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Logarithmic Sobolev inequality on free path space over a compact Riemannian manifold
Statistics and Probability Letters 98 (2015) 12–19
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Logarithmic Sobolev inequality on free path space over a compact Riemannian manifold Ling Pei Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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Article history: Received 23 September 2014 Received in revised form 2 December 2014 Accepted 5 December 2014 Available online 13 December 2014 Keywords: Logarithmic Sobolev inequality Integration by parts Free path space Martingale representation
abstract In this paper we use Bismut’s method to verify the formula of integration by parts on the free path space. By the formula, we verify the martingale representation for reference measure Pµ , the law of the Brownian motion on the base manifold with initial distribution µ. Then by the martingale representation we get the logarithmic Sobolev inequality on compact free Riemannian path space. © 2014 Published by Elsevier B.V.
1. Introduction Integration by parts is an important topic in infinite dimension analysis. The integration by parts formulas for path space and loop space on a compact manifold are firstly proved in Driver (1992, 1993). A simple proof for logarithmic Sobolev inequality on path spaces by using the martingale representation was given in Capitaine et al. (1997) and Fang (1994). When the initial law µ is the Riemannian measure, such a formula of integration by parts was proved in Leandre and Norris (1997). For the Riemannian manifold that is noncompact, the integration by parts was proved in Fang and Wang (2005), and in this paper the authors assumed that the logarithmic Sobolev inequality for the initial law µ was essential. After that they deduced the logarithmic Sobolev inequality on free Riemannian path spaces. The integration by parts and logarithmic Sobolev inequality on free compact Riemannian loop space were proved in Shi (2011). In our article, by using the martingale representation theorem we mainly prove the LSI on free Riemannian path spaces. Let (M , ⟨, ⟩) be a d-dimensional compact Riemannian manifold without boundary. From now on we assume that the Ricci curvature of (M , ⟨, ⟩) is bounded by K. Let O(M ) be the bundle of orthonormal frames of M, and µ be a probability measure on M having strictly positive density w.r.t. the volume measure, such that dµ = υ(x)dx
for υ strictly positive with |∇ log υ| ∈ L2 (µ).
We now consider the space Ω := W0 (Rd ) × M endowed with the product Borel σ -field F and the product measure P µ = P W × µ. Define bt : Ω → Rd by bt (ω) = ωt and ξ : Ω → M by ξ (ω) = x where ω = (ω, x), then (bt )t ≥0 is a Brownian motion. The distribution of ξ is µ, Ft := σ {bs (·); s ≤ t } ∨ ξ . In the sequel always this probability space is (Ω , P , F , Ft ). Let {H1 , H2 , . . . , Hd } be the canonical horizontal vector fields on O(M ). Consider the following Stratonovich stochastic differential equation
dUt =
d
Hi (Ut ) ◦ dbit
U = η. i 0 E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2014.12.004 0167-7152/© 2014 Published by Elsevier B.V.
L. Pei / Statistics and Probability Letters 98 (2015) 12–19
13
Let γt = π (Ut ), so stochastic process γ : Ω × [0, 1] → M satisfies
dγt = Ut ◦ dbit
γ0 = ξ .
It is well known that Ut is a horizontal Brownian motion on O(M ), γt is a Brownian motion on M, with random initial ξ . To define the gradient operator on the free Brownian path space, let us first recall the procedure made for the case with fixed initial point x. Let
H0 :=
h ∈ C ([0, T ] : Rd ) : h(0) = 0, |h|H0 :=
T
|h˙ (s)|2 < ∞
0
be the Cameron–Martin space on the flat path space. For a given Brownian path γ· with horizontal lift U· and for any h0 , the directional derivative along h0 of a function F on Px (M ) is defined by D0h0 F (γ. ) :=
d dε
F (expγt ε Ut h0 (t ))
ε=0
.
Now, we consider the free path space P (M ) := C ([0, T ]; M ). Due to the freedom of the initial point, it is natural for us to make use of the following Cameron–Martin space:
H :=
h ∈ C ([0, T ] : Rd ) :
T
|h˙ (s)|2 < ∞ ,
0
which is a Hilbert space under the inner product, for h1 ∈ H, h2 ∈ H
⟨h1 , h2 ⟩H := ⟨h1 (0), h2 (0)⟩Rd + ⟨h1 − h1 (0), h2 − h2 (0)⟩H0 . For given h ∈ H, the corresponding directional derivative of a good function F along h is defined by Dh F (γ ·) :=
d dε
F (expγ · [ε Ut h(t )])
ε=0
.
