Probab. Theory Relat. Fields (2011) 151:559–590
DOI 10.1007/s00440-010-0308-5
A concrete estimate for the weak Poincaré inequality
on loop space
Xin Chen · Xue-Mei Li · Bo Wu
Received: 3 November 2009 / Revised: 17 May 2010 / Published online: 12 June 2010
© Springer-Verlag 2010
Abstract The aim of the paper is to study the pinned Wiener measure on the loop
space over a simply connected compact Riemannian manifold together with a Hilbert
space structure and the Ornstein–Uhlenbeck operator d ∗ d. We give a concrete estimate for the weak Poincaré inequality, assuming positivity of the Ricci curvature of
the underlying manifold. The order of the rate function is s −α for any α > 0.
Keywords Brownian bridge measure · Loop space · Orstein–Uhlenbeck operator ·
Weak Poincaré inequality
Mathematics Subject Classification (2000)
60Hxx · 58J65
1 Introduction
A.
Let M be a smooth connected compact complete Riemannian manifold (through this
paper, we all assume M to be a smooth connected complete Riemannian manifold, so
X.-M. Li’s research was supported by the EPSRC (EP/E058124/1).
X. Chen (B) · B. Wu
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
e-mail: chenxingz217@hotmail.com
B. Wu
e-mail: bwu77@163.com
X.-M. Li
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK
e-mail: xuemei@xuemei.org
123
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X. Chen et al.
we do not state these conditions in the remaining part). For a, b ∈ M, we consider the
pinned path space "a,b over M,
"a,b = {ω ∈ C([0, 1], M); ω(0) = a, ω(1) = b},
which is a smooth Finsler manifold with compatible distance function
d∞ (ω, γ ) := sup d(ω(s), γ (s)).
s∈[0,1]
When a = b, we have the loop space over M, based at a. Let FCb∞ ("a,b ) be the
collection of smooth cylinder function on "a,b . Each F ∈ FCb∞ ("a,b ) is determined
by a smooth function f on M n and a partition 0 < s1 < · · · < sn < 1 of [0, 1]:
F(ω) = (ev s1 ,...,sn )∗ f = f (ω(s1 ), . . . , ω(sn )).
(1.1)
T
(also called the
For each T > 0, endow "a,b with the pinned Wiener measure Pa,b
Brownian bridge measure), which is derived by pushing forward the standard Brownian bridge measure on the space of the pinned curves of C([0, T ]; M) with starting
point a and ending point b, to C([0, 1]; M) through the rescaling map ω(t) %→ ω( Tt ).
T can be equally defined through its integration over smooth cylinThe measure Pa,b
drical functions of type (1.1):
!
T
(dω) =
F(ω)Pa,b
1
pT (a, b)
!
f (x1 , x2 , . . . xn )
Mn
· ps1 T (a, x1 ) p(s2 −s1 )T (x1 , x2 ) . . . p(1−sn )T (xn , b)
n
"
dxi .
i=1
1 for simplicity, and the
where pt (x, y) is the heat kernel on M. Write Pa,b for Pa,b
corresponding expectation is denoted by Ea,b .
B.
Let ω(s) be the canonical process on "a,b , Fs be the natural filtration and F = F1 .
T ), see [9]. Denote by // (ω) :
Then ω(s) is a semi-martingale with ("a,b , F, Fs , Pa,b
s,t
Tω(s) M → Tω(t) M the stochastic parallel translation along the continuous path ω(·),
T a.s. defined. Write // = // . Let H be the space of finite energy curves
which is Pa,b
s
0,s
n
in R ,

& !1

&
H = h : [0, 1] → Rn && |ḣ(s)|2 ds < ∞, h is absolutely continuous .




123
0
A concrete estimate for the weak Poincaré inequality on loop space
561
We identify Ta M with Rn and define the Bismut’s tangent space Hω0 in "a,b :
&
*
+
Hω0 = //· (ω)h · & h ∈ H, h 0 = 0, h 1 = 0 ,
which is a Hilbert space with the inner product:
(X, Y )Hω0 =
!1 ,
0
d
d
(//t,0 (ω)X t )), (//t,0 (ω)Yt ))
dt
dt
-
dt
Ta M
and corresponding norm | − |Hω0 .
Consider the differential operator d which sends a differentiable function on "a,b
(viewed as a Finsler manifold) to a differential 1-form. For F a smooth cylindrical
function, d F can be considered as a bounded linear map on Bismut tangent space. By
the Riesz representation theorem, there is the H 1 gradient D0 F(ω) ∈ Hω0 , given by
(D0 F(ω), X h )Hω0 = d F(X h ),
for all vectors X h of the form X h (s) = //s (ω)h s in Hω0 . In particular, for cylindrical
function F = (evs1 ,...,sn )∗ f of the form (1.1),
n
.
D0 F(ω)(t) =
i=1
//si ,t (ω)∇i f (ω(s1 ), ω(s2 ), . . . ω(sn )) · G 0 (si , t)
where ∇i f is the gradient of f for the ith variable and G 0 (s, t) = s ∧ t − s · t,
d2
0 ! s, t ! 1, is the Green’s function of the operator − ds
2 with Dirichlet boundary
conditions on (0, 1). Also we have,
|D0 F|2H0 =
ω
n
.
G 0 (si , s j )
i, j=1
·)//si ,s j (ω)∇i f (ω(s1 ), . . . ω(sn )), ∇ j f (ω(s1 ), . . . ω(sn ))*ω(s j )
(1.2)
For each T > 0, the quadratic form defined on smooth cylinder function by
T
E/a,b
(F, F) :=
!
T
|D0 F|2H0 Pa,b
(dω),
ω
"a,b
T , which is due to an integration by parts forcan be extended to a Dirichlet form Ea,b
T ) is the the same as the closure
mula, see [9]. The domain of the Dirichlet form D(Ea,b
of the gradient operator D0 . Follow the custom, we call this Dirichlet form the O–U
1 for simplicity.
Dirichlet form. And we denote Ea,b for Ea,b
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X. Chen et al.
If µ is a probability measure, we denote by E[F; µ] the average of a function
F ∈ L 2 (µ) with respect to this measure and Var(F; µ) = E(F 2 ; µ) − [E(F; µ)]2
the corresponding variance. The main theorem of the paper is:
Theorem 5.1 Let M be a simply connected compact manifold with strict positive
Ricci curvature. For any small α > 0, there exists a constant s0 > 0 such that the
following weak Poincaré inequality holds, i.e.
Var(F; Pa,a ) !
1
Ea,a (F, F) + s||F||2∞ , s ∈ (0, s0 ), F ∈ D(Ea,a ).
sα
And the constant s0 does not depend on the starting point a ∈ M.
C. Historical remark
The Ornstein–Uhlenbeck operator and the Ornstein–Uhlenbeck process play an important role in the development of the L 2 theory on path and loop spaces, cf. [13]. The
study of the functional inequalities for O–U Dirichlet form with respect to the Wiener
measure (on path space) and to the pinned Wiener measure (on loop space) goes back
a long way. For the Wiener measure on path space over a compact manifold, it turns
out that there is no fundamental topological or geometrical obstruction to the validity of the Poincaré inequality. See e.g. the work of Fang [15] for the existence of a
Poincaré inequality for O–U Dirichlet form and that of Aida and Elworthy [6], Hsu
[19], Capitaine et al. [7] for the existence of a logarithmic Sobolev inequality. There
is also the approach of gradient stochastic different equations, e.g. Elworthy and Li
[14], Elworthy et al. [13].
But for the loop space over a compact manifold M, the problem seems much
more complicated. We limit ourselves here to the case of functional inequalities with
respect to the pinned Wiener measure. Gross [17] pointed out that Logarithmic Sobolev
inequality does not hold when M = S 1 and he proved instead a Logarithmic Sobolev
inequality plus a potential term when M was a compact Lie group. In general the
geometry and the topology of the manifold will play a significant role. In particular, a
Poincaré inequality does not hold for the O–U Dirichlet form if the underlying manifold is not simply connected, since the indicator function of each connected component
of the loop space is on the domain of the O–U Dirichlet form, see Aida [3]. Furthermore, in [10], Eberle constructed a simply connected compact Riemannian manifold
on the loop space over which the Poincaré inequality for O–U Dirichlet form did not
hold. As transpired in his proof, the validity of the Poincaré inequality may depend
on the starting point of the based loop space. A Clark–Ocone formula with a potential
was deduced by Gong and Ma [16], which led to their discovery of a Logarithmic
Sobolev inequality with a potential on loop space over general compact manifold. See
also Aida [1]. In their results, the simply connected condition is not needed for the
underlying manifold. Aida [4], on the other hand, deduced a Clark–Ocone formula
which led to a Logarithmic Sobolev inequality for a modified Dirichlet form, under
suitable conditions on the small time asymptotics of the Hessian of the logarithm of the
heat kernel of the underlying manifold. Built on that, a Poincaré inequality is shown
123
A concrete estimate for the weak Poincaré inequality on loop space
563
to hold for the O–U Dirichlet form on the loop space over hyperbolic space, see Chen
et al. [8].
Another development in the positive direction comes from Eberle [11], where it
was shown that a local Poincaré ineqaulity hold for the O–U Dirichlet form on loop
space over compact manifold. A parallel result was given by Aida [2]: when M was
simply connected, the O–U Dirichlet form had the weak spectral gap property. By the
weak spectral gap property for a Dirichlet form E in L 2 (P) it is meant that Fn → 0 in
probability for any sequence of functions {Fn }∞
n=1 ⊂ D(E ) satisfying the following
conditions,
sup ||Fn || L 2 ! 1,
n
E(Fn ) = 0,
lim E (Fn , Fn ) = 0,
n→∞
see also Kusuoka [20]. Although we do not know the relation between Eberle’s local
Poincaré ineqaulity and Aida’s weak spectral gap property, it was noted in Röckner
and Wang in [22], the weak spectral gap property was equivalent to the following
weak Poincaré inequality:
Varµ ( f ) ! β(s)E ( f, f ) + s|| f ||2∞ , s ∈ (0, s0 ) f ∈ D(E ) ∩ L ∞ (µ)
Here β : R+ → R+ is a non-increasing function and s0 > 0 is a constant. And in [5],
Aida used such weak Poincaré inequality to give an estimate on the spectral gap of a
Schrödinger operator on the loop space. We refer the reader to Wang [24] for analysis,
development and historical references on such inequalities. Our contribution here is
the concrete estimate of β(s) in the inequality above. Here we need to find suitable
exhausting local sets replacing the role played by geodesic balls in the proof of weak
Poincaré inequality on finite dimensional manifolds (see [24]). We use a slightly different collection of local sets from that used by Eberle in [11] for technical reasons.
The main difficulty here is to get a suitable estimate of the constants for the local
Poincaré inequalities on such local sets as in the proof for finite dimensional case. In
fact, we do not derive a local Poincaré inequality, some additional term of the L ∞
norm will appear in the estimate, but finally we can control such terms to get a global
weak Poincaré inequality.
The paper is organised as follows. In Sect. 2, we introduce notation and state some
results, especially that of Eberle [11,12] on which our proof is based on. In Sect. 3,
we give some variance estimate for small time by using the estimates and ideas in
[11]. In Sect. 4, A weak Poincaré type inequality for the distribution of the Brownian
bridge evaluated at N equal time intervals is given. We use a combination of small time
asymptotics and Poincaré inequality for the Wiener measure to control the growth of
the constants with N . In particular, some of the methods in this section are inspired
by [12] and [18]. In Sect. 5, the main theorem is proved by reducing the variance of a
function on the loop space to the variance of a function on a product manifold which
is localized to subsets which are chains of small geodesic balls, and the variance of
functions on some sub-path with respect to pinned Wiener measure with small time
parameter.
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X. Chen et al.
2 Notations and known results
Let {Bs } be the Ta M valued stochastic anti-development of the canonical process ω(s),
T ). It is however not a Brownian
which is a semi-martingale with ("a,b , F, Fs , Pa,b
n
motion, see [9]. Denote by L(R ; Ta M) the set of all linear maps from Rn to Ta M.
Lemma 2.1 [11] Let {As (ω), ω ∈ "a,b , 0 ! s ! 1} be a L(Rn ; Ta M) valued adapted
process such that s → As (ω) is C 1 for every ω and
sup
sup |A-s (ω)| < ∞.
ω∈"a,b s∈[0,1]
Suppose that Hs (ω) = As (ω)h s for some h ∈ H with h 1 = 0 and X · (ω) = //· (ω)A·
(ω)h · . Define
δuT
X :=
!u 0
0
1
1
T −1 Hs- + //s−1 Ric#ω(s) (//s Hs ) d Bs , 0 ! u < 1
2
Then
δ T X := lim δuT X
u→1
T ) and the limit is in L 2 (" ; P T ).
exists in L 1 ("a,b ; Pa,b
a,b
a,b
If a, b ∈ M are not in the cut locus of each other, we take A· such that //· A· is the
damped stochastic parallel transport and take H· to be parallel push back of the Jacobi
fields along the unique geodesic connecting a and b with initial vector v. By a result of
T
Malliavin–Stroock, the variance of δ T X defined in above lemma with respect to Pa,b
are uniformly bounded for a, b, v, T in compact sets. In fact, we have the following
lemma,
Lemma 2.2 [11,21] Let b ∈ M \ Cut(a), v ∈ Ta M, and T > 0. There is a vector
X sT,a,b,v = //s Hs , with initial value X 0T,a,b,v = v and H as in Lemma 2.1, such that
for every T0 > 0 and r ∈ (0, inj M ),
sup ρ(T, r ) < ∞,
T ∈(0,T0 ]
where
ρ(T, r ) :=
sup
a,b∈M,d(a,b)!r, v∈Ta M,|v|=1
2
3
45
T
Var δ T X T,a,b,v ; Pa,b
.
(2.3)
The next lemma deals with the derivative with the starting point of the expectation
under pinned Wiener measure,
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A concrete estimate for the weak Poincaré inequality on loop space
565
Lemma 2.3 [11] Let v ∈ Ta M. For each X s (ω) = //s (ω)Hs (ω) with Hs (ω) as in
T a.s,
Lemma 2.1, and that X 0 = v, Pa,b
3
4
4
3
T
T
T
da E.,b
[F] [v] = Ea,b
[d F(X )] − Cov δ T X, F; Pa,b
(2.4)
for each smooth cylinder function F ∈ FCb∞ ("a,b ).
X s is a vector field on pinned path space if and only if v = 0. By the proof of
T [δ T X ] = 0 when v = 0, and in this case it is the integration
Theorem 3.2 in [11], Ea,b
by parts formula on pinned path space:
6
7
T
T
Ea,b
Fδ T X
[d F(X )] = Ea,b
For two paths ω1 , ω2 with ω1 (1) = ω2 (0), define ω1 ∨ ω2 as following:
ω1 ∨ ω2 (s) =
8
ω1 (2s)
if s ∈ [0, 1/2],
ω2 (2s − 1) if s ∈ [1/2, 1].
For each ω in "a,b , we can find one and only one pair of ω
91 , ω
92 to satisfy that
92 . For each fixed T > 0, ω ∈ "a,b with a, b ∈
/ Cut(ω(1/2)) and
ω = ω
91 ∨ ω
92 , v ∈ Tω(1/2) M, let
ω=ω
91 ∨ ω
:
X sT,v (ω) =

