, France (e-mail: [email protected]) 2 Department of Mathematics, Wuhan University, 430072 Wuhan, China (e-mail: [email protected]

(Received: 18 February 2005; accepted: 3 February 2006) Abstract. Under the BakryYEmery’s
2 -minoration condition, we establish the logarithmic Sobolev inequality for the Brownian motion with drift in the L2 metric instead of the usual CameronYMartin metric. The involved constant is sharp and does not explode for large time. This inequality with respect to the L2 -metric provides us the gaussian concentration inequalities for the large time behavior of the diffusion. Mathematics Subject Classifications (2000): 60E15, 60H07. Key words: Logarithmic Sobolev inequality (LSI), concentration inequality, path space, Malliavin calculus.

1. Introduction The log-Sobolev inequality (LSI in short), discovered by Gross ([16], 1975) for Gaussian measures and the Wiener measure, plays a prominent role in the infinite dimensional analysis as the Sobolev inequalities do in the finite dimensional analysis. Herbst first, Aida et al. [2] and Aida and Stroock [3] found that LSI implies the Gaussian concentration inequality for Lipschitzian functions, Bobkov and Go¨tze [6] characterized the Gaussian concentration inequality for Lipschitzian functions by means of the so-called T1 -transportation inequality, which is equivalent to the Gaussian integrability of the distance function by the recent work of Djellout et al. [10]. Otto and Villani [20] and Bobkov et al. [7] find that LSI implies the Talagrand T2 -transportation inequality. See Bakry [4], Ledoux [17] and Villani [21] for systematic treatment and further study. For introducing our question for diffusions, let us first recall the LSI of Gross on the Wiener space.
 Let
be the Wiener measure on W d ¼ C ½0; T
; Rd and h 2 H where H is the Cameron Martin space, i.e., the Hilbert space H of all absolutely continuous functions h : ½0; T
! Rd such that hð0Þ ¼ 0 and Z T 2 jh_ ðtÞj2 dt < 1: khkH :¼ 0

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MATHIEU GOURCY AND LIMING WU

The directional derivative along h of a smooth function f at  2 W d is defined on W d by: Dh f ðÞ ¼ lim

"!0

f ð þ "hÞ  f ðÞ : "

The linearity and continuity in h of the above expression gives the existence of an element Df ðÞ ¼ ðDf ð; tÞÞt2½0;T
in H (the Malliavin gradient) such that: 8h 2 H; Dh f ðÞ ¼ hDf ðÞ; hiH ¼

Z 0

T

_ _ Df ð; tÞ  h_ ðtÞdt

ð1:1Þ

_ _ where Df ð; tÞ ¼ dtd Df ð; tÞ. For the Wiener measure
on W d , Gross ([16], 1975) proved the following (see also the PhD of Gentil [13]): THEOREM 1.1. For all bounded and smooth functions F on W d ,   Ent
ðF 2 Þ r 2CE
kDFk2H

ð1:2Þ

where the constant C ¼ 1 is sharp. Here the entropy of 0 r f 2 L1 ð
Þ w.r.t.
is defined as    f
Ent
ðf Þ ¼ E f log : E
f When
satisfies (1.2), we shall say that
satisfies the LSI(C) w.r.t. the gradient in H. This inequality w.r.t. the CameronYMartin metric is extended to the Brownian motion over a Riemannian manifold M by Hsu [15], Aida [1], Capitaine et al. [8] under the boundedness of the Ricci curvature, and to general elliptic diffusions under the boundedness of the BakryYEmery curvature. Particularly Aida [1] proved that the constant in the LSI does not explode for large time T under the uniform positivity of the Ricci curvature. But as explained by the second author in [23], the LSI w.r.t. the CameronYMartin metric for diffusion ðXt Þ even with a constant independent of T (as in [1]) does not produce the concentration inequality of correct order in large time T for the functionals Z T  Z T 1 FðXÞ :¼ pffiffiffiffi gðXs Þds  E gðXs Þds ; g : M ! R smooth: T 0 0 The reason is that the Lipschitzian pffiffiffiffi coefficient of F above w.r.t. the CameronYMartin metric is of order T in general, which explodes. To get

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

79

useful informations for the ergodic behavior (i.e., the large time behavior) of ðXt Þ or more precisely for producing the Hoeffding’s type Gaussian concentration inequality for functionals like FðXÞ above for large time, as pointed out by Djellout, Guillin and the second author in [10], we should establish the LSI w.r.t. the L2 -metric instead of the CameronYMartin metric, with a constant which does not explode for large time. The famous
2 -minoration criterion of Bakry and Emery [5] gives the LSI for the single time law PT ðx; Þ ¼ Px ðXT 2 Þ and for the unique invariant measure m of ðXt Þ. The main purpose of this Note is to extend their LSI to the processlevel X½0;T
w.r.t. the L2 -metric. This work is also a continuation of [10] where the Talagrand T2 -transportation inequality for diffusions w.r.t. the L2 -metric is established via the Girsanov transformation, under a weaker condition than the BakryYEmery condition (see also [24] for T2 -inequality w.r.t. an uniform norm). This paper is organized as follows. In the next section we first study the Brownian motion with drift, i.e., solution of the SDE dXt ¼ dWt þ bðXt Þdt; X0 ¼ x

ð1:3Þ

in Rd , where the drift b satisfies the BakryYEmery condition: ðrbÞ r  KId; K 2 R where ðrbÞ ¼ ðð@j bi þ @i bj Þ=2Þ is the symmetrized gradient of the vector field b. Our method is very simple: To prove that the mapping  from the path w of the Brownian motion equipped with the CameronYMartin metric to the path X of the diffusion equipped of L2 ð½0; T
; Rd Þ-norm is Lipschitzian. The method above does not work if there is a non-constant volatility coefficient ðXt Þ in the SDE (1.3) above. In that case, we consider a general elliptic diffusion with generator 12  þ V on a Riemannian manifold. We shall employ the method of the martingale representation formula, as developed by Fang [12] for Poincare´ inequality and by Capitaine et al. [8] for log-Sobolev inequality. For the Brownian motion over a Riemannian manifold, when the Ricci curvature is bounded, we shall see in Section 3 how to derive the LSI quickly from the Fang’s martingale representation formula. But in the general case where the BakryYEmery’s curvature is not bounded, Fang’s version of martingale representation formula seems to be no longer true and that is the main difficulty. We shall use a localization technique to get a martingale representation formula for a class of test functions, and then the LSI. This is the task of Section 4. Finally in Section 5 we present several useful consequences of the LSI about the concentration of additive functionals of the diffusions for large time. In this note we denote the maximum of u and v by u _ v and their minimum Pd d ; x y ¼ x y and jxj denotes the Euclidian norm of x. by u ^ v. For x; y 2 R  i¼1 i i When there is no possible confusion, we will use L2 for L2 ð½0; T
; Rd Þ.

