- Email: [email protected]
Local Poincaré inequalities on loop spaces
C. R. Acad. Sci. Paris, t. 333, Série I, p. 1023–1028, 2001 Probabilités/Probability Theory
Local Poincaré inequalities on loop spaces Andreas EBERLE Math. Institute, 24–29, St. Giles, Oxford OX1 3LB, England E-mail: [email protected] (Reçu le 12 octobre 2001, accepté le 19 octobre 2001)
Abstract.
Let M be a compact connected Riemannian manifold, and fix x, y ∈ M . For a sufficiently small constant R > 0, Poincaré inequalities w.r.t. pinned Wiener measure with time parameter T > 0 are proven on the sets ΩR,N x,y , N ∈ N, consisting of all continuous paths ω : [0, 1] → M such that ω(0) = x, ω(1) = y, and d(ω(s), ω(t)) < R if s, t ∈ [(i−1)/N, i/N ] for some integer i. Moreover, the asymptotic behaviour of the best constants in the Poincaré inequalities as T goes to 0 is studied. It turns out that the asymptotic depends crucially on the Riemannian metric on M and, in particular, on the geodesics contained in ΩR,N x,y . Key ingredients in the proofs are a bisection argument, estimates for finite-dimensional spectral gaps, and a crucial variance estimate by Malliavin and Stroock. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Inégalités de Poincaré locales sur les espaces de lacets Résumé.
Soit M une variété riemannienne compacte connexe. Soient x et y des points de M et soit R > 0 une constante petite. Nous démontrons des inégalités de type Poincaré par rapport à la distribution du pont brownien sur les ensembles ΩR,N x,y , N ∈ N, qui se composent de tous les chemins continus ω : [0, 1] → M tels que ω(0) = x, ω(1) = y, et d(ω(s), ω(t)) < R si s, t ∈ [(i − 1)/N, i/N ] pour un certain entier i. De plus, nous étudions le comportement asymptotique des meilleures constantes dans les inégalités de Poincaré lorsque le paramètre de temps du pont brownien tend vers 0. Ce comportement asymptotique dépend fortement de la métrique riemannianne sur M , en particulier des géodésiques contenues dans ΩR,N x,y . 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Soit M une variété riemannienne compacte et connexe, et soient x, y ∈ M . Considérons l’espace Ωx,y des chemins continus ω : [0, 1] → M tels que ω(0) = x et ω(1) = y. En particulier, Ωx,x est un espace T de lacets. Pour T > 0, la distribution Px,y du pont brownien de x à y en temps T , reparamétrisée sur l’intervalle [0, 1], est une mesure de probabilité sur Ωx,y , cf. (1). Pour tout chemin ω ∈ Ωx,y , l’espace tangent Tω Ωx,y se compose des champs de vecteurs continus le long de ω qui s’annulent en 0 et en 1. En utilisant une version fixe τs (ω), s ∈ [0, 1], ω ∈ Ωx,y , du transport parallèle stochastique, on définit des sous-espaces hilbertiens Tω1 Ωx,y de champs de vecteurs H1 , cf. (2) et (3). La relation (4) définit le gradient Note présentée par Paul M ALLIAVIN. S0764-4442(01)02174-7/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
1023
A. Eberle T correspondant D0 sur les fonctions cylindriques F : Ωx,y → R. La forme bilinéaire Ex,y donnée par (6) est 2 T 1,2 T fermable sur L (Ωx,y ; Px,y ), de domaine de fermeture H (Ωx,y ; Px,y ). On a démontré dans [2], qu’en T ne satisfait pas à une inégalité de Poincaré globale du type (7). général, Ex,y T T ) et λ∗ (U; Px,y ) Considérons maintenant un ouvert U ⊆ Ωx,y . Définissons les constantes λ(U; Px,y 1,2 T par (8), l’infimum étant pris sur tous les fonctions non constantes F ∈ H (Ωx,y ; Px,y ) telles que F = 0 T p.s. sur Ωx,y \ U , et, pour la définition de λ∗ , U F dPx,y = 0. Soit ΩR x,y , R > 0, l’ensemble des chemins T ¯ R) = inf{λ(ΩR ω ∈ Ωx,y tels que d(ω(s), ω(t)) < R pour tous s, t ∈ [0, 1], et soit λ(T, x,y ; Px,y ); x, y ∈ M }.