Thus, the gradient DF (γ ·) can be fixed as an H-valued random variable through
⟨DF (γ ·), h⟩H = Dh F (γ ·). In particular, if F is a cylinder function (denoted by F ∈ F C0∞ ), that is, if there exist 0 ≤ t1 < · · · < tN ≤ T and f ∈ ∞ C0 (M N ) such that F (γ· ) = f (γt1 , . . . , γtN ),
γ ∈ Px (M ).
For F ∈ F C0∞ , one has Dh F (γ ·) =
N
⟨∇i f , Uti h(ti )⟩,
h∈H
i=1
and hence, DF (γ ·)(t ) =
N (1 + (t ∧ ti ))Ut−i 1 ∇i f + D0 F (γ ·)(t ). i=1
Since Ut is Ft -measurable, DF ∈ L2 (P (M ) → H : P µ ) with
2 N E (F , F ) := |DF |2H dPµ = E (1 + (t ∧ ti ))Ut−i 1 ∇i f + |D0 F (γ ·)|2H0 . i=1 P (M )
Let us introduce TP (M ) := {ψ ∈ C ([0, T ] × P (M ); TM ) : ψ(t , γ ) ∈ Tγt M is Ft -adapted, U −1 ψ ∈ L2 (P (M ) → H : P µ )}. We shall establish the integration by parts formula for directional derivatives along a large enough class of vectors in TP (M ). Let X0 (M ) be the space of smooth vector fields on M, and let H0 := {h0 ∈ L2 (Ω → H0 ; P µ ) : h0 (t ) is Ft -adapted}. To (h0 , X ) ∈ H0 × X0 (M ), we associate
ψh0 ,X (t ) = h0 (t ) + U0−1 X (γ0 ). We have U ψh0 ,X ∈ TP (M ).
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L. Pei / Statistics and Probability Letters 98 (2015) 12–19
2. The pull-back formula and the integration by parts In this part we mainly proof the integration by parts on free path space by the pull-back formula. We mainly refer to Shi (2011) for the pull-back formula and the integration by parts. Recall vector field Zt , Zt = Ut ψh0 ,X (t ) = Ut ho (t ) + Ut U0−1 X (x0 ), ξ ϵ ,ξ
For Z , we let γt SDE following:
where h0 ∈ H0 , X ∈ X0 (M ).
(ϵ) be Bismut’s transition of γt along the vector fields Z , which means
ξ ϵ ,ξ dγt (ϵ) ϵ=0 dϵ
|
= Zt , satisfying the
ξ ϵ ,ξ
dγt
ξ ϵ ,ξ
γ0
(ϵ) = Ut (ϵ) ◦ dbt (ϵ) (ϵ) = ξ ϵ
(2.1)
where ξ ϵ = expξ (ϵ X (ξ )), 0 ≤ ϵ < δM / maxx∈M |X (x)|, δM > 0 is the inject radium of compact manifold M. U0ϵ = ηϵ is the horizontal lift of ξ ϵ on O(M ) with initial distribution η0 = η. Denote µϵ = P ◦ (ξ ϵ )−1 . The Rd -valued Brownian semimartingale (bt (ϵ)) is still defined by: bt (ϵ) =
t
exp{ϵ Os }dbs + ϵ
t
0
αs ds.
(2.2)
0
Lemma 2.1. Ot and αt satisfy the conditions above, so 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ), 2 t −1 Ot = −U0 (∇ U0 )X (x0 ) + ΩUs (ψh0 ,X (s), Us−1 ◦ dγs ). 0
Proof. According to the result of Driver (1992), we know that Stratonovich stochastic differential calculus on M is same as the classical differential calculus on M. Hence, by differentiating (2.1) with respect to ϵ at ϵ = 0 we obtain d
ξ ϵ ,ξ
dϵ
dγt
(ϵ)
= Ut ◦ d
ϵ=0
d dϵ
bt (ϵ)
+
ϵ=0
d dϵ
Ut (ϵ)
◦dbt
(2.3)
r =0
Note that, by (2.2) d
d dϵ
bt (ϵ)
ϵ=0
= Ot dbt + αt dt .