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2. Brownian Motion with Drift on Rd 2.1. GRADIENT r IN L2 AND THE LSI W.R.T. r We are given a standard Rd -valued Brownian motion Wt on Rd , a fixed initial point x in Rd and a continuously differentiable mapping b : Rd ! Rd . Let us consider the Rd -valued diffusion Xt ðxÞ defined by the stochastic differential equation: dXt ¼ dWt þ bðXt Þdt ; X0 ¼ x 2 Rd :

ð2:1Þ

We assume the following condition on the drift b ðrbÞ r  K Id; K 2 R

ð2:2Þ

where Id is the identity matrix in Rd , rb ¼ ð@j bi Þij is the Jacobian matrix, and A denotes the symmetrization ðA þ At Þ=2 of the matrix A; and for two symmetric matrices A; B, A r B means that B  A is non-negative definite. This condition, when K > 0, is exactly the BakryYEmery’s
2 -minoration condition for the logarithmic Sobolev inequality of the unique invariant measure m on Rd . Details about this condition, and applications to hypercontractive diffusions can be found in [4, 5]. We are interested here in the logarithmic Sobolev inequality on the path space w.r.t. the gradient in L2 ð½0; T
; Rd Þ. More precisely, regarding W d ¼ Cð½0; T
; Rd Þ as a dense subspace of L2 ð½0; T
; Rd Þ, we can introduce: DEFINITION 2.1. For a function f on W d , differentiable with respect to the L2 -norm, let rf ðÞ be the element in L2 ð½0; T
; Rd Þ such that Dg f ðÞ ¼ hrf ðÞ ; giL2 ð½0;T
;Rd Þ ; 8g 2 H: We shall write rf ðÞ ¼ ðrt f ðÞÞt2½0;T
. Its relation with the gradient Df in the CameronYMartin space H (introduced in (1.1)) is given by LEMMA 2.2. For a function f on W d , differentiable with respect to the L2 _ € ðÞ is in L2 ð½0; T
; Rd Þ in the sense of norm, we have for every  2 W d , Df distribution and _ € rt f ðÞ ¼ Df ð; tÞ; dt  a:e: Proof. Such function f is differentiable w.r.t. the norm of H. Let C01 ðð0; TÞ; R Þ be the space of all Rd -valued C1 -functions with compact support in ð0; TÞ, d

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

81

_ € which is dense in L2 ð½0; T
; Rd Þ. For every h 2 C01 ðð0; TÞ; Rd Þ, writing Df ð; tÞ ¼ d2 dt2 Df ð; tÞ in the sense of distribution, we have, Z T rs f ðÞ  hðsÞds ¼ hrf ðÞ; hiL2 ¼ Dh f 0 Z T _ _ Df ð; sÞ  h_ ðsÞds ¼ 0Z T _ € ¼ Df ð; sÞ  hðsÞds; 0

where the desired result follows.

Ì

We denote by Cb1 ðW d =L2 Þ be the space of all bounded functions f on W d , differentiable w.r.t. the L2 ð½0; T
; Rd Þ-norm, such that rf is also continuous and bounded from ðW d ; kkL2 Þ to L2 ð½0; T
; Rd Þ. The aim of this section is to prove THEOREM 2.3. Let Px be the law of the diffusion X:ðxÞ defined by the SDE (2.1). Assume (2.2). Then for all functions f 2 Cb1 ðW d =L2 Þ, Z T 2 Px EntPx ð f Þ r 2CðTÞE jrt f j2 dt; ð2:3Þ 0

where 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  > > 1  2eKT  e2KT ; if 2eKT  e2KT > 0; K 6¼ 0; > 2 > K > < 2 T CðTÞ :¼ ; if K ¼ 0; > 2 > >
 > > : 1 e2KT  2eKT þ 1 ; if 2eKT  e2KT r 0; K 6¼ 0: 2 K ð2:4Þ In particular if K > 0, CðTÞ r 1=K 2 ( for all T > 0) is non-explosive. Note that our estimate of CðTÞ is sharp for T ! þ1 for the OrnsteinY Uhlenbeck process dXt ¼ dWt  ðXt =2Þdt by the discussion in [10, Remarks 5.7].

2.2. ANALYTIC PREPARATION In this paragraph we recall and prove several known facts, for self-containedness. Let Md ðRÞ be the space of all real d  d-matrices. For A 2 Md ðRÞ, it is clear that 8x 2 Rd , hAx ; xi ¼ hA x ; xi: We first give a Gronwall Lemma of matrix type. LEMMA 2.4. If JðtÞ 2 Md ðRÞ satisfies the equation J_ ðtÞ ¼ AðtÞJðtÞ with Jð0Þ ¼ Id;

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MATHIEU GOURCY AND LIMING WU

and if A r BId for a real number B in the order of the nonnegative definiteness, then we have jJðtÞyj r eBt jyj; 8y 2 Rd ; 8t U 0: Proof. Indeed we have:  
d
2Bt jJðtÞxj2 ¼ e2Bt 2BjJðtÞxj2 þ 2hJ_ ðtÞx; JðtÞxi e dt 
¼ e2Bt 2BjJðtÞxj2 þ 2hAðtÞJðtÞx; JðtÞxi 
¼ e2Bt 2BjJðtÞxj2 þ 2hA ðtÞJðtÞx; JðtÞxi ¼ 2e2Bt hðBId  A ðtÞÞJðtÞx; JðtÞxi r0 so jJðtÞxj r eBt jxj:

Ì

We now give a result to control the norm of a bounded self-adjoint operator from L2 to L2 . LEMMA 2.5. Let
be a bounded self-adjoint operator from L2 ðE;
Þ to L2 ðE;
Þ where
is -finite. Assume that k
k1;1 :¼

supT 1

06¼f 2L

k
f kL1 ð
Þ L2

k f kL1 ð
Þ

< þ1:

2 Then k
k2;2 ðthe norm in LT ÞÞ r k
k1;1 . 1 L1  L2 , we have by the symmetry of
, Proof. For any f 2 L

k
f k1 ¼

T sup 1

h
f ; gi

g2L1

T sup 1

h f ;
gi

g2L1

¼

L ;kgk1 r 1

L ;kgk1 r 1

r k
k1;1 k f k1 ; i.e., k
k1;1 r k
k1;1 . Hence by the RieszYThorin’s interpolation theorem, Ì k
k2;2 r k
k1;1 .

