Commun. Math. Phys. 189, 9 – 16 (1997)
Communications in
Mathematical Physics c Springer-Verlag 1997
Logarithmic Sobolev Inequalities on Path Spaces Over Riemannian Manifolds? Elton P. Hsu Department of Mathematics, Northwestern University, Evanston, IL 60208, USA. E-mail:
[email protected] Received: 3 September 1996 / Accepted: 6 February 1997
Abstract: Let Wo (M ) be the space of paths of unit time length on a connected, complete Riemannian manifold M such that γ(0) = o, a fixed point on M , and ν the Wiener measure on Wo (M ) (the law of Brownian motion on M starting at o). If the Ricci curvature is bounded by c, then the following logarithmic Sobolev inequality holds: Z F 2 log |F |dν ≤ e3c kDF k2 + kF k2 log kF k. Wo (M )
1. Introduction Logarithmic inequalities were introduced in Gross [5] as a tool for studying hypercontractivity of symmetric Markov semigroups. Let (X, ν) be a probability space and E a densely defined nonnegative quadratic form on L2 (X, ν). We say that the logarithmic Sobolev inequality holds for E if Z F 2 log |F |dν ≤ E(F, F ) + kF k2 log kF k, ∀F ∈ Dom(E). X
Gross [5] proved that it holds for the standard gaussian measure on RN for any N with Z |∇F |2 dν. E(F, F ) = RN
This implies immediately by a simple argument that the logarithmic Sobolev inequality holds for the quadratic form Z |DF |2 dν, E(F, F ) = Wo (RN )
?
Research was supported in part by NSF grant DMS-9406888.
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E.P. Hsu
on the probability space (Wo (RN ), ν), where Wo (RN ) is the path space on RN and ν the Wiener measure, and D is the gradient operator on Wo (Rn ) (see the definition below). It was proved in Gross [6] that the logarithmic Sobolev inequality holds for the Wiener measure on Wo (G), where G is a connected Lie group with the gradient operator derived from the Cartan (±)-connection. Note that for these connections the curvature vanishes. It has been conjectured by the author that a logarithmic Sobolev inequality on the path space over a general complete, connected Riemannian manifold holds with a bounding constant which can be estimated in terms of the Ricci curvature. This conjecture is completely borne out by the main result of the present work. Let M be a complete, connected Riemannian manifold of dimension n. Throughout this work we assume that M has bounded Ricci curvature. We write |RicM | ≤ c if sup {|Ric(v, v)| : v ∈ Tx M, |v| = 1, x ∈ M } ≤ c. Fix a point o ∈ M and let Wo (M ) = {γ ∈ C([0, 1], M ) : γ(0) = o} , be the space of pinned paths from o. We will work with the Wiener measure ν on Wo (M ), which can be defined as follows. Let O(M ) be the bundle of orthonormal frames over M and π : O(M ) → M the canonical projection. Fix an orthonormal frame uo at o and let {Us } be the solution of the following stochastic differential equation on O(M ): dUs = HUs ◦ dωs ,
U 0 = uo ,
(1.1)
where {ωs } is an Rn -valued Brownian motion. Here H = {Hi , 1 ≤ i ≤ n} are the canonical horizontal vector fields on O(M ). The projected process γs = πUs is a Brownian motion on M starting from o. The Wiener measure ν is just the law of the Brownian motion {γs }. We now introduce the gradient operator D on the path space. Let H be the Rn -valued Cameron-Martin space, i.e., the space of Rn -valued functions h such that h0 = 0 and h˙ ∈ L2 ([0, 1]; Rn ). It is a Hilbert space with the norm Z 1 |h˙ s |2Rn ds. |h|2H = 0