T /2,ω(1/2),ω(0),v
 X 1−2s
(9
ω1 −1 ) if s ∈ [0, 1/2],
 X T /2,ω(1/2),ω(1),v (9
ω )
2s−1
2
if s ∈ [1/2, 1]
where X sT,a,b,v is as in Lemma 2.2 and ω
91 −1 (s) := ω
91 (1 − s), 0 ! s ! 1, is the time
reverse of the path ω
91 .
For F ∈ FCb∞ ("a,b ) and ω ∈ "a,b , let
T
! (F)(ω) =
8
sup{v∈Tω(1/2) M,|v|=1} (d F( :
X sT,v ))2 (ω), if a, b ∈
/ Cut(ω(1/2))
0,
otherwise.
(2.5)
For each smooth cylinder function F ∈ FCb∞ ("a,b ), there exists a unique function
;
/ ω1 , ω
/ ω1 , ω
92 ) , defined on z∈M "a,z ×"z,b , such that F(9
92 ) = F(9
ω1 ∨9
ω2 ) = F(ω)
F(9
for each ω in "a,b with ω = ω
91 ∨ ω
92 .
r = B (a) ∩ B (b)
Lemma 2.4 [11] For a, b ∈ M, T > 0, and r > 0, denote Ua,b
r
r
(Br (a) means the ball with center a and radius r) and
T
µa,b
(d x) =
pT /2 (a, x) pT /2 (x, b)
d x.
pT (a, b)
(2.6)
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X. Chen et al.
• There exists a positive number R1 , such that when r ∈ (0, R1 ),
T
T
Var(F; Pa,b
) ! 2q(T, r )Ea,b
[! T (F)]
+ (1 + 4q(T, r )ρ(T /2, r ))
! 2
6
3
47
T /2,1
T /2
/ ω1 , ω
Ea,x Var2 F(9
92 ); Px,b
r
Ua,b
6
75
T /2,2
T /2
T
/ ω1 , ω
Var1 ( F(9
92 ); Pa,x ) µa,b
(d x)
+ Ex,b
(2.7)
holds for every smooth cylinder function F : "a,b %→ R such that F(ω) = 0
r . Here ! T (F), ρ(T /2, r ) are defined by (2.5) and (2.3)
if ω(1/2) is not in Ua,b
respectively. ET /2,i , Vari indicates that the corresponding expectation or variance is taken with respect to the ith-subpath /
ωi , i = 1, 2,
• The constant q(T, r ) in above inequality does not depend on a, b ∈ M and satisfies
lim T −1 q(T, r ) !
T ↓0
1 + Kr2
∀r ∈ (0, R1 ).
4
(2.8)
for some K > 0.
3 Estimates on the variance with small time parameter
The following lemma gives a short time asymptotics of the variance. It is crucial for
the proof of the main result in this section. For simplicity, in the remaining part of
the paper, the constants, e.g. C, T , and N , will change according to different situation
but we will clarify which parameter such C depends on. At first, we state a lemma
deriving from Lemma 2.4 by a cut-off procedure,
Lemma 3.1 There exists a number R1 > 0 such that for all a, b ∈ M with d(a, b) <
r < R1 , if T < T1 (η, r ) for a positive number T1 (η, r ), the following inequality holds
T ),
for any number 0 < η < 1 and smooth cylindrical function F on ("a,b , Pa,b
T
Var(F; Pa,b
)
6
7
T
r }
! 4q(T, r )Ea,b
! T (F)1{ω(1/2)∈Ua,b
! 2
6
3
47
T /2,1
T /2
/ ω1 , ω
+ (1 + 4q(T, r )ρ(T /2, r ))
Ea,x
Var2 F(9
92 ); Px,b
r
Ua,b
6
3
475
T /2,2
T /2
T
/ ω1 , ω
Var1 F(9
µa,b
92 ); Pa,x
(d x)
+ Ex,b
<
=
128q(T, r ) − (1−4η)r 2
2T
+ 6+
e
||F||2∞ .
η2 r 2
(3.9)
T (d x) is the distribution of the mid-point of the Brownian Bridge,
Here the measure µa,b
r := B (a) ∩ B (b), q(T, r ) is some constant satisfying (2.8), and
given by (2.6), Ua,b
r
r
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A concrete estimate for the weak Poincaré inequality on loop space
567
ET /2,i , Vari indicates that the expectation or variance is taken with respect to the
subpath /
ωi .
Remark The constants R1 and q(T, r ) are the same as that in Lemma 2.4, and R1 is
smaller than the injectivity radius of M.
Proof Step 1 For a positive r smaller than the injectivity radius of M, a, b ∈ M with
d(a, b) < r , define a function Ψa,b : "a,b → R by
Ψa,b (ω) := ϕ(d(a, ω(1/2))) · ϕ(d(b, ω(1/2))).
(3.10)
Here the smooth function ϕ : R+ → R satisfies the following conditions,
ϕ(s) =
8
1, if s ! (1 − η)r,
0, if s " r
and |ϕ - | !
2
.
ηr
(3.11)
T ), the domain of the O–U Dirichlet form, and
Then the function Ψa,b is on D(Ea,b
|D0 Ψa,b (ω)|Hω0 ! ηr4 . Furthermore we show below that for all small η > 0 there is
constant T1 (η, r ) such that if T < T1 (η, r ),
T
Pa,b
(Ψa,b 2= 1) ! 2e−
(1−4η)r 2
2T
.
(3.12)
We begin with estimating the probability
>
?
T
T
Pa,b
(B(1−η)r (a))c .
(d(a, ω(1/2)) > (1 − η)r ) = µa,b
By Varadhan’s estimate [23],
limT ↓0 T log pT (a, b) = −
d 2 (a, b)
uniformly on M × M.
2
Hence for any 0 < η < 1, there exists a constant T1 (η, r ) > 0, such that for every
0 < T < T1 (η, r ),
−
d 2 (a, b) η2 r 2
d 2 (a, b) η2 r 2
−
! log pT (a, b) ! −
+
.
2T
4T
2T
4T
(3.13)
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568
X. Chen et al.
In the calculations that follows we assume that 0 < T < T1 (η, r ). Note that d(a, b) <r ,
T
Pa,b
(d(a, ω(1/2)) > (1 − η)r )
!
1
=
pT /2 (a, x) pT /2 (x, b)d x
pT (a, b)
{d(a,x)>(1−η)r }
!e
!e
d 2 (a,b) η2 r 2
2T + 4T
r2
2T
2 2
+ η4Tr
!
e−
d 2 (a,x) η2 r 2
+ 2T
T
{d(a,x)>(1−η)r }
e
2 r 2 η2 r 2
− (1−η)
+ 2T
T
! e−
(1−4η)r 2
2T
pT /2 (x, b)d x
.
Similarly,
T
Pa,b
(d(b, ω(1/2)) > (1 − η)r ) ! e−
(1−4η)r 2
2T
.
Hence
T
T
T
Pa,b
(Ψa,b 2= 1) ! Pa,b
(d(a, ω(1/2)) > (1 − η)r ) + Pa,b
(d(b, ω(1/2)) > (1 − η))
! 2e−
(1−4η)r 2
2T
.
Step 2 Let R1 be the constant in Lemma 2.4. Assume that r < R1 and we first observe
that
4
3
42
3
T
T
T
= Ea,b
F 2 − Ea,b
F
Var F; Pa,b
>
?2
>
?
T
T
! Ea,b
FΨa,b + ||F||2∞ Pa,b
Ψa,b 2= 1
3
42
T
− Ea,b
[F − FΨa,b + FΨa,b ]
4
3
T
T
+ 3||F||2∞ Pa,b
! Var FΨa,b ; Pa,b
(Ψa,b 2= 1)
4
3
2
(1−4η)r
T
+ 6e− 2T ||F||2∞
! Var FΨa,b ; Pa,b
(3.14)
r , Lemma 2.4 applies to F+
Since Ψa,b (ω) = 0 when ω(1/2) ∈
/ Ua,b
a,b and we have,
T
T
) ! 2q(T, r )Ea,b
[! T (FΨa,b )]
Var(FΨa,b ; Pa,b
+ (1 + 4q(T, r )ρ(T /2, r ))
! 2
T /2,1
T /2
!
×
Ea,x [Var2 ( FΨ
ω1 , ω
92 ); P )]
a,b (9
x,b
r
Ua,b
123
5
T /2,2
T /2
T
!
ω1 , ω
92 ); Pa,x )] µa,b
(d x),
+Ex,b [Var1 ( FΨ
a,b (9
A concrete estimate for the weak Poincaré inequality on loop space
569
T /2
!
We next deal with the terms Vari ( FΨ
ω1 , ω
92 ); Px,b ). Since Ψa,b (ω)=
a,b (9
ϕ(d(a, ω(1/2))) · ϕ(d(b, ω(1/2))) is determined by ω(1/2), for i = 1, 2.
3
4
3
4
T /2
T /2
/ ω1 , ω
!
ω1 , ω
92 ); Px,b = ϕ(d(a, x))2 · ϕ(d(x, b))2 · Vari F(9
92 ); Px,b
Vari FΨ
a,b (9
3
4
T /2
/ ω1 , ω
r }.
! Vari F(9
92 ); Px,b I{x∈Ua,b
Consequently
T
T
) ! 2q(T, r )Ea,b
[! T (FΨa,b )]
Var(FΨa,b ; Pa,b
+ (1 + 4q(T, r )ρ(T /2, r ))
! 2
T /2,1
T /2
/ ω1 , ω
r }]
×
Ea,x [Var2 ( F(9
92 ); Px,b )I{x∈Ua,b
r
Ua,b
5
T /2,2
T /2
T
/ ω1 , ω
r }] µ
92 ); Px,b )I{x∈Ua,b
+Ex,b [Var1 ( F(9
a,b (d x),
(3.15)
Let Sa M := {v ∈ Ta M, |v| = 1}. For each ω ∈ "a,b , by the definition of ! T in (2.5),
! T (FΨa,b )(ω)
5
2
X T,v )]2 (ω); v ∈ Sω(1/2) M
= sup [d(FΨa,b )( :
2
5
2
! 2 sup [d F( :
X T,v )]2 Ψa,b
; v ∈ Sω(1/2) M
2
5
+2 sup F 2 [dΨa,b ( :
X T,v )]2 (ω); v ∈ Sω(1/2) M
r }
! 2! T (F)(ω)I{ω(1/2)∈Ua,b
2
5
+2||F||2∞ sup )v, ∇x [ϕ(d(a, x)) · ϕ(d(x, b))]*2 ; v ∈ Sx M
r } +
! 2! T (F)(ω)I{ω(1/2)∈Ua,b
32
||F||2∞ I{Ψa,b (ω)2=1} .
η2 r 2
The required inequality (3.9) follows from (3.14), (3.15) and (3.16).
(3.16)
4
3
Proposition 3.2 There is a constant R0 such that for each 0 < η < 1/4 the followT ) provided that d(a, b) < r < R and 0 < T <
ing inequality holds on ("a,b , Pa,b
0
T0 (η, r ) for some T0 (η, r ) > 0:
3
4
T
T
Var(F; Pa,b
|D0 F|2H0
) ! T C(r )Ea,b
ω
+C(η, r )e
2
− (1−4η)r
2T
||F||2∞ , F ∈ FCb∞ ("a,b )
Here C(η, r ), C(r ) are independent of T .
123
570
X. Chen et al.
Remark Note that r must be less than the injective radius here, so we can not make
the constant in front of ||F||2∞ tend to 0, and the above estimate is not a weak Poincaré
inequality.
Proof By approximation procedure, it suffices to show the inequality holds for all
smooth cylindrical functions with the following form,
< m
==
< < = < =
2
2 −1
1
,
F(ω) = f ω m , ω m , . . . ,ω
2
2
2m
f ∈ C ∞ (M 2
m −1
), m ∈ N+ .
(3.17)
For any ω ∈ "a,b , let
<
i −1+s
ωi (s) = ω
2k
=
, s ∈ [0, 1], k ∈ N+ , 1 ! i ! 2k
For simplicity, we did not reflect the index k in the definition of the new path ωi . For
each smooth cylinder function F and positive integer k we define a unique function
@2 k
;
F [k] of 2k sub-paths. It is defined on z=(x ,...x )∈M 2k −1 i=1
"xi−1 ,xi (here x0 = a
and x2k = b), such that for each ω ∈ "a,b
1
2k −1
F [k] (ω1 , . . . , ω2k ) = F(ω).
A
@2k T /2k
In fact, F [k] (ω1 , . . . ω2k ) i=1
Pxi−1 ,xi (dωi ) is a smooth version of the conditional
T [F|ω(1/2k ) = x , . . . ω(1 − 1/2k ) = x
[1] is the same as
expectation Ea,b
1
2k −1 ] and F
/
F in Lemmas 2.4 and 3.1.
N ,T
in M N −1 as,
For N " 1 and T > 0 we define the probability measure µa,b
N ,T
µa,b
(dx) :=
p T (b, x N −1 ) p T (x N −1 , x N −2 ) . . . p T (x1 , a)
N
N
N
pT (a, b)
dx N −1 . . . dx1 . (3.18)
Fix a number 0 < r < R1 for R1 as in Lemma 3.1, 0 < η < 1/4 and a positive
/ as a function of any of the two
number T < T1 (η, r ). For the variance terms for F
subpaths ω
91 , ω
92 on the right side of inequality (3.9), we can apply (3.9) again, on
each sub-path while keeping the other fixed, to obtain an estimate on the variance of
F in terms of the variances and the operation ! T /2 for F [2] as a function of any of
r , so we can use Lemma 3.1 here). Repeat this
the four subpaths (note that x ∈ Ua,b
procedure by mid-dividing the path and applying (3.9). The variance terms will finally
vanish after a repetition of m times for the smooth cylinder function of type (3.17),
123
A concrete estimate for the weak Poincaré inequality on loop space
571
and we have,
T
Var(F; Pa,b
)
m−1
.
!4
G(k, T, r )q(T /2k , r )
k=0