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LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

2.3. PROOF OF THEOREM 2.3 2.3.1. Outline of the Proof The SDE (2.1) has a unique strong solution (the reader is referred to [18], p 194) X ¼ ðWÞ; where  : W d ! W d . Indeed  is continuous on W d and for each  2 W d ,  ¼ ðÞ is the unique solution of the ordinary differential equation t ¼ x þ t þ

Z

t

bðs Þds: 0

We start with the LSI (1.2) of Gross. If f   2 D2 ðDÞ (the domain of D in L2 ðW d ;
Þ as defined in the Malliavin calculus), we have EntPx f 2 ¼ Ent
½ð f  Þ2
r 2E
kDð f  Þk2H :

ð2:5Þ

Now for every h 2 H, we shall prove that ð þ "hÞ  ðÞ "!0 "

AðÞh :¼ lim

ð2:6Þ

holds in the sup-norm of W d (then in that of L2 ). Thus ð þ "hÞ ¼ ðÞ þ "

dð þ "hÞ j"¼0 þ ð"Þ ¼ ðÞ þ "AðÞh þ ð"Þ d"

and consequently

 f ð þ "hÞ  f ðÞ Dh ð f  ÞðÞ ¼ lim "!0 "

 f ðÞ þ "AðÞh þ ð"Þ  f ðÞ ¼ lim "!0 "
 ¼ ðDAðÞh f Þ ðÞ : If we can prove that kAðÞhkL2 r

pffiffiffiffiffiffiffiffiffiffiffi CðTÞkhkH

ð2:7Þ

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MATHIEU GOURCY AND LIMING WU

where CðTÞ is given in the theorem, we shall get kDð f  ÞðÞkH ¼

sup jDh ð f  ÞðÞj khkH r 1

¼


 jðDAðÞh f Þ ðÞ j

sup h2H ; khkH r 1

¼

sup


 hðrf Þ ðÞ ; AðÞhiL2

h2H ; khkH r 1

sup

r

h2H ; khkH r 1

r


 kðrf Þ ðÞ kL2 kAðÞhkL2

pffiffiffiffiffiffiffiffiffiffiffi
 CðTÞkðrf Þ ðÞ kL2 :

Thus by (2.5), we obtain EntPx f 2 r 2E
kDð f  Þk2H

 r 2CðTÞE
kðrf Þ   k2L2 Z T ¼ 2CðTÞEPx jrt f j2 dt 0

the desired inequality. 2.3.2. Proof of Theorem 2.3 To make the above proof rigorous, it remains to verify three points: (2.6), (2.7) and f   2 D2 ðDÞ. The last point is a consequence of (2.6), (2.7), because (2.6), (2.7), imply jDh ð f  ÞðÞj r

pffiffiffiffiffiffiffiffiffiffiffi CðTÞkhkH kðrf ÞððÞÞkL2

for all h 2 H (this is a well known fact in the Malliavin calculus, cf. Nualart [19]). Equation (2.6) is an elementary fact in the ODE theory, so omitted. We now prove (2.7). Since  ¼ ð þ "hÞ verifies Z t
  t ¼ x þ t þ "hðtÞ þ b s" ds; 0

then, g :¼ AðÞh ¼ gðtÞ ¼ hðtÞ þ

d d"

Z

t 0

" satisfies the linear differential equation:
 rb s gðsÞds

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

85

where  ¼ ðÞ. This equation has a unique solution and it can be given explicitly by Z t At As 1 h_ ðsÞds gðtÞ ¼ 0

where At satisfies: A0 ¼ Id ;

d At ¼ rbðt ÞAt : dt

Under the condition (2.2), we obtain using Lemma 2.4: jAt As1 xj r eKðtsÞ jxj; 80 r s r t: We state now the desired inequality (2.7) as LEMMA 2.6. We have kAðÞhk2L2 ¼ kgk2L2 r CðTÞkhk2H where CðTÞ is given in Theorem 2.3 (2.4). Rt _ Proof. From the following expression gðtÞ ¼ 0 At A1 s hðsÞds, we can compute: 2 Z T Z t   2 1  At A h_ ðsÞds dt kgkL2 ¼ s   0

0

Z T Z

t

e

r 0

KðtsÞ

jh_ ðsÞjds

2 dt

0

Now, we have to control the last term. A CauchyYSchwarz control is easy, but the constant so obtained explodes. In order to avoid this explosion, let us write Z T Z

t

e 0

KðtsÞ

jh_ ðsÞjds

2 dt

0

¼

Z T Z

jh_ ðuÞjdu

0

KðtuÞ

e ZZ

Z

KðtvÞ

e

 _ jhðvÞjdv dt

 KðtvÞ _ _ jhðuÞje jhðvÞjdu dv dt

0 r u;v r t

Z

T

e 0 r u;v r T

t 0

Z T ZZ 0

¼

KðtuÞ

e 0

¼

t

u_v

2KtþKðuþvÞ



dt jh_ ðuÞjjh_ ðvÞjdu dv:

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MATHIEU GOURCY AND LIMING WU

Since 2ðu _ vÞ  ðu þ vÞ ¼ ju  vj, we have if K 6¼ 0, Z T
ðu; vÞ :¼ e2KtþKðuþvÞ dt u_v KðuþvÞ

¼

e


2Kðu_vÞ  e2KT e

2K eKjuvj  eKð2TuvÞ ¼ 2K and if K ¼ 0, Z T
ðu; vÞ :¼ e2KtþKðuþvÞ dt ¼ ðT  u _ vÞ: u_v

Then we obtain Z 2 kgkL2 r

T

Z

0

T


ðu; vÞjh_ ðuÞjjh_ ðvÞjdudv:

ð2:8Þ

0

Define an operator
: L2 ð0; TÞ ! L2 ð0; TÞ by Z T
f ðvÞ ¼
ðu; vÞf ðuÞdu: 0

It is a self-adjoint operator on L2 ð0; TÞ. Then the term to control become h
jh_ j ; jh_ ji 2 : L

But it is clear that (since
ðu; vÞ U 0):  Z T Z T    dv  k
f kL1 ¼
ðu; vÞf ðuÞdu   0

0

Z r

T

sup v2½0;T


!Z

T

j f ðvÞjdv:


ðu; vÞdu 0

0

Thus Lemma 2.5 yields k
kL2 r AðTÞ where Z T AðTÞ ¼ sup
ðu; vÞdu: v2½0;T


0

Finally by an elementary calculus, we get AðTÞ ¼ CðTÞ given in Theorem 2.3.