¯ T HÉORÈME 1. – Il existe R0 , C ∈ ]0, ∞[ tels que pour tout R ∈ ]0, R0 [, on ait λ(T, R) > 0 pour tous ¯ T > 0, et lim inf T ↓0 T · λ(T, R) 1 − CR2 . La démonstration de ce théorème est esquissée dans la section 3 de la version anglaise. En combinant le théorème 1 avec des résultats sur des espaces de chemins discrets démontrés dans [4], nous obtenons aussi des inégalités de Poincaré sur des ensembles de chemins avec des sauts limités. Soit ΩR,N x,y , R > 0, N ∈ N, T ; P ) > 0 pour tous R ∈ ]0, R [ et N ∈ N. De plus : défini par (11). Nous obtenons λ(ΩR,N 0 x,y x,y T HÉORÈME 2. – Si y n’est pas conjugué à x, et si Ωx,y ne contient qu’une seule géodésique de longueur T NR qui soit un minimum local de l’énergie, alors lim inf T ↓0 T λ∗ (ΩR,N x,y ; Px,y ) > 0 . T HÉORÈME 3. – Si Ωx,y contient plusieurs minima locaux de l’énergie de longeur < NR, alors : R,N T T −∞ < lim inf T log λ ΩR,N x,y ; Px,y < lim sup T log λ Ωx,y ; Px,y < 0. T ↓0
T ↓0
1. Framework Let M be a compact connected Riemannian manifold, and fix x, y ∈ M . Let Ωx,y = ω ∈ C [0, 1], M ; ω(0) = x, ω(1) = y , denote the corresponding pinned path space over M . In particular, Λx = Ωx,x is the based loop space at x. T For T > 0, the distribution Px,y of the Brownian bridge from x to y with time parameter T is a probability measure on the Borel σ-algebra of Ωx,y . It is uniquely determined by T f ω(s1 ), ω(s2 ), . . . , ω(sk ) Px,y (dω) Ωx,y
= Mk
f (x1 , x2 , . . . , xk ) ps1 T (x, x1 )p(s2 −s1 )T (x1 , x2 ) · · ·
p(sk −sk−1 )T (xk−1 , xk )p(1−sk )T (xk , y)
k
V (dxi )/pT (x, y)
(1)
i=1
for all k ∈ N, f ∈ C∞ (M k ), and 0 < s1 < s2 < · · · < sk < 1. Here pt (x, y) denotes the heat kernel of ∆/2 on M . For ω ∈ Ωx,y , the tangent space Tω Ωx,y consists of all continuous vector fields X along ω that vanish at 0 and 1. Now fix T > 0 and x, y ∈ M . The M -valued stochastic process (Πs )0s1 , Πs (ω) = ω(s), is T a semimartingale w.r.t. the probability measure Px,y and the augmentation (FsT,x,y )0s1 of the filtration Fs = σ(Πu ; 0 u s) generated by the process. Let τs , 0 s 1, be a fixed version of the stochastic parallel transport along the paths of this semimartingale, and let τs,u = τu τs−1 , 0 s, u 1. Recall that
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Local Poincaré inequalities on loop spaces
(τs )0s1 is uniquely determined up to equivalence. We may assume that τs (ω) is the usual parallel transport if ω is piecewise smooth, and that τs,u (ω) is an isometry from Tω(s) M to Tω(u) M for every ω ∈ Ωx,y and 0 s, u 1. For ω ∈ Ωx,y we define the H1 tangent space
τs (ω)−1 Xs ∈ H1,2 (0, 1; TxM ) . Tω1 Ωx,y = X ∈ Tω Ωx,y ; s →
(2)
It is a Hilbert space with inner product 1 (X, Y )ω = 0
∇Y ∇X (s), (s) ds, ds ds ω(s)
(3)
d 1 where ∇X ds (s) := dε (τs+ε,s (ω)Xs+ε )|ε=0 . For smooth ω, Tω Ωx,y is the usual tangent space at ω in the 1 Hilbert manifold of H paths from x to y. Let F (ω) = f (ω(s1 ), ω(s2 ), . . . , ω(sk )), k ∈ N, f ∈ C∞ (M k ), be a smooth cylinder function on Ωx,y . We define the H1 gradient D0 F : Ωx,y → T 1 Ωx,y by D0 F (ω) ∈ Tω1 Ωx,y and
0 D F (ω), X ω = XF where XF =
k
(i) i=1 Xsi f (ω(s1 ), . . . , ω(sk ))
∀X ∈ Tω1 Ωx,y ,
(4)
is the derivative of F in direction X. Explicitly,
k 0 D F (ω) (t) = τsi ,t (ω) grad(i) f ω(s1 ), . . . , ω(sk ) · (si ∧ t − si t).
(5)
i=1
In particular, the function ω → (D0 F (ω), D0 G(ω))ω is bounded for smooth cylinder functions F, G. Up T to Px,y -equivalence, this function is independent of the choice of the stochastic parallel transport (τs )0s1 T made above. Hence the quadratic form Ex,y given on smooth cylinder functions by T (F, Ex,y
G) =
0 T D F (ω), D0 G(ω) ω Px,y (dω)
(6)
does not depend on the choice of the stochastic parallel transport either. By using the integration by parts T T identity w.r.t. pinned Wiener measure, it can be shown that Ex,y is closable on L2 (Ωx,y ; Px,y ), cf. [1]. The 0 1,2 T gradient operator D extends to functions in the domain H (Ωx,y ; Px,y ) of the closure. 2. Local Poincaré inequalities Let Var(F ; P ) = (F − F dP )2 dP denote the variance of a function F w.r.t. a probability measure P . If M is not simply connected then the indicator functions of the homotopy classes in Ωx,y are non-constant T )). But even in the simply connected case, the functions in the kernel of the operator (D0 , H1,2 (Ωx,y ; Px,y 1 existence of a non-trivial closed geodesic σ : S → M such that the curvature is strictly negative on σ(S 1 ) can destroy the validity of a global Poincaré inequality of type T T T Cx,y · Ex,y (F, F ) Var F ; Px,y
T for all F ∈ H1,2 Ωx,y ; Px,y
(7)
T with a finite constant Cx,y , cf. [2]. This is contrary to the situation on the based path space Ωx = {ω ∈ C([0, 1], M ); ω(0) = x}, where a global Poincaré inequality w.r.t. Wiener measure always holds by a result of S. Fang [7].