2k ! 
6
7
k
.
T /2k , j
×
Ex j−1 ,x j ! T /2 , j (F [k] )(ω1 , . . . ω2k )I{ω j (1/2)∈Uxr ,x }
j−1 j

j=1U
j
×
"
i2= j
m−1
.
+
k=0
T /2k
Pxi−1 ,xi (dωi )



k
µ2 ,T (d x)
 a,b
=
<
128q(T /2k , r ) k − 2k (1−4η)r 2
2T
2 e
G(k, T, r ) 6 +
||F||2∞
η2 r 2
(3.19)
@k
where G(0, T, r ) = 1, G(k, T, r ) = i=1
(1 + 4q(T /2i−1 , r )ρ(T /2i , r )) for each
k
k > 0, U j = {x = (x1 , . . . , x2k −1 ) ∈ M 2 −1 : d(x j−1 , x j ) < r } for j = 1, 2 . . . , 2k
T /2k , j
(x0 = a and x2k = b). We denote by Ex j−1 ,x j and ! T /2
k, j
(F [k] ) the corresponding
k
expectation and the operation ! T /2
(defined in (2.5)) with respect to the jth sub-path
for function F [k] .
By (5.8) in the proof of lemma 5.1 in [11] if T is small enough,
(3.20)
sup G(k, T, r ) < C(r ).
k∈N
By this and Lemma 3.3 below we can find a positive number R0 < R1 , such that for
each 0 < r < R0 , there is a T2 (r ) > 0, when T < T2 (r ) the following holds for all
positive integer m:
m−1
.
G(k, T, r )q(T /2k , r )
k=0


2k ! 
6
7
.
k

T /2k , j
·
Ex j−1 ,x j ! T /2 , j (F [k] )(ω1 , . . . ω2k )1{ω j (1/2)∈Uxr ,x }
j−1 j

×
j=1U
j
"
i2= j
T /2k


2k ,T

Pxi−1 ,xi (dωi ) µa,b (d x)

T
! T C(r )Ea,b
|D0 F|2H0
ω
(3.21)
Note that by Lemma 2.4 there is T0 (η, r ) < min(T2 (r ), T1 (η, r )) such that if T < T0
(η, r ), then |q(T, r )| ! C(r ) for some constans C(r ) and (3.20) holds. Using this
123
572
X. Chen et al.
bound we can see that for 0 < η < 1/4, T < T0 (η, r ),
=
<
128q(T /2k , r )
sup G(k, T, r ) 6 +
! C(r, η)
η2 r 2
k∈N
and
m−1
.
k=0
=
∞
k
2
.
128q(T /2k , r ) k − 2k (1−4η)r 2
k − 2 (1−4η)r
2T
2T
2
G(k, T, r ) 6 +
e
!
C(r,
η)
2
e
η2 r 2
<
k=0
! C(r, η)e−
(1−4η)r 2
2T
(3.22)
4
3
We conclude the proof from (3.19)–(3.22).
Finally we complete the proof of the Proposition by proving the lemma below.
k
Lemma 3.3 Let U j = {x = (x1 , . . . , x2k −1 ) ∈ M 2 −1 : d(x j−1 , x j ) < r } for
j = 1, 2 . . . 2k (x0 = a and x2k = b). We can find a R2 > 0, for each 0 < r < R2 ,
there is a number T (r ) > 0, when T < T (r ), we have
m−1
.
k=0


2k ! 
6
7
.
k

T /2k , j
q(T /2k , r ) · 
Ex j−1 ,x j ! T /2 , j (F [k] )(ω1 , . . . ω2k )1{ω j (1/2)∈Uxr ,x }
j−1 j

j=1U
j
×
"
i2= j
T /2k


2k ,T

Pxi−1 ,xi (dωi ) µa,b (d x)

T
! T C(r )Ea,b
|D0 F|2H0
ω
Here C(r ) is independent of T .
Proof Following the notation from [11], let {h k, j ; k " 0, 1 ! j ! 2k } be the
orthonormal basis of H01,2 ([0, 1]; R) consisting of Schauder functions, i.e. h 0,1 (s)
= s ∧ (1 − s),
8
h k, j (s) = 2−k/2 h 0,1 (2k s − ( j − 1)) if s ∈ [( j − 1)2−k , j2−k ],
h k, j (s) = 0
otherwise
for k " 1 and 1 ! j ! 2k . Let d = dim(M). We choose {ei , 1 ! i ! d}, a family of
measurable vector fields on M with {ei (z); 1 ! i ! d} an orthonomal basis on Tz M
for every z ∈ M. These give rise to an orthonormal basis of Hω0 :
k, j,i
Zs
(ω) = h k, j (s)//1/2,s (ω)ei (ω(1/2)), s ∈ [0, 1], k " 0, 1 ! j ! 2k , 1 ! i ! d.
123
A concrete estimate for the weak Poincaré inequality on loop space
573
For each F ∈ FCb∞ ("a,b ), let
d 6
d 3
42
72 .
.
k, j,i
,k, j (F)(ω) =
d F(Z
D0 F, Z k, j,i 0 .
) =
i=1
Hω
i=1
Then we have
k
|D0 F(ω)|2H0
ω
=
∞ .
2
.
k=0 j=1
,k, j (F)(ω), ∀ω ∈ "a,b .
By Lemma 4.3 in [11], there exist constants R2 > 0, such that for each r ∈ (0, R2 ),
/(r ) > 0 such that when T < T
/(r )
there is a T
T
r } ! C(r ),0,1 (F)(ω) +
! (F)(ω)1{ω(1/2)∈Ua,b
∞
.
l=0
l
−l
C(r )(T + 2 )
2
.
,l,n (F)(ω),
n=1
for each smooth cylinder function F and ω ∈ "a,b ,
Thus, we obtain
! T /2
k, j
(F [k] )(ω1 , . . . ω2k )1{ω j (1/2)∈Uxr
k, j
! C(r ),0,1 (F [k] )(ω1 , . . . ω2k ) +
2l
×
.
j−1 ,x j
∞
.
l=0
}
C(r )(T /2k + 2−l )
k, j
,l,n (F [k] )(ω1 , . . . ω2k ).
(3.23)
n=1
k, j
Here ,l,n (F [k] ) means the corresponding operation ,l,n is taken with respect to the
jth subpath for function F [k] . Note that
k, j
,l,n (F [k] )(ω1 , . . . ω2k ) =
=
d 6
72 >
.
?
d F [k] (Z l,n,i ) ω1 , . . . , ω j−1 , •, ω j+1 , . . . ω2k
i=1
d
.
i=1
6
72
l
2k d F(Z l+k,( j−1)2 +n,i ) (ω)
= 2k ,l+k,( j−1)2l +n (F)(ω).
(3.24)
123
574
X. Chen et al.
The second equality above is due to the defintion of Z l,n,i and some time rescaling
procedure. By (3.23) and (3.24) we obtain,
m−1
.
k
k
q(T /2 , r )
k=0
!
2
.
j=1
m−1
.
k=0
8
k
! T /2 ,j (F [k] )(ω1 , . . . ω2k )1{ω j (1/2)∈Uxr
k
k
C(r )q(T /2 , r )2 +
k
.
C(r )(2
l=0
k−2l
j−1 ,x j
+T )q(T /2
k−l
}
,r )
G
k
2
.
,k,j (F)(ω).
j=1
(3.25)
Hk
Let g(k, T, r ) := C(r )q(T /2k , r )2k + l=0
C(r )(2k−2l + T )q(T /2k−l , r ), by the
/(r ), such that for each T < T (r ),
estimate (2.8) for q(T, r ) we can find a T (r ) < T
supk∈N g(k, T, r ) ! T C(r ), where C(r ) is a constant independent of T and k.
So by (3.25), for T < T (r ), we have,
m−1
.
k=0


2k ! 
6
7
.
k

T /2k , j
q(T /2k , r ) · 
Ex j−1 ,x j ! T /2 , j (F [k] )(ω1 , . . . ω2k )1{ω j (1/2)∈Uxr ,x }
j−1 j

j=1U
j
×
!
"
i2= j
m−1
.
k=0
!