Ì REMARK 2.7. In the proof above, we see that ð; hÞ ! AðÞh ¼ g is continuous from W d  H to L2 ð½0; T
; Rd Þ. Thus for all f 2 Cb1 ðW d =L2 Þ, f   is continuously differentiable on W d w.r.t. the norm of H, and we have shown pffiffiffiffiffiffiffiffiffiffiffi kDð f  ÞðÞkH r CðTÞkrf ÞððÞÞkH :

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

87

Now let Q 2 Md ðRÞ and F a continuously differentiable function on W d w.r.t. the norm of H, applying the Gross theorem to FðQÞ, we get Z T _ _ 2 t EntðF ðQW ÞÞ r 2max ðQ QÞE j DFj2 ðQW ; tÞdt 0

where max ðQt QÞ is the maximal eigenvalue of Qt Q. Thus by the estimate above, we see that dXt ¼ bðXt Þdt þ QdWt ; X0 ¼ x satisfies : 8f 2 Cb1 ðW d =L2 Þ, 2

t

Entð f ðXÞÞ r 2CðTÞmax ðQ QÞE

Z

T

jrt f j2 ðXÞdt:

0

In other words our approach here is also well adapted for eventually degenerate noise QdWt . 3. Brownian Motion on Manifold When one studies the diffusion dXt ¼ ðXt ÞdWt þ bðXt Þdt with a non-constant volatility coefficient , the application  mapping the path of the BM to the path pffiffiffiffiffiffiffiffiffiffi ffi of X does not satisfy: krh kL2 r CðTÞkhkH . So the approach in the previous section losses its pertinence. Instead, we shall explore another method, based on the martingale representation formula, as developed in [12] and [8]. We shall study only the elliptic diffusion on a Riemannian manifold generated by 12  þ V. To illustrate clearly the idea, we begin with the BM over a Riemannian manifold with bounded Ricci curvature. In that case, using the martingale representation formula, Fang [12] obtained the Poincare´ inequality, Hsu [14, 15] and Capitaine et al. [8] obtained the logarithmic Sobolev inequality on the path space, both w.r.t the CameronYMartin metric. The constant of the LSI in those works explodes for large time t. Aida [1] obtained the LSI w.r.t. the CameronYMartin metric with a bounded constant for large time T once the Ricci curvature is bounded from below by a positive constant. But as explained in the Introduction the LSI w.r.t. the CameronYMartin metric even with a nonexplosive constant does not produce concentration R T inequality of correct order (in large time T) for functionals such as FðXÞ ¼ 0 gðXs Þds. 3.1. MARTINGALE REPRESENTATION AND GRADIENT W.R.T. L2 -NORM We follow the exposition of [8]. Let M be a complete and connected manifold of dimension d equipped with the Levi-Civita connection r. Denote by  the LaplaceYBeltrami operator on M, OðMÞ the bundle of orthonormal frames, and  : OðMÞ ! M the canonical projection. Fix a point x0 2 M. Each frame u 2

88

MATHIEU GOURCY AND LIMING WU

OðMÞ is a linear isometry u : Rd ! TðuÞ ðMÞ. We assume throughout this section the boundedness of the Ricci curvature sup kRicu k < þ1;

ð3:1Þ

u2OðMÞ

and denote also Ricu its scalarization. Let Wx0 ðMÞ be the space of continues paths from ½0; T
to M starting at x0 . Let U be the horizontal Brownian motion. We recall that U is solution of the Stratonovich SDE: dUt ¼

d X

Hi ðUt Þ  dWti ; U0 ¼ u0 2 OðMÞ

ð3:2Þ

i¼1

where Wt is a standard Brownian motion on Rd defined on some probability space ð; F ; PÞ, and Hi ; 1 r i r n are the canonical horizontal vector fields on OðMÞ. Details about this construction can be found in [14] and [15]. Then X ¼ ðUÞ is the Brownian motion valued in M starting from x0 , whose law
is the Wiener measure on Wx0 ðMÞ. Hence, for a Brownian path X, and h in the Cameron Martin space, Ut ht is a tangent vector at Xt . It determines a vector field Dh by Dh Xt ¼ Ut ht . For smooth cylindrical functions f on Wx0 ðMÞ, that is f ðÞ ¼ f ðt1 ; . . . ; tn Þ with  2 Wx0 ðMÞ, the Malliavin derivative operator is defined by: Df ð; tÞ ¼

n X i¼1

Ut1 i

df ðt ; . . . ; tn Þti ^ t: dxi 1

Notice that from the path of the BM X over the Riemannian manifold and U0 ¼ u0 , one can reconstruct ðUt Þ and ðWt Þ a.s. (w.r.t. the law of X). Thus the above definition is intrinsic (cf. [14]). Let ðD; D2 ðDÞÞ be its closure in L2 ð
Þ. For h 2 H, Dh f ðÞ and Df ðÞ 2 H are such that: Z T _ _ Df ð; sÞ  h_ ðsÞds: 8h 2 H; Dh f ðÞ ¼ hDf ðÞ ; hiH ¼ 0

Let F : Wx0 ðMÞ ! R be a real function in the domain D2 ðDÞ of D. Under the boundedness condition (3.1), the Fang’s version of ClarkYOconeYHaussmann formula for the Brownian motion on a manifold is read as Z T
 hHt ; dWt i ð3:3Þ FðXÞ ¼ E FðXÞ þ 0

where   Z T _ _ _ _ 1 1 ðA*s Þ RicUs DFðsÞdsjBt : Ht ¼ E DFðtÞ  A*t 2 t

ð3:4Þ

89

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

In this expression, ðBt Þ is the filtration generated by ðWt Þ, ðAt Þ0 r t r T is a matrix valued process defined with: dAt 1  At RicUt ¼ 0; A0 ¼ Id: 2 dt

ð3:5Þ

The above formula relies on the BismutYDriver integration by parts formula: For any h 2 H, the adjoint Dh* of Dh in L2 ð
Þ is given by D*h ¼ Dh þ

Z 0

T

h

i

1 h_ t þ RicUt ht ; dWt : 2

ð3:6Þ

According to Lemma 2.2, we introduce our gradient w.r.t. the L2 -metric. _ € DEFINITION 3.1. For F 2 D2 ðDÞ such that DF 2 L2 ð½0; T
; Rd Þ we define the gradient of F w.r.t. L2 -norm by _ € rt FðÞ ¼ DFð; tÞ dt  a:e: In other words, rFðÞ 2 L2 ð½0; T
; Rd Þ is determined by hrFðÞ ; hiL2 ¼ Dh FðÞ ; 8h 2 C01 ð
0; T½; Rd Þ: We assume now 8u 2 OðMÞ ;

1 Ricu U KId; K 2 R: 2

ð3:7Þ

The constant of our LSI will depend only on this lower bound of the Ricci curvature. LEMMA 3.2. Let A solution of (3.5). We have
  8s U t ; kA*t A1 s k r exp Kðs  tÞ : Proof. With a derivation of At A1 t ¼ Id, we have d 1 1 ðAs At Þ ¼  RicUs A1 s At : ds 2 By Lemma 2.4 and (3.7), we have the desired result.