1025
A. Eberle T For a non-empty open subset U ⊆ Ωx,y let H1,2 0 (U; Px,y ) denote the closed subspace of all F ∈ 1,2 T T H (Ωx,y ; Px,y ) with F = 0 Px,y -a.e. on Ωx,y \ U , and let
T = inf λ U; Px,y F ∈D
T (D0 F, D0 F ) dPx,y , T (F − ETx,y [F |U])2 dPx,y U U
(8)
T where ETx,y [·|·] denotes conditional expectation w.r.t. Px,y , and D consists of all non-constant F ∈ 1,2 T ∗ T H0 (U; Px,y ). Below, we will also consider λ (U; Px,y ) which is defined similarly but with the infimum T taken only over F ∈ D with U F dPx,y = 0. For R > 0 let ΩR x,y denote the open set consisting of all ω ∈ Ωx,y such that d(ω(s), ω(t)) < R for all s, t ∈ [0, 1]. For T > 0 let
¯ (T, R) = inf λ ΩR ; P T ; x, y ∈ M . λ x,y x,y
(9)
¯ (T, R) > 0 for all T > 0, T HEOREM 1. – There exist R0 , C ∈ (0, ∞) such that for every R ∈ (0, R0 ), λ and ¯ lim inf T · λ(T, R) 1 − CR2 T ↓0
(10)
Next, we want to exhaust Ωx,y by sets on which a Poincaré type inequality holds. Let E(ω) denote the (possibly infinite) energy of a path ω ∈ Ωx,y , i.e., E(ω) =
k 1 d(ω(si−1 ), ω(si ))2 , sup 2 si − si−1 i=1
where the supremum is taken over all partitions 0 = s0 < s1 < · · · < sk = 1, k ∈ N. In differential geometry it is common to exhaust the H1 loop space by sets of type {E a}, a > 0, cf., e.g., [10]. These sets have measure 0, but we can use the following larger sets instead: for R > 0 and N ∈ N let ΩR,N x,y = ω ∈ Ωx,y ;
max
sup
i=0,1,...,N −1 s,t∈[i/N,(i+1)/N ]
d ω(s), ω(t) < R .
(11)
R,N 2 Clearly, Ωx,y = N ∈N ΩR,N x,y for all R > 0, and {ω ∈ Ωx,y ; E(ω) < a} ⊆ Ωx,y if 1/N R /2a. Since 1,2 T 1,2 T T ∗ T H0 (Ωx,y ; Px,y ) = H (Ωx,y ; Px,y ) by definition, λ (Ωx,y ; Px,y ), λ (Ωx,y ; Px,y ), and the inverse of the T optimal constant Cx,y in (7) coincide. By using appropriate cut-off functions, we can show: 1,2 R,N T L EMMA. – For every R, T > 0, the union N ∈N H0 (Ωx,y ; Px,y ) is a dense subspace of 1,2 T R,N T ∗ R,N T ), N ∈ N, both converge H (Ωx,y ; Px,y ). In particular, the sequences λ(Ωx,y ; Px,y ) and λ (Ωx,y ; Px,y T to λ(Ωx,y ; Px,y ) as N → ∞. We fix R0 as in Theorem 1. Using this theorem, the following is not difficult to show: C OROLLARY. – For all T > 0, R ∈ (0, R0 ), N ∈ N, and x, y ∈ M , T λ ΩR,N x,y ; Px,y > 0. In the form stated, the assertion does not depend on the geometric and topological properties of M . In fact, it even holds if M is not simply connected. The reason is that the indicator functions of the homotopy classes do not satisfy Dirichlet boundary conditions on ΩR,N x,y . The geometry becomes crucial, however, if T ; P ). we make statements about the size of λ(ΩR,N x,y x,y
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Local Poincaré inequalities on loop spaces
Recall that the point y is called conjugate to x along a geodesic γ ∈ Ωx,y if dγ (0) expx is not of maximal rank. By Sard’s theorem, for almost every x, y ∈ M , y is not conjugate to x along any geodesic. We fix N ∈ N and R ∈ (0, R0 ) with N R > d(x, y). T HEOREM 2. – Suppose that y is not conjugate to x along any geodesic. If there exists only one geodesic in Ωx,y of length N R which is a local minimum of the energy functional then T lim inf T · λ∗ ΩR,N x,y ; Px,y > 0. T ↓0
T In sharp contrast to Theorem 2, λ(ΩR,N x,y ; Px,y ) decays exponentially fast if E has several local minima:
T HEOREM 3. – If there exist two geodesics of length < N R in Ωx,y that are strict local minima of the energy functional then R,N T T −∞ < lim inf T log λ ΩR,N x,y ; Px,y lim sup T log λ Ωx,y ; Px,y < 0. T ↓0
T ↓0
Moreover, it can be shown that if there exists a non-trivial closed geodesic γ : S 1 → M such that the curvature is strictly negative on γ(S 1 ), and x and y are close to γ(S 1 ), then the exponential decay rate of T λ(ΩR,N x,y ; Px,y ) as T ↓ 0 becomes arbitrarily large as N → ∞. On the other hand, the same rate stabilizes for large N if the Ricci curvature on M is strictly positive. 