For each h ∈ H, let Dh be the vector field on the path space Wo (M ) defined by D ht(γ)s = 1U (γ)s hs , where U (γ) is the horizontal lift of γ with initial value uo . Let ζh , t ∈ R be the flow on the path space Wo (M ) generated by the vector field Dh . Let F be a real-valued function on Wo (M ). We define F (ζht γ) − F (γ) , t→0 t
Dh F (γ) = lim
if the limit exists in L2 (ν). The gradient DF is defined to be the H-valued function DF such that hDF, hiH = Dh F for all h ∈ H. Suppose that F is a cylindrical function given by (1.2) F (γ) = f (γs1 , · · · , γsl ), where f : M × · · · × M → R1 is smooth and 0 ≤ s1 < · · · < sl ≤ 1. Then it is easy to verify that l X (i) (s ∧ si )U (γ)−1 (1.3) DF (γ) = si ∇ F (γ), i=1
Logarithmic Sobolev Inequalities
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where ∇(i) F denotes the gradient of f with respect to the ith variable. From definition we have for F in (1.2) 2 l l X X 2 −1 (i) (si − si−1 ) U si ∇ F . |DF |H = j=i i=1
(1.4)
This formula will be useful later. We state the main result of the present work. Theorem 1.1. Suppose that M is a complete, connected manifold such that |RicM | ≤ c. Then the following logarithmic Sobolev inequality on the path space Wo (M ): Z F 2 log |F |dν ≤ e3c kDF k2 + kF k2 log kF k, ∀F ∈ Dom(D). Wo (M )
We conclude this introduction by stating a few well known consequences of the logarithmic Sobolev inequality (see Gross [7]). The self-adjoint operator L = −D∗ D is a generalization of the usual Ornstein-Uhlenbeck operator for a euclidean path space. Let Qt = eLt/2 be the associated Markovian semigroup. Theorem 1.2. Let M be a complete, connected Riemannian manifold such that |RicM | ≤ c. Let λM = e−3c . (i) The semigroup {Qt } on the path space Wo (M ) is hypercontractive: kQt kLp (ν)→Lq (ν) ≤ 1
if eλM t ≥
q−1 . p−1
(ii) The spectral gap of L exists and is at least λM , namely the following Poincar´e inequality holds: if F ∈ Dom(D), then 2 kF − EF k2 ≤ λ−1 M kDF k .
(iii) If F ∈ L2 (ν), then kQt F − EF k ≤ e−λM t/2 kF k. Remark 1.3. Using a Clark-Haussman-Ocone formula for path spaces, Fang [4] proved directly the existence of a spectral gap for the Ornstein-Uhlenbeck operator L on the path space over a connected, compact Riemannian manifold. 2. Gradient of a Wiener Functional The key to the present proof to the logarithmic Sobolev inequality is a formula for ∇Ex F , the gradient of the expected value of a cylindrical function F . Define the matrix-valued process {φs } by Z 1 s φτ RicUτ dτ, (2.1) φs = I − 2 0 where Ricu : Rn → Rn denotes the Ricci curvature transform read at the frame u and I is the identity matrix.
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Proposition 2.1. Let F be a cylindrical function given by (1.2). Then ) ( l X −1 (i) φ si U s i ∇ F . ∇Ex F = Ux Ex
(2.2)
i=1