Ì

90

MATHIEU GOURCY AND LIMING WU

The key observation is: _ € LEMMA 3.3. For F 2 D2 ðDÞ such that
 a:s, DFðÞ 2 L2 ð½0; T
; Rd Þ (then _ _ _ _ ð; TÞ ¼ 0, DFðÞ can and will be chosen to be absolutely continuous) and DF we have in the representation formula (3.3), Z T  1 Ht ¼ E A*t A*s rs FdsjBt : t

Proof. Taking the adjoint in the proof of lemma 3.2 gives  d  1 1 1 A*t A*s ¼  A*t A*s RicUs : ds 2 By integration by parts in the expression (3.4) of Ht , we have
 almost surely, 1  A*t 2

Z

T

_ _ ðA*s Þ RicUs DFðsÞds ¼

Z

T

_ d 1 _ ðA*t A*s Þ DFðsÞds t ds

s¼T Z T _ € 1 _ 1 _  A*t A*s DFðsÞds ¼ A*t A*s DFðsÞ

1

t

s¼t

_ _ ¼ v  DFðtÞ þ

Z

T

t

1 A*t A*s rs Fds

t

Ì

Then the desired result follows by (3.4). 3.2. POINCARE´ INEQUALITY FOR THE L2 NORM

Now, we follow Fang [12]. For each F 2 D2 ðDÞ such that
 almost surely. _ € _ _ DFðÞ 2 L2 ð½0; T
; Rd Þ and DFð; TÞ ¼ 0, we have by Lemma 3.3, Z T
 2 Var
F ¼ EðFðXÞ  EFðXÞÞ ¼ E jHt j2 dt Z

¼E

0

rE

 T  Z E

Z T Z

rE

2  1 * * At As rs FðXÞdsjBt  dt

T t

2 kA*t A*s k jðrs FÞðXÞjds dt 1

t

0


T

0

Z T Z

T KðstÞ

e 0

t

2 jrs Fjds dt

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

91

where the last inequality is given by lemma 3.2. We write the last quantity as (by Fubini) Z T Z TZ T
KðutÞ e jru Fj eKðvtÞ jrv Fjdudvdt E t

0

¼E




t

Z

TZ

0

Let
ðu; vÞ :¼ in Lemma 2.6

T

dudvjru Fj  jrv Fj

0

R u^v 0

Z

u^v

eKðutÞKðvtÞ dt

0

eKðutÞKðvtÞ dt and
f ðvÞ :¼ Z

k
k2;2 r k
k1;1 ¼ sup

RT 0


ðu; vÞf ðuÞdu. We have as

T


ðu; vÞdv r CðTÞ

u2½0;T


0

where CðTÞ is given in Theorem 2.3. Thus we get, VarðFÞ r k
k2;2 E




Z

T

2

jrt Fj dt r CðTÞE

0




Z

T

jrt Fj2 dt;

0

i.e., we have shown THEOREM 3.4. Assume (3.1) and (3.7). For each F 2 D2 ðDÞ such that _ _ _ €
 almost surely, DFðÞ 2 L2 ð½0; T
; Rd Þ and DFð; TÞ ¼ 0, Z T
 Var
F r CðTÞE
jrt Fj2 dt 0

where CðTÞ is given in Theorem 2.3. 3.3. THE LOGARITHMIC SOBOLEV INEQUALITY FOR L2 -NORM THEOREM 3.5. Assume (3.1) and (3.7). For each F 2 D2 ðDÞ such that _ _ € _
 almost surely, DFðÞ TÞ ¼ 0, 2 L2 ð½0; T
; Rd Þ and DFð; Z T
2
Ent
F r 2CðTÞE jrt Fj2 dt 0

where CðTÞ is given in Theorem 2.3. Proof. We follow the ingenious method of Capitaine et al. [8]. Assume at first 1 UF U">0 "

ð3:8Þ

92

MATHIEU GOURCY AND LIMING WU

for some " 2 ð0; 1Þ. Consider the continuous martingale Ms ¼ E½F 2 ðXÞ j Bs
, such that M0 ¼ EF 2 and MT ¼ F 2 ðXÞ. Let Ht ¼ E

Z

T

 1 2 * * At As rs ðF ÞdsjBt :

t

Using the martingale representation in Lemma 3.3 and applying the Ito formula, we get after taking expectation EðMT log MT Þ  EðM0 log M0 Þ ¼ E

1 2

Z

T

0

dhMis 1 ¼E 2 Ms

Z

T 0

jHs j2 ds: Ms

Since rs ðF 2 Þ ¼ 2Frs F, ds 
 a:e:, by the CauchyYSchwarz inequality, we have

1 EntðF Þ ¼ E 2 2

Z

T

Z

T

r 2E 0

r 2E

T

t

0




 Z  E  "Z  E 

Z T Z 0

2  A*t A*s rs ðF ÞðXÞds j Bt  1

2

E½F 2 ðXÞ j Bt


# 2  A*t A*s rs FðXÞds j Bt dt

T

1

t

T

dt

2 kA*t A*s k  jrs Fjds dt: 1

t

We can conclude by the proof of the Poincare´ inequality. We now remove the restriction (3.8). For general F, let pffiffiffiffiffiffiffiffiffiffiffiffiffi " þ x2 : :¼ L arctan L

F";L :¼ f";L ðFÞ; where f";L

It is easy to see that F";L 2 D2 ðDÞ and _ _ _ _ 0 ðFÞ DFðtÞ; DF ";L ðtÞ ¼ f";L

0 rt F";L ¼ f";L ðFÞrt F:

Then F";L satisfies again the assumption of the theorem and especially (3.8). Then F";L satisfies the LSI. Moreover by the expression above we also have E