(2.4)
By the definition of torsion T = 0 Ut −1
d dϵ
ξ ϵ ,ξ
dγt
(ϵ)
= U t −1 d
ϵ=0
d dϵ
ξ ϵ ,ξ
γt
(ϵ)
ϵ=0
= ψh˙0 ,X dt .
(2.5)
∀a ∈ Rn , let Jt = −Ut
d
−1
dϵ
Ut (ϵ)
ϵ=0
a. ξ ϵ ,ξ
Note that {Ut (ϵ)a}0≤t ≤1,−ε<ϵ<ε is parallel along {γt (ϵ)}0≤t ≤1,−ε<ϵ<ε for any fixed ϵ ∈ (−ε, ε), i.e. d(Ut (ϵ)a) = 0. Hence, by the definition of the curvature of the connection ∇ on M we have
d Ut (ϵ)a dϵ d
ϵ=0
= d Ut (ϵ)a dϵ d
ϵ=0
− dUt (ϵ)a dϵ d
ϵ=0
= ∇dγt ∇Zt (Ut a) − ∇Zt ∇dγt (Ut a) = −R(γt )(Zt , dγt )Ut a.
So dJt = Ut −1 R(γt )(Zt , dγt )Ut a
= ΩUt (ψh0 ,X (t ), Ut −1 ◦ dγt )a = ΩUt (ψh0 ,X (t ), ◦dβt )a,
L. Pei / Statistics and Probability Letters 98 (2015) 12–19
15
and
− Ut
−1
d dϵ
Ut (ϵ)
ϵ=0
= −U0−1 (∇ U0 )X (x0 ) +
t
ΩUs (ψh0 ,X (s), ◦dβs ).
(2.6)
0
Note that t
ΩUs (ψh0 ,X (s), ◦dβs ) ◦ dβt =
t
ΩUs (ψh0 ,X (s), ◦dβs )dβt + 0
0
1 2
RicUt ψh0 ,X (t )dt ,
(2.7)
where RicUt =
n
ΩUt (·, ei )ei .
i =1
By (2.3)–(2.7) we obtain: Ut −1 dZt = Ot dβt + αt dt + U0−1 (∇ U0 )X (x0 ) −
t
ΩUs (ψh0 ,X (s), ◦dβs ) ◦ dβt , 0
then
ψ˙ h0 ,X (t )dt = Ot dβt + αt dt + U0−1 (∇ U0 )X (x0 ) −
t
ΩUs (ψh0 ,X (s), ◦dβs )dβt − 0
1 2
RicUt ψh0 ,X (t )dt .
We have 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ), 2 t −1 Ot = −U0 (∇ U0 )X (x0 ) + ΩUs (ψh0 ,X (s), Us−1 ◦ dγs ). 0
This completes the proof.
Theorem 2.2. Let Zt = Ut ψh0 ,X (t ), Zt ∈ Ft , for any F ∈ F C ∞ (M ), Pµ
Pµ
Pµ
E [∂Z F ] = −E [Fdivµ X (ξ )] + E
1
F
⟨αt , dβt ⟩ ,
0
where divµ X (x) = div X (x) + ⟨∇ ln ρ0 (x), X (x)⟩Tx M , 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ). 2
Proof. By Lemma 2.1, we have 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ), 2 t Ot = −U0−1 (∇ U0 )X (x0 ) + ΩUs (ψh0 ,X (s), Us−1 ◦ dγs ). 0
Let bt (ϵ) =
t
exp{ϵ Os }dbs + ϵ 0
t
αs ds :, b˜ t (ϵ) + ϵ 0
t
αs ds, 0
and Nt (ϵ) = exp
t
0
1 ⟨−ϵαs , db˜ s (ϵ)⟩ − 2
t
|ϵαs |2 ds .
0
By Shi (2011), we know that {Nt (ϵ)}0≤t ≤1 is a uniform integrable martingale. From Girsanov’s theorem, for any ϵ ∈ [0, max δM |X(x)| ), (bt (ϵ))0≤t ≤1 is a Rd -valued Brownian motion under (Ω , Ft , N1 (ϵ)dP ). x∈ M
ξ ϵ ,ξ
For F (γt Pµ
(ϵ)) a function on (Ω , Ft , P ), we write out the distribution directly
E [F (γ ξ ϵ
ϵ ,ξ
(ϵ))] = EM [ρ0 (x)Exϵ [N1 (−ϵ)F (γt )]]
where x = expx ϵ X(x).