Proof. The case l = 1 is due to Bismut (see Bismut [2], p.82). We give a proof here based solely on Itˆo’s formula for the horizontal Brownian motion {Us }, see (1.1). Let f be a smooth function on M and consider the function J(τ, u) = Eπu f (γτ ). It satisfies the equation 1 (2.3) ∂τ J(s − τ, u) + 1H J(s − τ, u) = 0, 2 P n where 1H = i=1 Hi2 is Bochner’s horizontal Laplacian on O(M ). This implies that {J(s − τ, Uτ ), 0 ≤ τ ≤ s} is a martingale. We now apply Itˆo’s formula to the horizontal gradient ∇H J(s − τ, Uτ ), using the fact that {Uτ } is a diffusion generated by 1H : dτ ∇H J(s − τ, Uτ ) = ∂τ ∇H J(s − τ, Uτ )dτ + h∇H ∇H J(s − τ, Uτ ), dωτ i 1 + 1H ∇H J(s − τ, Uτ )dτ 2 1 H H 1 , ∇ J(s − τ, γτ )dτ = h∇H ∇H J(s − τ, Uτ ), dωτ i + 2 1 H H +∇ ∂τ J(s − τ, Uτ ) + 1 J(s − τ, Uτ ) dτ 2 1 = h∇H ∇H J(s − τ, Uτ ), dωτ i + RicUτ ∇H J(s − τ, Uτ )dτ. 2 Here we have used (2.3) and Bochner’s formula H H 1 , ∇ J(u) = Ricu ∇H J(u) for the lift J of a function on M . It follows that φτ ∇H J(s − τ, Uτ ), 0 ≤ τ ≤ s is a martingale. Equating the expected values at τ = 0 and τ = s we obtain ∇H J(s, uo ) = E φs ∇H J(0, Us ) , This is equivalent to what we wanted because by definition ∇H J(τ, u) = u−1 ∇Eπu f (γτ ). By the Markov property and the case l = 1 we have ∇g(γs1 ) , (2.4) ∇Ex F = ∇Ex g(γs1 ) = Ux Ex φs1 Us−1 1 where
g(y) = Ey {f (y, γs2 −s1 , · · · , γsl −s1 )} . Using the induction hypothesis we have ∇g(y) = Ey ∇(1) f (y, γs2 −s1 , · · · , γsl −s1 ) +Uy
l X
Ey φsi −s1 Us−1 ∇(i) f (y, γs2 −s1 , · · · , γsl −s1 ) , i −s1
i=2
where Uy is any orthonormal frame at y and {Us } starts at Uy . Substituting this into (2.4) and using the Marko property again we obtain the desired equality.
Logarithmic Sobolev Inequalities
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The following corollary to Proposition 2.1 is known and appeared as a special case of Theorem 6.4(iii) in Donnelly and Li [3]. Corollary 2.2. Suppose that RicM ≥ −c for a nonnegative constant c. Then |∇Ex f (γs )| ≤ ecs/2 Ex |∇f (γs )|. Proof. This follows from the preceding proposition (with l = 1) and the inequality |φs | ≤ ecs/2 , which is a consequence of the assumption on the Ricci curvature. 3. A Finite Dimensional Case Let µ be the gaussian measure on Rn given by n/2 2 1 e−|x| /2s dx. µs (dx) = 2πs Gross [6] proved the following logarithmic Sobolev inequality: Z Z f 2 log |f |dµs ≤ s |∇f |2 dµs + kf k2µs log kf kµs . Rn
Rn
The first step towards proving a logarithmic Sobolev inequality in path space is to generalize the above result to Riemannian manifolds with the gaussian measure µs replaced by νs,z (dx) = p(s, z, x)dx, where p(s, z, x) is the heat kernel. Note that νs,z is the distribution of Brownian motion (starting at z) at time s. The main result of this section is Theorem 3.1 below. As before, let Z f (x)p(s, z, x)dx Ps f (z) = νs,z (f ) = M
be the heat semigroup. Theorem 3.1. Suppose that RicM ≥ −c for a nonnegative constant c. Then for any smooth function f on M , Z ecs − 1 k∇f k2νs,z + kf k2νs,z log kf kνs,z . f 2 log |f |dνs,z ≤ c M Proof. We may assume that f is strictly p greater than a fixed positive constant on M . Otherwise consider the function f = f 2 + 2 and let → 0. Let g = f 2 and consider the function Hr = Pr φ(Ps−r g), where φ(t) = 2−1 t log t. Differentiating with respect to r and noting that 1 commutes with Pr we have 1 dHr 1 = Pr 1φ(Ps−r g) − Pr {φ0 (Ps−r g)1Ps−r g} dr 2 2 o 1 n 0 2 = Pr φ (Ps−r g)1Ps−r g + φ00 (Ps−r g) |∇Ps−r g| 2 1 − Pr {φ0 (Ps−r g)1Ps−r g} 2 o 1 n 00 2 = Pr φ (Ps−r g) |∇Ps−r g| . 2
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E.P. Hsu
Now using Corollary 2.2 and then the inequality 2
{Ps−r |∇g|} ≤ 4Ps−r gPs−r |∇f |2 , we have
( 2 ) Ps−r |∇g| dHr 1 c(s−r) ≤ e Pr dr 4 Ps−r g c(s−r) Pr Ps−r |∇f |2 ≤e = ec(s−r) Ps |∇f |2 .