T /2k
2k ,T

Pxi−1 ,xi (dωi ) µa,b (d x)

g(k, T, r )
k
2
.
j=1
T
Ea,b
[,k, j (F)]
T
|D0 F|2H0
T C(r )Ea,b
ω
4
3
4 An estimate over discretized loop space
r,N
of M N −1 as,
For each r ∈ R+ and integer N " 1, define the subset Ua,b
2
5
r,N
Ua,b
:= (x1 , . . . x N −1 ) ∈ M N −1 ; d(xi−1 , xi ) < r, 1 ! i ! N , x0 = a, x N = b .
(4.26)
N ,T
And recall that µa,b
is the probability measure on M N −1 defined in (3.18), which is
T . We have the
also the joint distribution of (ω(i/N ), i = 1, 2, . . . N − 1) under Pa,b
N ,1
.
following estimate of the variance with respect to µa,b
Proposition 4.1 Let M be a compact simply connected manifold with strict positive Ricci curvature. For any 0 < η < 1/8, 0 < r < R0 , there exists an integer
123
A concrete estimate for the weak Poincaré inequality on loop space
575
N1 (η, r ) > 0, such that for l > N1 (η, r ) there exists an integer N2 (η, l, r ) with the
property that if N > N2 (η, l, r )
N ,1
Var( f ; µa,a
)
! C(l, r )N C(l,r ) e2N ηr
2
N
−1
.
i=1
+C(l, η, r )N C(l,r ) e
!
M N −1
2
− N (1−8η)r
2
N ,1
|∇i f |2 dµa,a
|| f ||2∞
r,N
for all f ∈ C ∞ (M N −1 ) with supp( f ) ⊂ U a,a .
Proof Step 1 For any integers N > l > 1 and a function f ∈ C ∞ (M N −1 ). Define a
: : "a,a %→ R as,
smooth cylinder function F
:
F(ω)
:= f (ω(1/N ), . . . , ω(1 − 1/N )).
(4.27)
For x = (x1 , . . . , x N −l ) ∈ M N −l we define a function fl : M N −l → R as following,
fl (x1 , . . . , x N −l ) =
!
l,
l
f (x1 ,. . . ,x N −l , y1 ,. . . ,yl−1 ) µx NN−l ,a (dy1 . . . dyl−1 ) (4.28)
M l−1
In fact
fl (x1 , . . . x N −l ) = Ea,a
0
:
F(ω)
| ω(1/N ) = x1 , . . . , ω
<
N −l
N
=
1
= x N −l .
For such x define a function on "x N −l,a by
/l (x1 , . . . x N −l , ω) := f (x1 , . . . x N −l , ω(1/l), . . . , ω(1 − 1/l)), ω ∈ "x N −l ,a .
F
(4.29)
Clearly
l
I
J
/l (x1 , . . . x N −l , •) .
fl (x1 , . . . x N −l ) = ExNN −l ,a F
N ,l,T
Let ℘l = σ {ω(i/N ), 1 ! i ! N − l}, an σ -algebra on "a,a and µa,a
be the
probability measure on M N −l :
N ,l,T
µa,a
(d x1 , . . . , d x N −l )
p lT (a, x N −l ) p T (x N −l , x N −l−1 ) · · · p T (x1 , a)
N
N
d x N −l . . . d x1 . (4.30)
= N
pT (a, a)
123
576
X. Chen et al.
We have
N ,1
)
Var( f ; µa,a
: Pa,a )
= Var( F;
6
7
6
7
: − Ea,a [ F|℘
: l ])2 + Ea,a (Ea,a [ F|℘
: l ] − Ea,a [ F])
: 2
= Ea,a ( F
=
<
!
4
3
l
N ,l,1
N ,l,1
/l ; PxNN −l ,a µa,a
(4.31)
Var F
(d x) + Var fl , µa,a
=
M N −l
N ,l,1
Step 2 Now we are going to estimate Var( fl , µa,a
). Let Pa1 be the distribution of a
standard Brownnian motion on compact manifold M starting from a with time parameter 1, which is a probability measure on the path space "a over M with starting point
a and time 1. Let
γaN ,l,1 (d x1 , . . . d x N −l ) := p 1 (a, x1 ), . . . p 1 (x N −l−1 , x N −l )dx1 , . . . dx N −l
N
N
be a probability measure on M N −l , which is the joint distribution of (ω(i/N ), ω ∈
"a , i = 1, 2, . . . N − l) under Pa1 . By the Poincaré inequality for Pa1 on the path space
over compact manifold [15] we get,
Var( fl ; γaN ,l,1 ) = Var(F l ; Pa1 ) ! CEa1 |DF l |2Hω ! C N
N −l !
.
i=1
M N −l
|∇i fl |2 dγaN ,l,1
where F l (ω) := fl (ω(1/N ), . . . ω(1 − l/N )) for ω ∈ "a and D is the gradient operator related to Bismut tangent norm |.|Hω in path space over compact manifold M. We
also use the relation
|DF l (ω)|2Hω ! (N − l)
N −l
.
i=1
|∇i fl (ω(1/N ), . . . ω(1 − l/N ))|2 , ω ∈ "a ,
in above inequality which can be checked by direct computation. Thus, we have
N ,l,1
Var( fl ; µa,a
)
= Var
K
fl ;
p l (a, x N −l )
N
p1 (a, a)
γaN ,l,1
L
N −l !
4
3
.
! C · osc p l (a, ·) · N ·
N
123
i=1
M N −l
N ,l,1
|∇i fl |2 dµa,a
. (4.32)
A concrete estimate for the weak Poincaré inequality on loop space
supx∈M g(x)
inf x∈M g(x)
Here for g a function on M, osc(g(·)) :=
Varadhan’s estimate (3.13), if
l
N
577
is the oscillation of g. By
< T1 (η, r ) then
4
3
N 2 2 D2
osc p l (a, ·) ! e l (η r + 2 )
N
for D the diameter of the compact manifold M. So by this and (4.32) when
l
N < T1 (η, r ), we have
Var
3
N ,l,1
fl ; µa,a
4
! C Ne
N
l
2
(η2 r 2 + D2 )
N −l !
.
i=1
M N −l
N ,l,1
|∇i fl |2 dµa,a
.
(4.33)
Step 3 Now we are going to estimate |∇i fl | in terms of f . It is easy to see that for
i < N − l,
|∇i fl |2 (x1 , . . . x N −l ) !
!
M l−1
l,
l
|∇i f |2 (x1 , . . . x N −l , y1 , . . . yl−1 )µx NN−l ,a (dy)
(4.34)
and for i = N − l,
|∇ N −l fl |2 (x1 , . . . x N −l )
= sup d N −l fl (v)
|v|=1
!2
!
M l−1
l,
l
|∇ N −l f |2 (x1 , . . . x N −l , y1 , . . . yl−1 )µx NN−l ,a (dy)
& < l
&2
=
&
&
N
&
/
+2 sup &dz Ez,a ( Fl ) |z=x N −l (v)&& .
(4.35)
|v|=1
/l (ω) := f (x1 , . . . x N −l , ω(1/l), . . . ω(1 − 1/l)) is as defined in (4.29). As
Here F
d(a, x N −l ) ! r may not hold, we can not choose the vector constructed in Lemma 2.2
when we apply Lemma 2.3 to estimate the differentiation. To estimate the differentiation of the expectation with respect to the starting point we choose another vector
field X l,v (s) := //s (1 − ls)+ v, 0 ! s ! 1. Recall that (Bs ) is the stochastic antil
development, in the expression for δ N X in Lemma 2.1, and
!s 3
4
//u−1 ∇log p (1−u)l (ω(u), a) du, 0 ! s < 1
Bs = βs +
N
0
123
578
X. Chen et al.
for some process (βs ) whose distribution is the Brownnian motion with time parameter
l
N
l
under the probability measure PxNN −l ,a (see [9]). We get,
=
<
3 l
42
l
l
l
Var δ N X l,v ; PxNN −l ,a ! ExNN −l ,a δ N X l,v
!E
l
N
x N −l ,a

1
=
!l <
1

−N v + Ricω(s) (//s (1 − ls)v)