Z 0

T

2

jrt F";L j dt r E




Z

T

jrt Fj2 dt:

0

2 By letting L ! þ1 first F";L " " þ F 2 , and " ! 0þ next, we have EntP 2 ðF";L ðXÞÞ ! EntP ðF 2 ðXÞÞ, so the desired LSI for F follows. Ì

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

93

Notice that the smooth cylindrical functions f ðt1 ;    ; tn Þ does not satisfy the assumptions of Theorems 3.4 and 3.5R(this is quite natural as for t fixed, t has no T meaning in L2 ). However FðÞ :¼ 0 f ðt; t Þdt with bounded and smooth f : ½0; T
 M ! R satisfies the condition in Theorem 3.4 and Theorem 3.5, because it is easy to prove that F 2 D2 ðDÞ and _ _ DFðt; Þ ¼

Z t

T

Us1 rf ðs; s Þds; rt FðÞ ¼ Ut1 rf ðt; t Þ

where rx f ðt; xÞ is the gradient on the space variable x 2 M. Before moving to our true object, let us make some comments on the condition (3.1) of the boundedness of the Ricci curvature, whereas the final results in this section seem do not depend on it Fin appearance._ Remarks 3.6. If the Ricci curvature is only lower bounded but not bounded, we have several technical questions related with the stochastic calculus used in this section: (a) The BismutYDriver integration by parts formula (3.6) has a question with the unbounded term RicUt ht . (b) Without the integration by parts formula (3.6), we have neither the Fang’s martingale representation formula (which also contains an unbounded term), nor the closability of D in L2 ð
Þ (more exactly we do not know). The last point is particularly annoying, since even D2 ðDÞ is no longer well defined. Now let us see how to bypass those delicate difficulties for general diffusions generated by 12  þ V, whose BakryYEmery curvature is only assumed to be lower bounded. 4. Brownian Motion with Drift Over a Riemannian Manifold In this section we extend the preceding logarithmic Sobolev inequality to the law of a diffusion process ðXt Þ with generator L ¼ 12  þ V over a complete connected Riemannian manifold M with lower bounded BakryYEmery curvature, i.e., 1 Ricu  ðrVÞ U KId; K 2 R: 2

ð4:1Þ

Here V is a smooth vector field and ðrVÞ is the symmetrized gradient. Recall that Wang [22] has obtained the Talagrand FT2 _-transportation inequality. We assume that the BM is non-explosive. In that situation, the diffusion with generator L ¼ 12  þ V is non-explosive (under (4.1)) and the law
V of the diffusion is absolutely continuous w.r.t. the law of the BM
.

94

MATHIEU GOURCY AND LIMING WU

4.1. A SPACE OF TEST-FUNCTIONS D We introduce now a space D of test-functions which is rich enough for applications in the concentration phenomena of the diffusion for large time. It allow us to avoid the question of D2 ðDÞ mentioned in Remark 3.6. DEFINITION 4.1. Let  D¼

FðÞ ¼ 

Z

T

f1 ðs; s Þds ; . . . ;

Z

0



T

fm ðs; s Þds

0

1 m where m 2 N *,  2 C1 b ðR ; RÞ and fi 2 C 0 ð½0; T
 MÞ. We define the gradient r w.r.t. L2 -norm by

rt

Z

T 0

f ðs; s Þds ¼ Ut1 rx f ðt; t Þ

and the fact that it obeys the rule of composition: rt 

Z

T

f1 ðs; s Þds ; . . . ;

0

Z



T

fm ðs; s Þds

0

:¼

X

@i ð  ÞUt1 rx fi ðt; t Þ:

i

Here rx f ðt; xÞ is the gradient w.r.t. the space variable x 2 M. Notice that the stochastic parallel transport Ut ¼ Ut ðÞ is
 almost surely well defined, then
V  a:s: well defined. 4.2. MARTINGALE REPRESENTATION FOR TEST-FUNCTIONS IN D For u 2 OðMÞ, the bundle of orthonormal frames, let VðuÞ ¼ u1 V ððuÞÞ 2 R , and let rVðuÞ ¼ u1 rV ððuÞÞ 2 Rd  Rd . They are the scalarization of the tensors V and rV, respectively. Recall that our diffusion ðXt Þ with X0 ¼ x0 can be constructed as follows: Xt ¼ ðUt Þ where ðUt Þ is the diffusion valued in OðMÞ, solution of d

dUt ¼

n X

~ ðUt Þdt Hi ðUt Þ  dWti þ V

ð4:2Þ

i¼1

~ is the horizontal lift of V, defined on ð; ðBt Þ; PÞ where ðBt Þ is the where V filtration of the Rd -valued Brownian motion ðWt Þ. Notice that solution Ut of (4.2) coincides in law with Ut ðXÞ where this last Ut is used in the definition of rt F.

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

95

LEMMA 4.2 (due to [8]). Assume that Ricu and rV are both bounded. Then for any smooth cylindrical functional F ¼ Fðt1 ;    ; tn Þ on Wx0 ðMÞ, FðXÞ ¼ EFðXÞ þ

Z

T

hHt ; dWt i;

ð4:3Þ

0

where   RT _ _ _ _ 1 Ht ¼ E DFðtÞ  t A*t ðA*s Þ M*s DFðsÞdsjBt ; 1 RicUt  rVðUt Þ; 2 dAt ¼ At Mt : A0 ¼ I; dt Mt ¼

ð4:4Þ

In the actual bounded case, it is easy to see that F 2 D satisfies again (4.3). By integration by parts, we obtain as in Lemma 3.3, LEMMA 4.3. Assume that Ricu and rV are both bounded. Then for any F 2 D, FðXÞ ¼ EFðXÞ þ

Z

T

Z hE

T

 1 * * At ðAs Þ rs FðXÞjBt ; dWt i;

t

0

where ðAt Þ is given in Lemma 4.2. Now we remove the boundedness condition. THEOREM 4.4. Assume (4.1). Then the conclusion of Lemma 4.3 remains true. As corollaries of the above martingale representation, we obtain by repeating the arguments in Section 3: THEOREM 4.5. For any F 2 D, Var
V ðFÞ r CðTÞE

Z

T

jrt Fj2 dt;

ð4:5Þ

0

Ent
V ðF 2 Þ r 2CðTÞE

Z

T

jrt Fj2 dt;

0

where CðTÞ is given in Theorem 2.3.