3. Idea of the proofs We give a very brief outline of some key steps in the proofs of the results described above. For details, we refer to [3] and [5]. Theorems 2 and 3 can be obtained by combining Theorem 1 with results on discretized loop spaces proven in [4]. A major new input is the proof of Theorem 1, which enables us to reduce the situation to a finite-dimensional one. We now sketch this proof very roughly. The line of reasoning is closer to E. Hsu’s first proof [8] of the logarithmic Sobolev inequality on path spaces than to recent attempts to show Poincaré inequalities on loop spaces. R = {z ∈ M ; d(x, z) < R and d(y, z) < R }. Let F Let R > 0, x, y ∈ M with d(x, y) < R, and let Ux,y R . For z ∈ M we define be a smooth cylinder function on Ωx,y such that F (ω) = 0 if ω(1/2) is not in Ux,y a function F on Ωz,x × Ωz,y by F (ω, ω ¯ ) = F (ω ∨ ω ¯ ) where (ω ∨ ω ¯ )(s) = ω(1 − 2s) for s ∈ [0, 1/2] and (ω ∨ ω ¯ )(s) = ω ¯ (2s − 1) for s ∈ [1/2, 1]. An elementary computation yields T T /2 T /2 T T /2 T /2 = Var F ; Pz,x µx,y (dz) + Var E·,x E·,y F (W, W ) ; µTx,y Var F ; Px,y ⊗ Pz,y T T /2 T /2 T /2 T /2 + Ez,y Var F (W, W ); Pz,x µx,y (dz) Ez,x Var F (W, W ); Pz,y 2 T /2 T /2 (12) + cT (R) · gradz E·,x E·,y F(W, W ) µTx,y (dz). T T , ETx,y denotes expectation w.r.t. Px,y , W and W Here µTx,y is the distribution of ω → ω(1/2) w.r.t. Px,y are the canonical projections on the first and second component in Ωz,x × Ωz,y , and E, Var means that the expectation or variance is taken w.r.t. W , whereas otherwise it is taken w.r.t. W . Moreover,
Var(f ; µTa,b ) ∞ r cT (r) = sup sup ; f ∈ C (M ) \ {0}, f = 0 on M \ Ua,b , | grad f |2 dµTa,b a,b∈M
1027
A. Eberle
where sup ∅ := −∞. By techniques from Simon [12] and heat kernel estimates, we can show for sufficiently small R, that lim sup T −1 cT (R) < ∞. (13) T ↓0
For t > 0, a, b ∈ M , and v ∈ Ta M let Hs (ω), s ∈ [0, 1], ω ∈ Ωa,b , be Ta M valued and adapted w.r.t. the filtration (Fst,a,b ) such that H0 (ω) = v, H1 (ω) = 0, and s → Hs (ω) is C1 for all ω. Let Xs (ω) = τs (ω)Hs (ω), and let 1 1 t −1
t δ X= -a.s., (14) t Hs + Ricτs (Hs ) dBs Pa,b 2 0 where Bs , 0 s 1, denotes the Ta M valued lifting (anti-development) of the Brownian bridge s → ω(s) t w.r.t. Pa,b , cf., e.g., [6], and Ricτs (v) = τs−1 Ric(τs v). In spite of the singularity of the Brownian bridge t as s ↑ 1, the existence of the Itô integral δ t X as an element in L2 (Ωa,b ; Pa,b ) can be shown, cf. [9]. Let Cov(F1 , F2 ; P ) denote the covariance of two random variables F1 , F2 w.r.t. a probability measure P . To commute the derivative and the expectation on the right hand side of (12), we use the identity t t t v, grad E·,b [G] a = Ea,b [XG] − Cov δ t X, G; Pa,b
(15)
which holds for all smooth cylinder functions G on C([0, 1], M ). By a crucial observation of Malliavin and Stroock [11], we can choose the vector field X depending on t, a, b and v in such a way that we have both good control of X itself and of the variance of δ t X t . In particular, in spite of the factor 1/t appearing in (14), for small R the variances can be w.r.t. Pa,b bounded independently of t ∈ (0, 1], a, b ∈ M with d(a, b) < R, and v ∈ Ta M with |v| = 1. Taking into account this fact, as well as (15) and (13), we can prove the local Poincaré inequality by subsequently applying the estimate (12) to all subintervals of the sequence of dyadic partitions of the interval [0, 1]. References [1] Driver B., Röckner M., Construction of diffusions on path and loop spaces of compact Riemannian manifolds, C. R. Acad. Sci. Paris, Série I 315 (1992) 603–608. [2] Eberle A., Absence of spectral gaps on a class of loop spaces, Preprint 00-076 SFB 343, Universität Bielefeld, 2000. [3] Eberle A., Poincaré inequalities on loop spaces, Habilitationssschrift Universität Bielefeld, Preprint 01-07-049 BiBoS, Universität Bielefeld, 2001. [4] Eberle A., Spectral gaps on discretized loop spaces, Preprint. [5] Eberle A., Local spectral gaps on loop spaces and their asymptotics, Preprint. [6] Emery M., Stochastic Calculus on Manifolds, Springer, Berlin, 1989. [7] Fang S., Inégalité du type de Poincaré sur l’espace des chemins riemanniens, C. R. Acad. Sci. Paris, Série I 318 (1994) 257–260. [8] Hsu E.P., Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds, Comm. Math. Phys. 189 (1997) 9–16. [9] Hsu E.P., Analysis on path and loop spaces, in: Probability Theory and Applications (Princeton, NJ, 1996), IAS/Park City Math. Ser., Vol. 6, American Mathematical Society, Providence, RI, 1999, pp. 277–347. [10] Klingenberg W., Closed Geodesics, Springer, Berlin, 1978. [11] Malliavin P., Stroock D., Short time behaviour of the heat kernel and its logarithmic derivatives, J. Diff. Geom. 44 (1996) 550–570. [12] Simon B., Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymptotic expansions, Ann. Inst. H. Poincaré Phys. Théor. 38 (1983) 295–307.