2
0
3
4
2

× dβs + //s−1 ∇log p (1−s)l (ω(s), a)ds 
N
! C(l)N 4 , v ∈ Sx N −l M.
+
In the last step we used the estimate |∇log ps (x, a)| ! C[ d(x,a)
s
compact manifolds. Also note that
have,
X l,v ( il )
√1 ](s
s
> 0) for
= 0 for 1 ! i ! l. By Lemma 2.3, we
&
= &2
< l
&
&
/l ) (v)&
sup &&d N −l ExNN −l ,a ( F
&
|v|=1
8&
& 0
=11/2 0
<
=11/2 G2
<
l
l
&
& Nl
l
l,v &
l,v
N
N
&
/
/
! sup &Ex N −l ,a [d Fl (X )]& + Var δ N X ; Px N −l ,a
Var Fl ; Px N −l ,a
|v|=1
<
=
l
/l ; PxNN −l ,a .
! C(l)N 4 Var F
(4.36)
Using (4.34)–(4.36) we derive the following estimate
N −l
.
i=1
|∇i fl |2 (−) ≤
N −l !
.
i=1
M l−1
=
<
l
l, l
/l ; PxNN −l ,a .
|∇i f |2 (−, y)µx NN−l ,a (dy) + C(l)N 4 Var F
then from that Combining this with (4.31) and (4.33), we obtain the following,
N ,1
Var( f ; µa,a
)
! C(l)N exp
0
<
N
l
<
η2 r 2 +
+ 1 + C(l)N 5 exp
123
<
D2
2
== .
N −l !
i=1
M N −1
N ,1
|∇i f |2 µa,a
(d x)
<
==1 !
N
D2
η2 r 2 +
l
2
M N −l
=
<
l
N ,l,1
/l ; PxNN −l ,a µa,a
Var F
(d x)
(4.37)
A concrete estimate for the weak Poincaré inequality on loop space
579
l
/l ; PxNN −l ,a ) by the variance with respect to the
Step 4 We estimate the variance Var( F
Brownian bridge measure. Note that
<
l,
l
l
/l ; PxNN −l ,a
Var F
=
= Var
l,
<
l, l
f (x1 , . . . x N −l , •, . . . , •); µx NN−l ,a
=
(4.38)
.
l
Let µx NN−l ,a be normalization of µx NN−l ,a on the subset Uxr,lN −l ,a of M l−1 , i.e.
l,
l
l,
l
l,
l
µx NN−l ,a (A) = µx NN−l ,a (A)/µx NN−l ,a (Uxr,lN −l ,a ),
A ⊆ Uxr,lN −l ,a .
r,l
For each smooth function g with support in U x N −l ,a , we have,
<
<
=
=
3
4
l, l
l, l
l, l
Var g; µx NN−l ,a ! µx NN−l ,a Uxr,lN −l ,a Var g; µx NN−l ,a
<
=
l, l
1 − µx NN−l ,a (Uxr,lN −l ,a )
+
||g||2∞ .
l, Nl
r,l
µx N −l ,a (Ux N −l ,a )
By asymptotic property (3.13), when
l,
l
l,
l
l
N
(4.39)
< T1 (η, r ), it satisfies that,
1 − µx NN−l ,a (Uxr,lN −l ,a )
= µx NN−1 ,a ({z : d(z i , z i+1 ) > r for some 0 ! i ! l − 1})
A
l−1
.
d(z i ,z i+1 )>r p N1 (x N −l , z 1 ) . . . p N1 (z l−1 , a)dz 1 , . . . dz l−1
!
p l (x N −l , a)
i=0
!l·
N
2
exp(− (1−4η)Nr
)
2
.
N 2 2
exp(− 2l (η r + D 2 ))
2 2
(4.40)
2
If we choose l sufficient large, e.g. η r l+D < 2(1 − 4η)r 2 , there is an integer
/(η, l, r ), such that whenever N > N
/(η, l, r )
N
l,
l
µx NN−l ,a (Uxr,lN −l ,a ) >
1
2
(4.41)
r,N
Since the support of f is a subset of U a,a , then for each fixed x1 , . . . x N −l ,
r,l
supp( f (x1 , . . . , x N −l , •, . . . , •)) ⊂ U x N −l ,a .
123
580
X. Chen et al.
Hence by (4.39)–(4.41), for each integer l sufficiently big, there exists an integer
/(η, l, r ), for each N > N
/(η, l, r ), we have,
N
=
<
l, l
Var f (x1 , . . . x N −l , •, . . . , •); µx NN−l ,a
!
1
l, Nl
x N −l ,a
λ(Uxr,lN −l ,a ; µ
+ 2l ·
)
×
l−1 !
.
l,
l
|∇ N −i f |2 (x1 , . . . x N −l , z 1 , . . . zl−1 )µx NN−l ,a (dz)
i=1
2
exp(− (1−4η)Nr
)
2
N 2 2
exp(− 2l
(η r + D 2 ))
|| f ||2∞ ,
(4.42)
where
=
<
l, Nl
r,l
λ Ux,a ; µx,a :=
inf
r,l
g∈C0∞ (Ux,a
)
A
l,
l
|∇g|2 dµx,aN
= .
<
l, l
Var g; µx,aN
r,l
Ux,a
(4.43)
/(η, l, r ) and
Therefore, by (4.37) and (4.42), we have for each l big enough, N > N
l
<
T
(η,
r
),
1
N
3
4
N ,1
Var f ; µa,a