Integrating over r from 0 to s we obtain the desired inequality.
4. Proof of the Main Theorem We are in a position to prove our main result Theorem 1.1. We divide the proof into two steps, Lemma 4.1 and Lemma 4.3 below. Lemma 4.1. Let M be a Riemannian manifold such that RicM ≥ −c for a nonnegative constant c. Suppose that F is a cylindrical F given by (1.2). Define {φs0 ,s , s ≥ s0 } by 1 d φs ,s = − φs0 ,s RicUs , ds 0 2
φs0 ,s0 = I.
(4.1)
Then Z
2 X l 2 c −1 (j) F log |F |dν ≤ e (si − si−1 )E φsi ,sj Usj ∇ F Wo (M ) j=i i=1 l X
+kF k2 log kF k. Proof. For the sake of simplicity we assume that F (γ) = f (γs1 , γs2 , γs3 ). Using the Markov property and Theorem 3.1 we have kF k2 log kF k (4.2) 1 = EEγs1 f (γ0 , γs2 −s1 , γs3 −s1 )2 log EEγs1 f (γ0 , γs2 −s1 , γs3 −s1 )2 2 1 = Ef1 (γs1 )2 log Ef1 (γs1 )2 2 ecs1 − 1 E|∇f1 (γs1 )|2 + Ef1 (γs1 )2 log |f1 (γs1 )|, ≥− c p where f1 (x) = Ex f (γ0 , γs2 −s1 )2 . Let g(x) = Ex f (γ0 , γs2 −s1 , γs3 −s1 )2 . We have |∇f1 |2 =
|∇g|2 . 4g
Logarithmic Sobolev Inequalities
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Now compute ∇g by Proposition 3.1 and use the Cauchy-Schwarz inequality. We have 2 ∇(1) F + φs1 ,s2 Us−1 ∇(2) F + φs1 ,s3 Us−1 ∇(3) F . E|∇f1 (γs1 )|2 ≤ E Us−1 1 2 3 Using this and the inequality (ecs − 1)/c ≤ sec for c ≥ 0 and 0 ≤ s ≤ 1 in (4.2) we have kF k2 log kF k 2 ≥ −ec s1 E Us−1 ∇(1) F + φs1 ,s2 Us−1 ∇(2) F + φs1 ,s3 Us−1 ∇(3) F 1 2 3
(4.3)
+Ef1 (γs1 )2 log |f1 (γs1 )|. Repeating the same calculation for the last term f1 (x)2 log |f1 (x)| =
1 Ex f (γ0 , γs2 −s1 , γs3 −s1 )2 log Ex f (γ0 , γs2 −s1 , γs3 −s1 )2 , 2
we have Ef1 (γs1 )2 log |f1 (γs1 )| 2 ≥ −ec (s2 − s1 )E Us−1 ∇(2) F + φs2 ,s3 Us−1 ∇(3) F 2 3
(4.4)
+Ef2 (γs1 , γs2 )2 log |f2 (γs1 , γs2 ), where f2 (x, y) =
p Ex Ey f (x, y, γs3 −s2 )2 . We have for the same reason Ef2 (γs1 , γs2 )2 log |f2 (γs1 , γs2 )| 2 ≥ −ec (s3 − s2 )E Us−1 ∇(3) F + E F 2 log |F | . 3
The desired inequality follows immediately from (4.3)–(4.5).