1028
Local Poincaré inequalities on loop spaces Andreas EBERLE Math. Institute, 24–29, St. Giles, Oxford OX1 3LB, England E-mail: [email protected] (Reçu le 12 octobre 2001, accepté le 19 octobre 2001)
Abstract.
Let M be a compact connected Riemannian manifold, and fix x, y ∈ M . For a sufficiently small constant R > 0, Poincaré inequalities w.r.t. pinned Wiener measure with time parameter T > 0 are proven on the sets ΩR,N x,y , N ∈ N, consisting of all continuous paths ω : [0, 1] → M such that ω(0) = x, ω(1) = y, and d(ω(s), ω(t)) < R if s, t ∈ [(i−1)/N, i/N ] for some integer i. Moreover, the asymptotic behaviour of the best constants in the Poincaré inequalities as T goes to 0 is studied. It turns out that the asymptotic depends crucially on the Riemannian metric on M and, in particular, on the geodesics contained in ΩR,N x,y . Key ingredients in the proofs are a bisection argument, estimates for finite-dimensional spectral gaps, and a crucial variance estimate by Malliavin and Stroock. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Inégalités de Poincaré locales sur les espaces de lacets Résumé.
Soit M une variété riemannienne compacte connexe. Soient x et y des points de M et soit R > 0 une constante petite. Nous démontrons des inégalités de type Poincaré par rapport à la distribution du pont brownien sur les ensembles ΩR,N x,y , N ∈ N, qui se composent de tous les chemins continus ω : [0, 1] → M tels que ω(0) = x, ω(1) = y, et d(ω(s), ω(t)) < R si s, t ∈ [(i − 1)/N, i/N ] pour un certain entier i. De plus, nous étudions le comportement asymptotique des meilleures constantes dans les inégalités de Poincaré lorsque le paramètre de temps du pont brownien tend vers 0. Ce comportement asymptotique dépend fortement de la métrique riemannianne sur M , en particulier des géodésiques contenues dans ΩR,N x,y . 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée Soit M une variété riemannienne compacte et connexe, et soient x, y ∈ M . Considérons l’espace Ωx,y des chemins continus ω : [0, 1] → M tels que ω(0) = x et ω(1) = y. En particulier, Ωx,x est un espace T de lacets. Pour T > 0, la distribution Px,y du pont brownien de x à y en temps T , reparamétrisée sur l’intervalle [0, 1], est une mesure de probabilité sur Ωx,y , cf. (1). Pour tout chemin ω ∈ Ωx,y , l’espace tangent Tω Ωx,y se compose des champs de vecteurs continus le long de ω qui s’annulent en 0 et en 1. En utilisant une version fixe τs (ω), s ∈ [0, 1], ω ∈ Ωx,y , du transport parallèle stochastique, on définit des sous-espaces hilbertiens Tω1 Ωx,y de champs de vecteurs H1 , cf. (2) et (3). La relation (4) définit le gradient Note présentée par Paul M ALLIAVIN. S0764-4442(01)02174-7/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
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A. Eberle T correspondant D0 sur les fonctions cylindriques F : Ωx,y → R. La forme bilinéaire Ex,y donnée par (6) est 2 T 1,2 T fermable sur L (Ωx,y ; Px,y ), de domaine de fermeture H (Ωx,y ; Px,y ). On a démontré dans [2], qu’en T ne satisfait pas à une inégalité de Poincaré globale du type (7). général, Ex,y T T ) et λ∗ (U; Px,y ) Considérons maintenant un ouvert U ⊆ Ωx,y . Définissons les constantes λ(U; Px,y 1,2 T par (8), l’infimum étant pris sur tous les fonctions non constantes F ∈ H (Ωx,y ; Px,y ) telles que F = 0 T p.s. sur Ωx,y \ U , et, pour la définition de λ∗ , U F dPx,y = 0. Soit ΩR x,y , R > 0, l’ensemble des chemins T ¯ R) = inf{λ(ΩR ω ∈ Ωx,y tels que d(ω(s), ω(t)) < R pour tous s, t ∈ [0, 1], et soit λ(T, x,y ; Px,y ); x, y ∈ M }.