< 3
=
4


N 2 2
N4
=
<
! C(l)N exp
η r + D 2 /2 
1+


l
l
l,
r,l
inf x∈M λ Ux,a
; µx,aN
×
N
−1
.
i=1
!
M N −1
N ,1
|∇i f |2 dµa,a
==1
0
< <
3η2 r 2 + 2D 2
(1 − 4η)r 2
5
+
|| f ||2∞ .
+ C(l)N exp N −
2
l
l,
l
(4.44)
r,l
Step 5 Finally, by (4.44) and the uniform estimate of λ(Ux,a
; µx,aN ) in x derived
in Lemma 4.2 below, for each integer l sufficiently big, there exists an integer
N2 (η, l, r ) > 0, such that if N > N2 (η, l, r ), then
4
3
N ,1
Var f ; µa,a
==1
0
< <
L(ε) + η2 r 2 + D 2 /2
+ 4Dε
! C(l, r )N C(l,r ) exp N
l
!
N
−1
.
N ,1
×
|∇i f |2 dµa,a
i=1
123
M N −1
A concrete estimate for the weak Poincaré inequality on loop space
581
0
< <
==1
3η2 r 2 + 2D 2
(1 − 4η)r 2
+
+ C(l, η, r )N C(l,r ) exp N −
|| f ||2∞ .
2
l
(4.45)
Note that the constants C and L in the inequality above do not depend on N , and L
does not depend on l and the starting point a. So for any fixed η > 0, 0 < r < R0 ,
2
2
we first choose a ε = ηr
4D to make 4Dε = ηr , then take a l big enough such that
L(ε)+η2 r 2 +D 2 /2
l
2 2
2
2
< ηr 2 and 3η r l+2D < ηr 2 for the chosen ε = ηr
4D (i.e. l > N0 (η, r )
for some constant N0 (η, r ) which only depends on η and r ). Hence by (4.45), there is
a constants N1 (η, r ), such that for each integer l > N1 (η, r ), there exists an integer
N2 (η, l, r ) > 0, such that if N > N2 (η, l, r ) then we have, as required,
3
4
N ,1
Var f ; µa,a
! C(l,r )N C(l,r ) e2N ηr
2
N
−1
.
i=1
!
M N−1
N ,1
|∇i f |2 dµa,a
+C(l,η,r )N C(l,r ) e−
N (1−8η)r 2
2
|| f ||2∞ .
Lemma 4.2 Let M be a compact simply connected manifold with strict positive
l,
l
r,l
Ricci curvature. For a, x ∈ M, r < R0 and N ∈ N+ , λ(Ux,a
; µx,aN ) as defined
l
in (4.43), there exists a constant T (l, r ), such that when N < T (l, r ), for each ε > 0
small enough,
=
=
=
<
< <
C(l, r )
L(ε)
l, l
r,l
+ 4Dε · N .
inf λ Ux,a
; µx,aN " C(l,r ) exp −
x,a∈M
l
N
where the constant C(l, r ) only depends on l, r and the constant L(ε) only depends
on ε, not on l.
l,T
Proof Step 1 Following [12] define a measure νa,b
on M l−1 as,
3
4
l,T
l
(dz) = exp −E a,b
(z)/T dz,
νa,b
where z = (z 1 , . . . , zl−1 ) ∈ M l−1 , dz = 2l−1
j=1 dz j and
l−1
l
(z 1 , . . . zl−1 ) =
E a,b
l .
d(z i , z i+1 )2 , z 0 = a, zl = b,
2
i=0
l,T
r,l
l−1 normalized to
Let ν l,T
a,b (dz) be the restirction of νa,b (dz) on the subset Ua,b of M
have mass 1. From [12, lemma 3.2], for each fixed l > 0,
4
3
l,T
lim sup sup osc dµl,T
a,b /dνa,b ! C(l, r ),
T ↓0 a,b∈M
r,l
Ua,b
123
582
X. Chen et al.
So, there is a T (l, r ) > 0 such that for any
=
<
l, Nl
r,l
λ Ux,a ; µx,a "
l
N
< T (l, r ),
=
<
1
l, Nl
r,l
λ Ux,a ; ν x,a .
2C(l, r )
(4.46)
r,l
r,l
r,l
Step 2 As in [12], let Ua,b,8
:= U a,b / ∼ be the one point compactification of Ua,b
,
r,l
which is obtained by identifying the boundary ∂Ua,b
as a single point 8. And let
r,l
r,l
/
C([0, 1]; U a,b ) denote the path in U a,b which is restricted to a continuous path on the
r,l
. Define
space Ua,b,8
r,l
Ma,b
(z) := inf
l
sup E a,b
( p(s)) a, b ∈ M,
s∈[0,1]
p∈Ir,l
a,b
(4.47)
r,l
/
where Ir,l
a,b = { p ∈ C([0, 1]; U a,b ); p(0) = z, p(1) = z 0 } and z 0 is a minimum point
r,l
l on U
of E a,b
a,b . And define
3
4
r,l
l
m r,l
a,b := sup Ma,b − E a,b
(4.48)
r,l
U a,b
l
In fact, if we take the supremum only among the local minimum points of E a,b
on
r,l
U a,b in the above definition, the value of m r,l
a,b will not change, see lemma 2.1 in [12].
According to the proof of Theorem 2.2 in [12], for each x, a ∈ M, if Nl is less than
some T (x, a, l),
K
L
=
< =3(l−1)d−2
<
l
N m r,l
l, Nl
x,a
r,l
exp −
λ Ux,a ; ν x,a " C(x, a, l)
, x ∈ M,
N
l
where d is the dimension of M.
Now our goal is to confirm that the constants T (x, a, l), C(x, a, l) above can be
chosen to be independent of x, a ∈ M. From step by step checking the proof Theorem
2.2 in [12], if the following three conditions are true, then we can find such constants:
(a) Uniform estimate on the gradient of the energy function: there exists a constant
C(l) > 0 depending only on l such that
sup
l
sup |∇ E x,a
(z)|2 ! C(l).
x,a∈M z∈U r,l
x,a
r,l
(b) A lower bound on the size of the tube Ux,a
: there exists a constant θ (l) > 0, such
that for all R < 1,
sup
sup
x,a∈M z∈∂U r,l
x,a
123
r,l
V ol(B R (z)/Ux,a
)
" θ (l),
V ol(B R (z))
A concrete estimate for the weak Poincaré inequality on loop space
583
where V ol(A) denotes the Riemannian volume of a subset A of M l−1 .
r,l
(c) If T is smaller than some T (l) > 0, there are finite subsets 5T0 (x, a) ⊂ ∂Ux,a
r,l
and 5T (x, a) ⊂ U x,a such that
• 5T0 (x, a) ⊂ 5T (x, a)
l .
• 5T (x, a) contains a minimum point z 0 (x, a) of E x,a
;
;
r,l
r,l
⊆ z∈5 0 (x,a) BT (z), U x,a ⊆ z∈5T (x,a) BT (z).
• ∂Ux,a
T
• supx,a∈M #5T (x, a) ! C(l)T −(l−1)d for some constants C(l).
where # means the number of elements in a finite set.
Since R0 from proposition 3.2 is less than the injective radius of compact manifold
r,l
l
is differentiable in the domain Ux,a
and condition (a) can
M, when r ∈ (0, R0 ), E x,a
be checked by direct computation. From the proof of Corollary 3.3 in [12], condition
(b) is true.
For condition (c), note that there is a T (l) > 0, for each T < T (l), due
; to the com/T ⊆ M such that M l−1 ⊆ z∈5
pactness of M l−1 , we can find a finite subset 5
/T BT (z)
;
/T ! C(l)T −(l−1)d . Now since M l−1 ⊆ z∈5
and #5
B
(z),
we
start
to
construct
/T /2 T /2
the set 5T (x, a) as following:
r,l
/T /2 and BT /2 (z) ⊂ Ux,a
(i) if z ∈ 5
, then add such z into 5T (x, a);
r,l
r,l
/T /2 and BT /2 (z) ∩ ∂Ux,a
(ii) if z ∈ 5
2= ∅, then take a point z̃ ∈ BT /2 (z) ∩ ∂Ux,a
and add this point z̃ into 5T (x, a).
r,l
l
on U x,a into 5T (x, a).
(iii) add a minimum point z 0 (x, a) of E x,a
Since in (ii), BT (z̃) ⊇ BT /2 (z), we have
;
;
z̃∈5T (x,a)
BT (z̃)
r,l
z̃∈5T (x,a)∩∂Ux,a
⊇
BT (z̃) ⊇
S
r,l
/T /2
z∈5
BT /2 (z) ⊇ M l−1 ⊇ U x,a
/T /2 ;
z∈5
S
r,l
BT /2 (z)∩∂Ux,a
2=∅
r,l
BT /2 (z) ⊇ ∂Ux,a
, x, a ∈ M
/T /2 + 1 ! 2(l−1)d C(l)T −(l−1)d , so condition (c) is satisfied.
and #5T (x, a) ! #5
By the above argument, we can find constants T (l) and C(l), which are independent