(2.8)
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L. Pei / Statistics and Probability Letters 98 (2015) 12–19
Differentiate (2.8) with respect to ϵ ∈ [0,
EP
δM
maxx∈M |X(x)|
) at ϵ = 0, we obtain
k µ
d |ϵ=0 (N1 (−ϵ))F (γt ) ⟨∇ (i) f (γ0 , . . . , γtk ), Uti ψh0 ,X (ti )⟩ = EM ρ0 (x)Ex dϵ i=0 d + EM ρ0 (x) |ϵ=0 {Exϵ [F (γt )]} . dϵ
(2.9)
Let g (xϵ ) = Exϵ [F (γ· )], d dϵ
|ϵ=0 g (xϵ ) =
d dϵ
|ϵ=0 g (expx ϵ X (x))
= ∇ g (expx ϵ X (x))|ϵ=0 ,
d dϵ
|ϵ=0 {expx ϵ X (x)}
Tx M
= ⟨∇ g (x), X (x)⟩Tx M , hence,
EM ρ0 (x)
d dϵ
|ϵ=0 {Exϵ [F (γt )]} = EM [ρ0 (x)⟨∇ g (x), X (x)⟩Tx M ] = EM [∂X∗(x) (ρ0 (x))g (x)]
= EM [−ρ0 (x)divµ X (x)Ex [F (γt )]]. 1 From (2.9), (2.10) and ddϵ |ϵ=0 ( N 1(ϵ) ) = 0 αt dbt , we obtain 1 k µ µ Pµ (i) E ⟨∇ f (γ0 , . . . , γtk ), Uti ψh0 ,X (ti )⟩ = −EP [F (γ )divµ X (ξ )] + E P F (γ )
(2.10)
1
αt dBt .
0
i=0
It finishes the proof.
From Theorem 2.2, we have completed the proof of integration by parts formula of the direction derivatives about Ft vector fields Zt by using Bismut’s idea which has the same representation as Fang and Wang (2005) on a noncompact M. By the same argument as Fang and Wang (2005), we can prove the closability of the gradient operator. Theorem 2.3. The gradient operator (D, F C0∞ ) : L2 (P µ ) → L2 (P (M ) → H ; P µ ) is closable. Consequently the quadratic form (E , F C0∞ ) is closable in L2 (P µ ) and the closure (E , D(E )) is a conservative Dirichlet form. Now we have the integration by parts formula about the Malliavin gradient operator on free path space as follows. Theorem 2.4. Let (M , ∇) be a d-dimensional compact connect Riemannian manifold without boundary. For the adapted vector field Zt = Ut ψh0 ,X (t ) = Ut (h0 (t ) + U0−1 X (x)), where X ∈ X0 (M ), h0 ∈ H0 , then for any F ∈ D (D), Pµ
Pµ
E ⟨DF , Z ⟩H = E
1
F
µ ⟨αt , dβt ⟩ − EP [Fdivµ X (ξ )]
0
where 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ). 2
3. Martingale representation theorem and logarithmic Sobolev inequality We refer to Hsu (1995, 2002) for the martingale representation theorem and logarithmic Sobolev inequality on compact Riemannian path spaces, and deduce them by Theorem 2.4. Theorem 3.1. For any F ∈ D (D), there exists a (F )t -predictable process (Ht )0≤t ≤1 such that F =E
Pµ
1
[F ] + 0
⟨Ht , dβt ⟩Rd ,
P µ -a.s.,
L. Pei / Statistics and Probability Letters 98 (2015) 12–19
17
where H t = EP
µ
Dt F +
1 2
Mt−1
1
P µ × dt-a.s.,
Mτ RicUτ (Dτ F )dτ |Ft ,
t
and M is the solution of the equation dMs ds
1
+ Ms RicUs = 0,
M0 = I .