¯ T HÉORÈME 1. – Il existe R0 , C ∈ ]0, ∞[ tels que pour tout R ∈ ]0, R0 [, on ait λ(T, R) > 0 pour tous ¯ T > 0, et lim inf T ↓0 T · λ(T, R) 1 − CR2 . La démonstration de ce théorème est esquissée dans la section 3 de la version anglaise. En combinant le théorème 1 avec des résultats sur des espaces de chemins discrets démontrés dans [4], nous obtenons aussi des inégalités de Poincaré sur des ensembles de chemins avec des sauts limités. Soit ΩR,N x,y , R > 0, N ∈ N, T ; P ) > 0 pour tous R ∈ ]0, R [ et N ∈ N. De plus : défini par (11). Nous obtenons λ(ΩR,N 0 x,y x,y T HÉORÈME 2. – Si y n’est pas conjugué à x, et si Ωx,y ne contient qu’une seule géodésique de longueur T NR qui soit un minimum local de l’énergie, alors lim inf T ↓0 T λ∗ (ΩR,N x,y ; Px,y ) > 0 . T HÉORÈME 3. – Si Ωx,y contient plusieurs minima locaux de l’énergie de longeur < NR, alors : R,N T T −∞ < lim inf T log λ ΩR,N x,y ; Px,y < lim sup T log λ Ωx,y ; Px,y < 0. T ↓0
T ↓0
1. Framework Let M be a compact connected Riemannian manifold, and fix x, y ∈ M . Let Ωx,y = ω ∈ C [0, 1], M ; ω(0) = x, ω(1) = y , denote the corresponding pinned path space over M . In particular, Λx = Ωx,x is the based loop space at x. T For T > 0, the distribution Px,y of the Brownian bridge from x to y with time parameter T is a probability measure on the Borel σ-algebra of Ωx,y . It is uniquely determined by T f ω(s1 ), ω(s2 ), . . . , ω(sk ) Px,y (dω) Ωx,y
= Mk
f (x1 , x2 , . . . , xk ) ps1 T (x, x1 )p(s2 −s1 )T (x1 , x2 ) · · ·
p(sk −sk−1 )T (xk−1 , xk )p(1−sk )T (xk , y)
k
V (dxi )/pT (x, y)
(1)
i=1
for all k ∈ N, f ∈ C∞ (M k ), and 0 < s1 < s2 < · · · < sk < 1. Here pt (x, y) denotes the heat kernel of ∆/2 on M . For ω ∈ Ωx,y , the tangent space Tω Ωx,y consists of all continuous vector fields X along ω that vanish at 0 and 1. Now fix T > 0 and x, y ∈ M . The M -valued stochastic process (Πs )0s1 , Πs (ω) = ω(s), is T a semimartingale w.r.t. the probability measure Px,y and the augmentation (FsT,x,y )0s1 of the filtration Fs = σ(Πu ; 0 u s) generated by the process. Let τs , 0 s 1, be a fixed version of the stochastic parallel transport along the paths of this semimartingale, and let τs,u = τu τs−1 , 0 s, u 1. Recall that
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Local Poincaré inequalities on loop spaces
(τs )0s1 is uniquely determined up to equivalence. We may assume that τs (ω) is the usual parallel transport if ω is piecewise smooth, and that τs,u (ω) is an isometry from Tω(s) M to Tω(u) M for every ω ∈ Ωx,y and 0 s, u 1. For ω ∈ Ωx,y we define the H1 tangent space
τs (ω)−1 Xs ∈ H1,2 (0, 1; TxM ) . Tω1 Ωx,y = X ∈ Tω Ωx,y ; s →
(2)
It is a Hilbert space with inner product 1 (X, Y )ω = 0
∇Y ∇X (s), (s) ds, ds ds ω(s)
(3)
d 1 where ∇X ds (s) := dε (τs+ε,s (ω)Xs+ε )|ε=0 . For smooth ω, Tω Ωx,y is the usual tangent space at ω in the 1 Hilbert manifold of H paths from x to y. Let F (ω) = f (ω(s1 ), ω(s2 ), . . . , ω(sk )), k ∈ N, f ∈ C∞ (M k ), be a smooth cylinder function on Ωx,y . We define the H1 gradient D0 F : Ωx,y → T 1 Ωx,y by D0 F (ω) ∈ Tω1 Ωx,y and
0 D F (ω), X ω = XF where XF =
k
(i) i=1 Xsi f (ω(s1 ), . . . , ω(sk ))
∀X ∈ Tω1 Ωx,y ,
(4)
is the derivative of F in direction X. Explicitly,
k 0 D F (ω) (t) = τsi ,t (ω) grad(i) f ω(s1 ), . . . , ω(sk ) · (si ∧ t − si t).
(5)
i=1
In particular, the function ω → (D0 F (ω), D0 G(ω))ω is bounded for smooth cylinder functions F, G. Up T to Px,y -equivalence, this function is independent of the choice of the stochastic parallel transport (τs )0s1 T made above. Hence the quadratic form Ex,y given on smooth cylinder functions by T (F, Ex,y
G) =
0 T D F (ω), D0 G(ω) ω Px,y (dω)
(6)
does not depend on the choice of the stochastic parallel transport either. By using the integration by parts T T identity w.r.t. pinned Wiener measure, it can be shown that Ex,y is closable on L2 (Ωx,y ; Px,y ), cf. [1]. The 0 1,2 T gradient operator D extends to functions in the domain H (Ωx,y ; Px,y ) of the closure. 2. Local Poincaré inequalities Let Var(F ; P ) = (F − F dP )2 dP denote the variance of a function F w.r.t. a probability measure P . If M is not simply connected then the indicator functions of the homotopy classes in Ωx,y are non-constant T )). But even in the simply connected case, the functions in the kernel of the operator (D0 , H1,2 (Ωx,y ; Px,y 1 existence of a non-trivial closed geodesic σ : S → M such that the curvature is strictly negative on σ(S 1 ) can destroy the validity of a global Poincaré inequality of type T T T Cx,y · Ex,y (F, F ) Var F ; Px,y
T for all F ∈ H1,2 Ωx,y ; Px,y
(7)
T with a finite constant Cx,y , cf. [2]. This is contrary to the situation on the based path space Ωx = {ω ∈ C([0, 1], M ); ω(0) = x}, where a global Poincaré inequality w.r.t. Wiener measure always holds by a result of S. Fang [7].