(4.5)
Remark 4.2. The above proof is reminiscent of the Federbush–Faris–Gross addivity property of logarithmic Sobolev inequalities for product measure spaces (see Gross [7]). The independence property needed in the original proof is replaced by the weaker property of Markov dependence. The same idea appeared in the works of Stroock and Zegarlinski [9, 10], especially p. 118 in the second article. Lemma 4.3. Suppose that M is a Riemannian manifold such that |RicM | ≤ c. Then 2 X l (j) 2c 2 (si − si−1 ) φsi ,sj Us−1 ∇ F j ≤ e |DF |H . j=i i=1
l X
Proof. Let Zi = we have
Pl
j=i
(4.6)
Us−1 ∇(j) F. From (4.1) and the assumption on the Ricci curvature j
kφsi ,sj
c − φsi ,sj−1 k ≤ 2
Z
sj
ec(s−si )/2 ds.
sj−1
Note that this is the only place we have to use the absolute bound of the Ricci curvature instead of just the lower bound. Now we have
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E.P. Hsu
2 2 X l X l −1 (i) φsi ,sj Usj ∇ F = Zi + φsi ,sj − φsi ,sj−1 Zj j=i j=i+1 2 Z sj l c X ≤ |Zi | + |Zj | ec(τ −si )/2 dτ 2 sj−1 j=i+1 2 Z 1 c2 1 c(τ −si ) 2 ≤ (1 + λ) |Zi | + 1 + e gτ dτ , λ 4 si where gs = |Zj | if s ∈ [sj−1 , sj ). It follows that the left-hand side of (4.6) 2 Z Z Z 1 1 c2 1 1 c(τ −s)/2 2 gs ds + 1 + e gτ dτ ds ≤ (1 + λ) λ 4 0 0 s Z 1 1 1 (cec − ec + 1) 1+ gs2 ds. ≤ 1+λ+ 4 λ 0 R1 2 2 Note that |DF |H = 0 gs ds by (1.4). We complete the proof by using the inequality cec − ec + 1 ≤ c2 ec and choosing λ = (c/2)ec/2 . Acknowledgement. The author thanks Dominique Bakry for his proof of Theorem 3.1. Thanks are also due to Bruce Driver, Jingyi Chen, Mike Cranston, Len Gross, and Dan Stroock for helpful discussions at various stages of this work. The results presented here were announced in [8]. After the present work was completed, the author was informed that Aida and Elworthy [1] obtained an independent proof a logarithmic Sobolev inequality by embedding the manifold into a euclidean space. Our result shows that the bounding constant depends only on the Ricci curvature.
References 1. Aida, S. and Elworthy, K. D.: Differential Calculus on path and loop spaces, I. Logarthmic Sobolev inequalities on path spaces. C. R. Acad. Sci. (Paris), 321, Seri´e I, 97–102 (1995) 2. Bismut, J.-M.: Large Deviations and Malliavin Calculus. New York: Birkh¨auser, 1984 3. Donnelly, H. and Li, P.: Lower bounds for the eigenvalues of Riemannian manifolds. Michigan Math. J. 29, 149–161 (1982) 4. Fang, S.: Un in´equalit´e du type Poincar´e sur un espace de chemins. C. R. Acad. Sci. (Paris), 318, Seri´e I, 257–260 (1994) 5. Gross, L.: Logarithmic Sobolev inequalities. Am. J. of Math. 97, 1061–1083 (1975) 6. Gross, L.: Logarithmic Sobolev inequalities on Lie groups. Ill. J. Math. 36, 447–490 (1992) 7. Gross, L.: Lecture Notes in Mathematics, No. 1563. New York: Springer-Verlag, (1993), pp. 54–82 8. Hsu, E. P.: In´egalit´e de Sobolev logarithmiques sur un espace de chemins. C. R. Acad. Sci. (Paris), 320, Seri´e I, 1009–1012 (1995) 9. Stroock, D. W. and Zergalinski, B.: The equivalence of the logarithmic Sobolev inequality and the Dobrushin–Shlosman mixing condition. Commun. Math. Phys. 144, 303–323 (1992) 10. Stroock, D. W. and Zergalinski, B.: The logarithmic Sobolev inequality for discrete spin systems on a lattice. Commun. Math. Phys. 149, 175–193 (1992) Communicated by A. Jaffe