ð4:6Þ

96

MATHIEU GOURCY AND LIMING WU

4.3. PROOF OF THEOREM 4.4: TECHNIQUE OF LOCALIZATION If M is compact, this result is contained in Lemma 4.3. Assume below that M is non-compact. R  RT T Step 1: Localization. Let F ¼  0 f1 ðs; s Þds ; . . . ; 0 fm ðs; s Þds 2 D and fix some compact K  M such that the supports of fi ; i ¼ 1;    ; m are all contained in ½0; T
 K. Take next a sequence of compact subsets ðKn Þn U 1 of M such that [ o ; Kn ¼ M: K  K1o (interior of K1 Þ; Kn  Knþ1 n

For each n U 1, choose (1) a complete connected Riemannian manifold Mn Kn such that the Riemannian metric of Mn restricted on Kn coincides with the original one of M and the Ricci curvature of Mn is globally bounded; (2) a smooth vector field Vn on Mn such that Vn ðxÞ ¼ VðxÞ for x 2 Kn and Vn ; rVn are globally bounded; (3) ð1=2ÞRic  rVn U ðK  1ÞId over Mn for all n. On the same probability space ð; F ; PÞ equipped with the Rd -valued Brownian motion Wt , let ðUtn Þ be the solution of dUtn ¼

d X

~n ðUtn Þdt Hin ðUtn Þ  dWti þ V

i¼1

and Xtn ¼ ðUtn Þ, where Hin ; i ¼ 1;    ; d are the canonical horizontal vector ~n is the horizontal lift of Vn . fields on OðMn Þ, V Consider the stopping time = Kn g: n :¼ inf ft U 0; Xt 2 Then by the uniqueness of the SDE, we have with probability one, Utn ¼ Ut ; Xtn ¼ Xt ; 80 r t r n : Regarding fi as function on ½0; T
 Mn with the convention that f ðt; xÞ ¼ 0 for x2 = K and FðÞ as a functional on Wx0 ðMn Þ, we can apply Lemma 4.2 to get FðX n Þ  EFðX n Þ  Z T Z T ðnÞ 1 ðnÞ ðnÞ n ¼ hE ðAt Þ*½ðAs Þ*
rs FðX ÞjBt ; dWt i 0

t

ð4:7Þ

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

97

2 where rðnÞ s F is the gradient w.r.t. L -norm on Wx0 ðMn Þ, and

Mtn

RicUtn  rVn ðUtn Þ;

¼

1 2

¼

dAt ðnÞ ¼ At Mtn : I; dt

ðnÞ

ðnÞ A0

Finally we make the last localization (on the functional F): Z Fn ðXÞ :¼ 

T^n

f1 ðs; Xs Þds ; . . . ; 0

Z

T^n

 fm ðs; Xs Þds :

0

We have Fn ðXÞ ¼ Fn ðX n Þ. Warning: We can take the directional derivative Dh Fn on Wx0 ðMn Þ by using the variation of the path as Driver, due to the fact that the supports of fi are contained in ½0; T
 Kno (though n is not differentiable). But we cannot prove that Fn 2 D2 ðDÞ on Wx0 ðMn Þ which is defined as the closure of the cylindrical smooth functions. If this last point is true, we have ðrt Fn ÞðXÞ ¼ 1t and then the martingale representation Fn ðXÞ ¼ Fn ðX Þ ¼ EFn ðXÞ þ n

Z

T^n

Z hE

T^n

1 * * At ðAs Þ rs Fn ðXÞjBt ; dWt i

t

0

where the desired martingale representation follows by letting n ! þ1. We have much tried in this road, but do not succeed to prove the key point that Fn 2 D2 ðDÞ on Wx0 ðMn Þ. Step 2. Let us see how to bypass the difficulty above. At first n

jFn ðXÞ  FðX Þj r

m X

k@i k1 k fi k1 ðT  T ^ n Þ ! 0

i¼1

for Pðn " þ1Þ ¼ 1 by the non-explosion of our diffusion under condition (4.1). As Fn ðXÞ ! FðXÞ; a:s:, then FðX n Þ ! FðXÞ; a:s: consequently in Lp ðPÞ for all p 2 ½1; þ1Þ. Thus the l.h.s. of (4.7) converges to FðXÞ  EFðXÞ as desired. For the r.h.s. of (4.7), denoting EðjBt Þ by EBt , we have: Z

T

hEBt

Z

T

t

0

¼

Z 0



ðnÞ n * 1 ðnÞ ðAt Þ*½ðAðnÞ s Þ
rs FðX Þds

T^n

hEBt

Z

T^n t

; dWt i

 ðAt Þ*½ðAs Þ*
rs FðXÞ ; dWt i 1

98

MATHIEU GOURCY AND LIMING WU

þ

Z

T^n

hEBt

Z

T^n

0

þ

Z



T

T

hEBt

Z

T^n

T

t

ðnÞ n * 1 ðnÞ ðAt Þ*½ðAðnÞ s Þ
rs FðX Þds

ðnÞ n * 1 ðnÞ ðAt Þ*½ðAðnÞ s Þ
rs FðX Þds

; dWt i

 ; dWt i

ð4:8Þ ð4.8Þ

:¼ ðIÞn þ ðIIÞn þ ðIIIÞn : Letting n ! þ1, we have obviously in L2 ðPÞ, Z T  Z T ðIÞn ! hEBt ðAt Þ*½ðAs Þ*
1 rs FðXÞds ; dWt i t

0

the r.h.s. of (4.7). For ðIIÞn and ðIIIÞn , since by our assumption (3) in Step 1 and the proof of Lemma 3.2, ðnÞ

1 ðK1ÞðstÞ ; 8s U t kAt Þ*½ðAðnÞ s Þ*
k r e

and m X

jrðnÞ s FðÞj r

k@i k1 krfi k1 ¼: C;

i¼1

we have EðIIÞ2n

rE

Z

  

T^n Z T 0

T^n

2

 ðnÞ n  ðAt Þ*½ðAsðnÞ Þ*
1 rðnÞ s FðX Þds

dt

r CTeðjKjþ1ÞT EðT  T ^ n Þ2 ! 0: Similarly we have Z EðIIIÞ2n r E

T T^n

2 Z T   ðnÞ 1 ðnÞ ðnÞ n  ðAt Þ*½ðAs Þ*
rs FðX Þds dt  t

r CTeðjKjþ1ÞT EðT  T ^ n Þ ! 0: Consequently letting n go to infinity in (4.7) for FðX n Þ, we get the desired martingale representation for FðXÞ. 5. Some Applications Throughout this section let X be the diffusion generated by 12  þ V as given in Section 4 and the BakryEmery minoration condition (4.1) be satisfied with K > 0.