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A. Eberle T For a non-empty open subset U ⊆ Ωx,y let H1,2 0 (U; Px,y ) denote the closed subspace of all F ∈ 1,2 T T H (Ωx,y ; Px,y ) with F = 0 Px,y -a.e. on Ωx,y \ U , and let
T = inf λ U; Px,y F ∈D
T (D0 F, D0 F ) dPx,y , T (F − ETx,y [F |U])2 dPx,y U U
(8)
T where ETx,y [·|·] denotes conditional expectation w.r.t. Px,y , and D consists of all non-constant F ∈ 1,2 T ∗ T H0 (U; Px,y ). Below, we will also consider λ (U; Px,y ) which is defined similarly but with the infimum T taken only over F ∈ D with U F dPx,y = 0. For R > 0 let ΩR x,y denote the open set consisting of all ω ∈ Ωx,y such that d(ω(s), ω(t)) < R for all s, t ∈ [0, 1]. For T > 0 let
¯ (T, R) = inf λ ΩR ; P T ; x, y ∈ M . λ x,y x,y
(9)
¯ (T, R) > 0 for all T > 0, T HEOREM 1. – There exist R0 , C ∈ (0, ∞) such that for every R ∈ (0, R0 ), λ and ¯ lim inf T · λ(T, R) 1 − CR2 T ↓0
(10)
Next, we want to exhaust Ωx,y by sets on which a Poincaré type inequality holds. Let E(ω) denote the (possibly infinite) energy of a path ω ∈ Ωx,y , i.e., E(ω) =
k 1 d(ω(si−1 ), ω(si ))2 , sup 2 si − si−1 i=1
where the supremum is taken over all partitions 0 = s0 < s1 < · · · < sk = 1, k ∈ N. In differential geometry it is common to exhaust the H1 loop space by sets of type {E a}, a > 0, cf., e.g., [10]. These sets have measure 0, but we can use the following larger sets instead: for R > 0 and N ∈ N let ΩR,N x,y = ω ∈ Ωx,y ;
max
sup
i=0,1,...,N −1 s,t∈[i/N,(i+1)/N ]
d ω(s), ω(t) < R .
(11)
R,N 2 Clearly, Ωx,y = N ∈N ΩR,N x,y for all R > 0, and {ω ∈ Ωx,y ; E(ω) < a} ⊆ Ωx,y if 1/N R /2a. Since 1,2 T 1,2 T T ∗ T H0 (Ωx,y ; Px,y ) = H (Ωx,y ; Px,y ) by definition, λ (Ωx,y ; Px,y ), λ (Ωx,y ; Px,y ), and the inverse of the T optimal constant Cx,y in (7) coincide. By using appropriate cut-off functions, we can show: 1,2 R,N T L EMMA. – For every R, T > 0, the union N ∈N H0 (Ωx,y ; Px,y ) is a dense subspace of 1,2 T R,N T ∗ R,N T ), N ∈ N, both converge H (Ωx,y ; Px,y ). In particular, the sequences λ(Ωx,y ; Px,y ) and λ (Ωx,y ; Px,y T to λ(Ωx,y ; Px,y ) as N → ∞. We fix R0 as in Theorem 1. Using this theorem, the following is not difficult to show: C OROLLARY. – For all T > 0, R ∈ (0, R0 ), N ∈ N, and x, y ∈ M , T λ ΩR,N x,y ; Px,y > 0. In the form stated, the assertion does not depend on the geometric and topological properties of M . In fact, it even holds if M is not simply connected. The reason is that the indicator functions of the homotopy classes do not satisfy Dirichlet boundary conditions on ΩR,N x,y . The geometry becomes crucial, however, if T ; P ). we make statements about the size of λ(ΩR,N x,y x,y
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Local Poincaré inequalities on loop spaces
Recall that the point y is called conjugate to x along a geodesic γ ∈ Ωx,y if dγ (0) expx is not of maximal rank. By Sard’s theorem, for almost every x, y ∈ M , y is not conjugate to x along any geodesic. We fix N ∈ N and R ∈ (0, R0 ) with N R > d(x, y). T HEOREM 2. – Suppose that y is not conjugate to x along any geodesic. If there exists only one geodesic in Ωx,y of length N R which is a local minimum of the energy functional then T lim inf T · λ∗ ΩR,N x,y ; Px,y > 0. T ↓0
T In sharp contrast to Theorem 2, λ(ΩR,N x,y ; Px,y ) decays exponentially fast if E has several local minima:
T HEOREM 3. – If there exist two geodesics of length < N R in Ωx,y that are strict local minima of the energy functional then R,N T T −∞ < lim inf T log λ ΩR,N x,y ; Px,y lim sup T log λ Ωx,y ; Px,y < 0. T ↓0
T ↓0
Moreover, it can be shown that if there exists a non-trivial closed geodesic γ : S 1 → M such that the curvature is strictly negative on γ(S 1 ), and x and y are close to γ(S 1 ), then the exponential decay rate of T λ(ΩR,N x,y ; Px,y ) as T ↓ 0 becomes arbitrarily large as N → ∞. On the other hand, the same rate stabilizes for large N if the Ricci curvature on M is strictly positive. 