of x, a and N , such that if Nl < T (l), then
K
L
=
< =3(l−1)d−2
<
l
N m r,l
l, Nl
x,a
r,l
λ Ux,a ; ν x,a " C(l)
exp −
.
N
l
(4.49)
Step 3 In the following, we try to give some uniform estimate about m r,l
x,a . As in [12],
define the energy of a path γ ∈ "a,b (possibly infinite) as:
k−1
E(γ ) :=
. d(γ (si ), γ (si+1 ))2
1
sup
2
si+1 − si
i=0
123
584
X. Chen et al.
where the supremum is obtained over all partitions 0 = s0 < s1 < · · · sk = 1. Assume
a, b ∈ M and a is not conjugate to b, let 6a,b denote the set of all geodesics (i.e. critimin denote the subset of all local energy minimum.
cal points of E on "a,b ), and let 6a,b
Fix a global energy minimum geodesic γa,b ∈ "a,b , then for each geodesic γ ∈ 6a,b ,
we define:
Ma,b (γ ) := inf sup E ◦ H (s)
H ∈I s∈[0,1]
where I = {H ∈ C([0, 1], "a,b ); H (0) = γ , H (1) = γa,b }. And define
5
2
min
.
m a,b := sup Ma,b (γ ) − E(γ ); γ ∈ 6a,b
The item m a,b can be viewed as an infinite dimensional version of the item (4.48).
r,l
Furthermore, every point z ∈ Ua,b
corresponds to a piecewise geodesic in M, so intuitively we may have more choices to take supremum in defining Ma,b than in defining
r,l
Ma,b
as (4.47). In fact, according to the proof of Corollary 1.5 in [12], we have, if a
is not conjugate to b,
+
m r.l
a,b ! m a,b , r ∈ (0, injM), l ∈ N .
(4.50)
For 0 < r < R0 , choose a ε > 0, satisfying with r + ε < injM. For any x ∈ M,
r,l
a ∈ M and x̃ ∈ Bε (x), ã ∈ Bε (a), if z = (z 1 , . . . zl−1 ) ∈ Ux̃,
ã , then
d(z 1 , x) ! d(x, x̃) + d(z 1 , x̃) < r + ε
and d(z i , z i+1 ) < r, 1 ! i ! l − 2
d(zl−1 , a) ! d(a, ã) + d(zl−1 , ã) < r + ε
r,l
r +ε,l
r +ε,l
which means z ∈ Ux,a
, hence we have U x̃,ã ⊆ Ux,a
.
r,l
l
Suppose z 0 (x̃, ã) be a minimum point of E x̃,
ã on U x̃,ã , and z 0 (x, a) be a minr +ε,l
r,l
l
imum point of E x,a
on U x,a , by the definition of Ma,b
in (4.47), for each δ > 0
r,l
r +ε,l
r +ε,l
/
and each z ∈ U x̃,ã ⊆ Ux,a , there exists a path q1 ∈ C([0, 1]; U x,a ), such that
q1 (0) = z, q1 (1) = z 0 (x, a), and
l
l
+ε,l
E x,a
◦ q1 (s) ! E x,a
(z) + m rx,a
+ δ, 0 ! s ! 1
(4.51)
r +ε,l
/
As the same reason, we can find a a path q2 ∈ C([0,
1]; U x,a ) with q2 (0) =
z 0 (x̃, ã), q2 (1) = z 0 (x, a) and
l
l
+ε,l
E x,a
◦ q2 (s) ! E x,a
(z 0 (x̃, ã)) + m rx,a
+ δ, 0 ! s ! 1.
Let
8
q1 (2s)
if 0 < s ! 21 ,
q(s) =
q2 (2 − 2s) if 21 < s ! 1
123
(4.52)
A concrete estimate for the weak Poincaré inequality on loop space
585
r,l
r,l
and τ = inf{s; q(s) ∈ ∂Ux̃,
ã } ∧ 1, τ̂ = sup{s; q(s) ∈ ∂U x̃,ã } ∨ 1. Define
8
q(s) if s ∈ [0, τ ) ∪ (τ̂ , 1],
/
q (s) =
q(τ̂ ) if s ∈ [τ, τ̂ ].
r,l
r,l
/
Then q̃ ∈ C([0,
1]; U x̃,ã ) and /
q (0) = z, /
q (1) = z 0 (x̃, ã). Note that for each z ∈ U x̃,ã ,
l
l
|E x̃,
ã (z) − E x,a (z)|
&
&
& l(d(z 1 , x)2 − d(z 1 , x̃)2 ) l(d(zl−1 , a)2 − d(zl−1 , ã)2 ) &
&
&
+
=&
&
2
2
! (d(a, ã) + d(x, x̃))Dl ! 2l Dε
(4.53)
where D is the diameter of the manifold M. Then, by (4.51), (4.52), (4.53) and the
definition of q̃, we have
l
l
E x̃,
q (s) ! E x,a
◦/
q (s) + 2l Dε
ã ◦ /
2
5
l
l
+ε,l
! max E x,a
(z), E x,a
(z 0 (x̃, ã)) + m rx,a
+ δ + 2l Dε
2
5
l
l
r +ε,l
! max E x̃,
ã (z), E x̃,ã (z 0 ( x̃, ã)) + m x,a + δ + 4l Dε
l
r +ε,l
= E x̃,
ã (z) + m x,a + δ + 4l Dε, 0 ! s ! 1.
The equality in the last step above is due to the fact that z 0 (x̃, ã) is a minimum point
r,l
r,l
l
of E x̃,
ã on U x̃,ã . Thus, according to the above inequality and the definition of M x̃,ã ,
r,l
r +ε,l
l
and by the arbitrary of δ, we obtain Mx̃,
ã (z) ! E x̃,ã (z) + m x,a + 4l Dε. Hence, by
this (4.50) and the definition of m r,l
x̃,ã , when a is not conjugate to x, d(x, x̃) < ε and
d(a, ã) < ε, we have
r +ε,l
m r,l
x̃,ã ! m x,a + 4l Dε ! m x,a + 4l Dε.
(4.54)
By [12, Theorem 1.4], when M is a compact simply connected manifold with strict
positive Ricci curvature, we have m a,b < ∞ for each pair of a, b ∈ M if a is not
conjugate to b. Since for any ε > 0, ;
a ∈ M, there exists a finite set 8ε,a ⊆ {x ∈
M : x is not conjugate to a} such that x∈8ε,a Bε (x) ⊇ M, then by (4.54), for each
a, b ∈ M with d(a, b) < ε,
sup m r,l
y,b ! sup m x,a + 4l Dε.
y∈M
(4.55)
x∈8ε,a
As the same way, there is a finite set 8ε , such that
(4.55),
;
x∈8ε
Bε (x) ⊇ M, by (4.54) and
sup m r,l
y,b ! sup sup m x,a + 4l Dε.
y,b∈M
(4.56)
a∈8ε x∈8ε,a
123
586
X. Chen et al.
Let
L(ε) := sup sup m x,a < +∞.
a∈8ε x∈8ε,a
So, by (4.46), (4.49) and (4.56), if
l
N
less than some T (l, r ), then
=
=
=
<
< <
C(l, r )
L(ε)
l, l
r,l
inf λ Ux,a
+ 4Dε · N .
; µx,aN " C(l,r ) exp −
x,a∈M
l
N
(4.57)
where constant C(l, r ) only depends on l, r , by now we have completed the proof. 3
4
5 The main theorem
Theorem 5.1 Let M be a simply connected compact manifold with strict positive
Ricci curvature. For any small α > 0, there exists a constant s0 > 0 such that the
following weak Poincaré inequality holds, i.e.
Var(F; Pa,a ) !
1
Ea,a (F, F) + s||F||2∞ , s ∈ (0, s0 ), F ∈ D(Ea,a ). (5.58)
sα
The constants s0 does not depend on the starting point a ∈ M.
Proof It suffices to show that (5.58) holds for F ∈ FCb∞ ("a,a ). Let ωi (s) :=
∞
ω( i−1+s
N ) for each ω ∈ "a,a . For a function F ∈ FC b;("a,a ), as in the proof of PropoN "
[N
]
sition 3.2, there is a unique function F defined on (x1 ,...,x N −1 )∈M N −1 2i=1
xi−1 ,xi
such that,
F [N ] (ω1 , ω2 , . . . , ω N ) = F(ω), ω ∈ "a,a ,
Step 1 We fix N > N2 (η, l, r ) with l > N1 (η, r ), for N1 (η, r ) and N2 (η, l, r ) as given
in Proposition 4.1. We first assume F(ω) = 0 if (ω(1/N ), ω(2/N ), . . . ω(1 − 1/N ))
r,N
. For (x1 , . . . , x N −1 ) ∈ M N −1 ,
is not in Ua,a
f [N ] (x1 , x2 , . . . x N −1 ) :=
= Ea,a
123
0
!
F [N ]
N
"
i=1
1
PxNi−1 ,xi (dωi )
1
=
<
& <1=
N −1
&
F(ω) & ω
= x1 , . . . , ω
= x N −1 .
N
N
A concrete estimate for the weak Poincaré inequality on loop space
587
Let = N := σ {ω(i/N ), 1 ! i ! N − 1}, an σ - algebra on "a,a . Then we have,
Var(F; Pa,a )
6
7
6
7
= Ea,a (F − Ea,a [F|= N ])2 + Ea,a (Ea,a [F|= N ] − Ea,a [F])2
=
!
!
M N −1
=
<
4
3
1
N −1 N
N ,1
N ,1
Var F [N ] ; ⊗i=0
Pxi ,xi+1 dµa,a
+ Var f [N ] , µa,a