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

99

5.1. HOEFFDING’S TYPE CONCENTRATION IMPLIED BY LSI Let g 2 Cb1 ð½0; T
 MÞ and consider the functional related to the central limit theorem: 1 FT ðÞ ¼ pffiffiffiffi T

Z

T

ðgðt; t Þ  Egðt; Xt ÞÞ dt:

0

If g 2 C01 ð½0; T
 MÞ, then by the log-Sobolev inequality in Theorem 4.5 with CðTÞ r 1=K 2 and by the Herbst method developed in Ledoux [17], we have EeFT ðXÞ

! 2 krx sgk21 ; 8 2 R: r exp 2K 2

Approximating g 2 Cb1 ð½0; 1
 MÞ by gn 2 C01 ð½0; 1
 MÞ, we see that the above inequality holds again for g 2 Cb1 ð½0; 1
 MÞ. Thus by Chebychev inequality and an optimization over , we obtain in the standard way the following Hoeffding’s type concentration inequality: ! r2 K 2 ; 8r > 0: ð5:1Þ PðFT ðXÞ > rÞ _ PðFT ðXÞ < rÞ r exp  2krx gk21 5.2. TALAGRAND’S T2 -TRANSPORTATION INEQUALITY AND TSIREL’SON’S INEQUALITY

When treating the invariance principle (or functional central limit theorem) or functional moderate deviations ([9]), we have to study the functionals of type 1 FT; ðXÞ :¼ pffiffiffiffi T

Z

Tt

sup 0 r s r t r sþ r 1

ðgðu; Xu Þ  Egðu; Xu ÞÞdu

ð5:2Þ

Ts

where g 2 Cb1 ð½0; T
 MÞ. The concentration inequality of FT; could be deduced from COROLLARY 5.1. Let g 2 Cb1 ð½0; T
 M ! Rm Þ and Yt ¼ gðt; Xt Þ. Then Y satisfies the log-Sobolev inequality on L2 ð½0; T
; Rm Þ: krx gk21 EntðF ðYÞÞ r 2 E K2 2

Z

T

jrt Fj2 ðYÞdt

ð5:3Þ

0

for all functions F 2 Cb1 ðL2 ð½0; T
; Rm ÞÞ, where r is the gradient on L2 ð½0; T
; Rm Þ. In particular,

100

MATHIEU GOURCY AND LIMING WU

(a) the law PY of Y satisfies on L2 ð½0; T
; Rm Þ the Talagrand T2 -transportation kr gk2 inequality with constant Kx 2 1 , i.e., W22 ðQ; PY Þ

  krx gk21 dQ r2 Ent dPY K2

for all probability measures Q on L2 ð½0; T
; Rm Þ such that Q PY , where W2 ðQ; PY Þ :¼

!1=2

ZZ inf

ðL2 ð½0;T
;Rm ÞÞ2

kh 

~hk2 2 ~ L ð½0;T
;Rm Þ ðdh; dhÞ

;

is the Wasserstein L2 -distance between Q and PY , here the infimum is taken over all couplings of ðQ; PY Þ, i.e., over all the probability measures on ðL2 ð½0; T
; Rm ÞÞ2 such that their marginal laws are, respectively, Q and PY . (b) (Tsirel’son’s inequality) For any non-empty subset A  L2 ð½0; T
; Rm Þ, GðhÞ :¼ suphh  EY; kiL2 ð½0;T
;Rm k2A

satisfies E exp

K2 krx gk21 K2

r exp



! 1 sup hY  EY; kiL2 ð½0;T
;Rm Þ  kkk2L2 ð½0;t
;Rm Þ 2 k2A !

krx gk21

EGðYÞ :

(c) In particular for FT; ðXÞ given in (5.2), we have ! 2 2  kr gk x 1 ; 8 2 R; EeFT; ðXÞ r exp EFT; ðXÞ þ 2K 2 !
 r2 K 2 P FT; ðXÞ  EFT; ðXÞ > r r exp  ; 8r > 0: 2krx gk21 Proof. At first for FðhÞ ¼ f ðhh; e1 i;    ; hh; en iÞ where ei 2 C 1 ð½0; T
; Rm Þ ~ ðXÞ, where and f 2 Cb1 ðRn Þ, if g 2 C01 ð½0; T
 MÞ, then FðYÞ ¼ F ~ ðÞ ¼ f F

Z

T

e1 ðtÞgðt; t Þdt;    ; 0

Z



T

en ðtÞgðt; t Þdt 0

LOGARITHMIC SOBOLEV INEQUALITIES OF DIFFUSIONS FOR THE L2 METRIC

101

belongs to our space D of test-functions. Hence noting that ~ ðÞ ¼ ðrt FÞð  ÞUt1 rx gðt; t Þ rt F the log-Sobolev inequality (5.3) follows from Theorem 4.5. Approximating g by gn 2 C01 ð½0; T
 MÞ, we obtain (5.3) for all such smooth cylindrical functions F. Next for F 2 Cb1 ðL2 ð½0; T
; Rm ÞÞ, let ðei Þi2N  C 1 ð½0; T
; Rm Þ be an orthonormal basis of L2 ð½0; T
; Rm Þ and Fn ðhÞ :¼ FðPn hÞ where Pn is the orthogonal projection to the subspace spanned by ðei Þ0 r i r n . By the continuous differentiability of F, rFn ðhÞ ! rFðhÞ in L2 ð½0; T
; Rm Þ for each h. Hence we obtain (5.3) for such F by the dominated convergence. Both parts (a) and (b) are consequences of the log-Sobolev inequality (5.3), as shown by Bobkov et al. [7] (certainly they established them only on Rn , but their arguments work for the actual Hilbert space setting). Finally in the case of part (c), m ¼ 1, letting  A :¼

C krx gk21 pffiffiffiffi 1½Ts;Tt
; 0 r s r t r s þ  r 1 ; C :¼ ; K2 T

we get part (c) from the Tsirel’son inequality in part (b).

Ì

When rx g is not bounded, we quote the following consequence of the logSobolev inequality in Theorem 4.5 due to Bobkov and Go¨tze [6]: COROLLARY 5.2. For F 2 D, Ee

ðFðXÞEFðXÞÞ



 2 2 r E exp krFkL2 ðXÞ : K2

This type of inequality turns out to be very useful for the moderate deviations of the quadratic additive functionals, cf. [11].

References 1. 2. 3. 4. 5.

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