3. Idea of the proofs We give a very brief outline of some key steps in the proofs of the results described above. For details, we refer to [3] and [5]. Theorems 2 and 3 can be obtained by combining Theorem 1 with results on discretized loop spaces proven in [4]. A major new input is the proof of Theorem 1, which enables us to reduce the situation to a finite-dimensional one. We now sketch this proof very roughly. The line of reasoning is closer to E. Hsu’s first proof [8] of the logarithmic Sobolev inequality on path spaces than to recent attempts to show Poincaré inequalities on loop spaces. R = {z ∈ M ; d(x, z) < R and d(y, z) < R }. Let F Let R > 0, x, y ∈ M with d(x, y) < R, and let Ux,y R . For z ∈ M we define be a smooth cylinder function on Ωx,y such that F (ω) = 0 if ω(1/2) is not in Ux,y a function F on Ωz,x × Ωz,y by F (ω, ω ¯ ) = F (ω ∨ ω ¯ ) where (ω ∨ ω ¯ )(s) = ω(1 − 2s) for s ∈ [0, 1/2] and (ω ∨ ω ¯ )(s) = ω ¯ (2s − 1) for s ∈ [1/2, 1]. An elementary computation yields T T /2 T /2 T T /2 T /2 = Var F ; Pz,x µx,y (dz) + Var E·,x E·,y F (W, W ) ; µTx,y Var F ; Px,y ⊗ Pz,y T T /2 T /2 T /2 T /2 + Ez,y Var F (W, W ); Pz,x µx,y (dz) Ez,x Var F (W, W ); Pz,y 2 T /2 T /2 (12) + cT (R) · gradz E·,x E·,y F(W, W ) µTx,y (dz). T T , ETx,y denotes expectation w.r.t. Px,y , W and W Here µTx,y is the distribution of ω → ω(1/2) w.r.t. Px,y are the canonical projections on the first and second component in Ωz,x × Ωz,y , and E, Var means that the expectation or variance is taken w.r.t. W , whereas otherwise it is taken w.r.t. W . Moreover,
Var(f ; µTa,b ) ∞ r cT (r) = sup sup ; f ∈ C (M ) \ {0}, f = 0 on M \ Ua,b , | grad f |2 dµTa,b a,b∈M
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A. Eberle
where sup ∅ := −∞. By techniques from Simon [12] and heat kernel estimates, we can show for sufficiently small R, that lim sup T −1 cT (R) < ∞. (13) T ↓0
For t > 0, a, b ∈ M , and v ∈ Ta M let Hs (ω), s ∈ [0, 1], ω ∈ Ωa,b , be Ta M valued and adapted w.r.t. the filtration (Fst,a,b ) such that H0 (ω) = v, H1 (ω) = 0, and s → Hs (ω) is C1 for all ω. Let Xs (ω) = τs (ω)Hs (ω), and let 1 1 t −1
t δ X= -a.s., (14) t Hs + Ricτs (Hs ) dBs Pa,b 2 0 where Bs , 0 s 1, denotes the Ta M valued lifting (anti-development) of the Brownian bridge s → ω(s) t w.r.t. Pa,b , cf., e.g., [6], and Ricτs (v) = τs−1 Ric(τs v). In spite of the singularity of the Brownian bridge t as s ↑ 1, the existence of the Itô integral δ t X as an element in L2 (Ωa,b ; Pa,b ) can be shown, cf. [9]. Let Cov(F1 , F2 ; P ) denote the covariance of two random variables F1 , F2 w.r.t. a probability measure P . To commute the derivative and the expectation on the right hand side of (12), we use the identity t t t v, grad E·,b [G] a = Ea,b [XG] − Cov δ t X, G; Pa,b
(15)
which holds for all smooth cylinder functions G on C([0, 1], M ). By a crucial observation of Malliavin and Stroock [11], we can choose the vector field X depending on t, a, b and v in such a way that we have both good control of X itself and of the variance of δ t X t . In particular, in spite of the factor 1/t appearing in (14), for small R the variances can be w.r.t. Pa,b bounded independently of t ∈ (0, 1], a, b ∈ M with d(a, b) < R, and v ∈ Ta M with |v| = 1. Taking into account this fact, as well as (15) and (13), we can prove the local Poincaré inequality by subsequently applying the estimate (12) to all subintervals of the sequence of dyadic partitions of the interval [0, 1]. References [1] Driver B., Röckner M., Construction of diffusions on path and loop spaces of compact Riemannian manifolds, C. R. Acad. Sci. Paris, Série I 315 (1992) 603–608. [2] Eberle A., Absence of spectral gaps on a class of loop spaces, Preprint 00-076 SFB 343, Universität Bielefeld, 2000. [3] Eberle A., Poincaré inequalities on loop spaces, Habilitationssschrift Universität Bielefeld, Preprint 01-07-049 BiBoS, Universität Bielefeld, 2001. [4] Eberle A., Spectral gaps on discretized loop spaces, Preprint. [5] Eberle A., Local spectral gaps on loop spaces and their asymptotics, Preprint. [6] Emery M., Stochastic Calculus on Manifolds, Springer, Berlin, 1989. [7] Fang S., Inégalité du type de Poincaré sur l’espace des chemins riemanniens, C. R. Acad. Sci. Paris, Série I 318 (1994) 257–260. [8] Hsu E.P., Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds, Comm. Math. Phys. 189 (1997) 9–16. [9] Hsu E.P., Analysis on path and loop spaces, in: Probability Theory and Applications (Princeton, NJ, 1996), IAS/Park City Math. Ser., Vol. 6, American Mathematical Society, Providence, RI, 1999, pp. 277–347. [10] Klingenberg W., Closed Geodesics, Springer, Berlin, 1978. [11] Malliavin P., Stroock D., Short time behaviour of the heat kernel and its logarithmic derivatives, J. Diff. Geom. 44 (1996) 550–570. [12] Simon B., Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: Asymptotic expansions, Ann. Inst. H. Poincaré Phys. Théor. 38 (1983) 295–307.
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