! .
N !
r,N
Ua,a

Var j
j=1
<
F [N ] ; P
="
1
N
x j−1 ,x j
P
1
N
xi−1 ,xi
i2= j
4
3
N ,1
.
+Var f [N ] ; µa,a


N ,1
(dωi ) µa,a
(d x)

(5.59)
Here Var j indicates the variance is with respect to the jth subpath. Note that f [N ]
r,N
is smooth with support in U a,a and || f [N ] ||∞ ! ||F||∞ . From Proposition 4.1, if
N > N2 (η, l, r ), then
4
3
N ,1
Var f [N ] ; dµa,a
! C(l, r )N C(l,r ) e2N ηr
2
N
−1
.
i=1
+C(l, η, r )N C(l,r ) e
!
M N −1
−N (1−8η)r 2
2
N ,1
|∇i f [N ] |2 dµa,a
||F||2∞ .
(5.60)
According to the proof of lemma 6.1 and lemma 6.2 in [11] (since the support of f [N ]
r,N
is in U a,a , we can choose some vector with better asymptotic property in the estimate
of the derivative of expectation with pinned Wiener measure as in Lemma 2.2), there
exists a constant C(r ), such that
N
−1 !
.
i=1
N ,1
! C(r )N Ea,a |D0 F|2H0
|∇i f [N ] |2 dµa,a
+C(r )N
ω

! .
N !
r,N
Ua,a

j=1
Var j
<
F [N ] ; P
1
N
x j−1 ,x j
="
i2= j
P
1
N
xi−1 ,xi


N ,1
(dωi ) µa,a
(d x).

(5.61)
123
588
X. Chen et al.
By Proposition 3.2, if

! .
N !
r,N
Ua,a
!

Var j
j=1
C(r )
N
! .
N
j=1
1
N
<
< T0 (η, r ), then
F [N ] ; P
1
N
x j−1 ,x j
="
P
1
N
xi−1 ,xi
i2= j


N ,1
(dωi ) µa,a
(d x1 , . . . d x N −1 )

|D0,( j) F [N ] (ω1 , . . . ω N )|2ω j Pa,a (dω)
+N C(η, r )e−
(1−4η)Nr 2
2
||F||2∞ ,
(5.62)
where D0,( j) means the gradient D0 of F [N ] with respect to the jth subpath. According
to (6.16) in the proof of Lemma 6.3 in [11], we have the following relation,
N
.
j=1
|D0,( j) F [N ] (ω1 , . . . ω N )|2ω j ! N |D0 F|2H0 , ω ∈ "a,a .
ω
(5.63)
By (5.59)–(5.63), if N > N2 (η, l, r ) with l > N1 (η, r ), then
4
3
2
1
! C(l, r )N C(l,r ) e2N ηr Ea,a |D0 F|2H0
Var F; Pa,a
ω
+C(l, η, r )N
C(l,r )
e
−N (1−8η)r 2
2
||F||2∞ .
(5.64)
Step 2 Consider a general function F from FCb∞ ("a,a ). Define a smooth cut-off
function on "a,a as,
=
< = ==
< < <
N
"
i
i −1
, ω
ϕ d ω
Ψ N (ω) :=
N
N
i=1
where ϕ is defined as in the proof of Lemma 3.1. By the proof of Lemma 3.1, if
1
N < T0 (η, r ),
=
<
(1 − 4η)Nr 2
6N
, |D0 Ψ N (ω)|Hω0 !
.
Pa,a (Ψ N 2= 1) ! N exp −
2
ηr
(5.65)
Then note that FΨ N (ω) = 0 when (ω(1/N ), ω(2/N ), . . . ω(1 − 1/N )) is not in
r,N
, hence by (5.64) and (5.65), if N > N2 (η, l, r ) with l > N1 (η, r ) , we obtain
Ua,a
123
A concrete estimate for the weak Poincaré inequality on loop space
589
Var(F; Pa,a )
! Var(FΨ N ; Pa,a ) + 3Pa,a (Ψ N 2= 1)||F||2∞
2
! C(l, r )N C(l,r ) e2N ηr Ea,a |D0 (FΨ N )|2H0 + C(l, η, r )N C(l,r ) e
−N (1−8η)r 2
2
ω
2
! C(l, r )N C(l,r ) e2N ηr Ea,a |D0 F|2H0 + C(l, η, r )N C(l,r ) e−
ω
N (1−8η)r 2
2
||F||2∞
||F||2∞ .
(5.66)
(1−8η)Nr 2
Let s := C(l, η, r )N C(l,r ) e− 2
in (5.66), then s tends to zero when N tends to
infinity. In particular for any small α > 0 we can choose a η small enough so that
there is a constant s0 (η, r, l, ε, α), s0 does not depend on the starting point a of the
loop space, such that,
Var(F; Pa,a ) !
1
Ea,a (F, F) + s||F||2∞ , s ∈ (0, s0 ), F ∈ D(Ea,a ).
sα
4
3
By now we have completed the proof.
Remark (1) The algebraic rate of blowing up in this weak Poincaré inequality
is the consequence of the exponential growth in N of the constant ξ(N ) for
the Dirichlet form item in the variance estimate for the discretized Brownnian bridge measure on the product space M N −1 . (cf. Proposition 4.1, where
2
ξ(N ) = C(l, r )N C(l,r ) e2N ηr ). In one step of the proof of Proposition 4.1, we
derive an estimate by comparison with the path space where Poincaré inequality
is known to hold. This estimate is not sharp. In fact, we think that maybe for some
manifolds with special properties, a better estimate of ξ(N ) can be obtained by
using these special properties (e.g. symmetric property of the sphere). A better
estimate of ξ(N ) would result in a lower blowing up rate in the weak Poincaré
inequality.
(2) We take the exhausting local sets {Or,N } to prove the weak Poincaré inequality,
where
8
Or,N = ω ∈ "a,a
G
<
==
< < =
i +1
i
,ω
<r
: sup d ω
N
N
0!i !N −1
;
and ∞
N =N0 Or,N = "a,a for each N0 > 0. Note that these exhausting local sets
are slightly different from that taken by Eberle in [11]. We make this choice just
for technical reason. It is easier to estimate the probability of the complement of
these exhausting local sets.
(3) The positive Ricci curvature condition in the main theorem can be replaced by
m x,y < +∞ for each x, y ∈ M satisfying that x is not conjugate to y. This condition is crucial to obtain the estimate in Proposition 4.1. A better understanding
of the geometric meaning of m x,y would be useful to get a sharper estimate for
some explicit manifolds. And from the analysis of the Section 1.4 of [12], for a
compact simply connected manifold M and x, y in M not conjugate with each
other, m x,y < +∞ if there only exist a finite number of geodesics connecting
123
590
X. Chen et al.
x and y, which are the local minimum of the energy functional on "x,y . In particular, by the proof of Theorem 1.4 in [12], if M is compact simply connected
and has strict positive Ricci curvature, the above assumption is satisfied and
m x,y < +∞ for any x, y ∈ M satisfying that x is not conjugate to y.
Acknowledgments We would thank the referees for useful comments and we would thank Feng-Yu Wang
for inspiring discussions and Courant Institute for its hospitality during the completion of the work.
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