2
Proof. For the adapted vector fields Zt = Ut ψh0 ,X (t ) = Ut (h0 (t ) + U0−1 X (x)), where X ∈ X0 (M ), h0 ∈ H0 , we compute DZ F in two ways. On one hand µ
µ
µ
µ
EP DZ F = EP ⟨DF , Z ⟩H = EP ⟨DF (0), Z (0)⟩Rd + EP ⟨DF − DF (0), Z − Z (0)⟩H0 , on the other hand, by the integration by parts formula we have µ
EP DZ F = EP
µ
1
F
1
ψ˙ h0 ,X + RicUt ψh0 ,X , dbt 2
0 1
⟨Ht , dbt ⟩
=E
1
0
µ
− EP [Fdivµ X ]
1
ψ˙ h0 ,X + RicUt ψh0 ,X , dbt 2
0
µ
− EP [Fdivµ X ].
Hence µ
µ
EP ⟨DF (0), Z (0)⟩Rd + EP ⟨DF − DF (0), Z − Z (0)⟩H0 Pµ
1
=E
1 µ ˙ Ht , ψh0 ,X (t ) + RicUt ψh0 ,X (t ) dt − EP [Fdivµ X ].
(3.11)
2
0
The next step is to extract a formula for Ht from the above relation. Set
˙ h0 ,X (t ) + kt = ψ
1 2
RicUt ψh0 ,X (t ).
˙ h0 ,X (t ) in terms of kt and Mt . The result is Let {Ms } be defined as in the statement of the theorem. Then we can solve for ψ ψh0 ,X (t ) = U0−1 X (x) + MtĎ
t
0
MτĎ−1 kτ dτ ,
where AĎ means the rotation of A. Differentiating with respect to t, we obtain 1
ψ˙ h0 ,X (t ) = kt + RicUt MtĎ
s
2
0
MτĎ−1 kτ dτ .
Using this expression in the integral on the left side of (3.11) and changing the order of integration, we have µ
µ
EP ⟨DF (0), Z0 ⟩Rd + EP ⟨DF − DF (0), Z − Z (0)⟩H0 µ
= EP ⟨DF (0), Z0 ⟩Rd + EP
µ
1
1
⟨Dt F + Mt−1 2
0
1
t
Mτ RicUτ (Dτ F )dτ , kt ⟩dt .
After these manipulations (3.11) becomes Pµ
E ⟨DF1 (0), X (ξ )⟩Rd + E
= EP
µ
1
Pµ
1
Dt F +
0
1 2
−1
1
Mt
t
Mτ RicUτ (Dτ F )dτ , kt dt
µ
⟨Ht , kt ⟩dt − EP [Fdivµ X ]. 0
In this relation we take X = 0 and kt can be an arbitrary Ft -adapted process. The formula for Ht in the theorem follows immediately. Now we can prove the logarithmic Sobolev inequality on compact free Riemannian spaces by Theorem 3.1. Theorem 3.2. Let dµ = v(x)dx Pµ (F 2 log F 2 ) ≤ 2eK E (F , F ) + Pµ (F 2 ) log Pµ (F 2 ).
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L. Pei / Statistics and Probability Letters 98 (2015) 12–19
Proof. Without loss of generality we may assume that F is bounded and inf{F (ω)|ω ∈ P (M )} > ε . µ ϕ Let ϕ = F 2 and (Mt )0≤t ≤1 be a right continuous version of {E P [ϕ|Ft ]}0≤t ≤1 , then by Theorem 3.1, dMt = ⟨Ht , dβt ⟩. By the Itô formula, we obtain 1
d(Mt ln Mt ) = (1 + ln Mt )dMt +
2Mt
d⟨M ⟩t ϕ
|Ht |2
ϕ
= ⟨(1 + ln Mt )Ht , dβt ⟩ +
2Mt
dt .
Integrate and take expectation on two sides, we have µ
µ
1
µ
E P [ϕ ln ϕ] − E P [ϕ] ln E P [ϕ] =
2
EP
µ
ϕ
1
|Ht |2 Mt
0
dt ,
hence
F2
2
F ln P (M )
1
µ
∥F 2 ∥L2 (P µ )
P (dγ ) =
2
E
Pµ
1
ϕ
|Ht |2 Mt
0
dt .
By Theorem 3.1 we have Ht = E P
µ
1
Dt F +
2
Mt−1
1
Mτ RicUτ (Dτ F )dτ |Ft .
t
For simplicity jt = Dt (F 2 ) = FDt F ,
Ht = E F
Dt F +
1 2
1
−1
Mτ RicUτ Dτ Fdτ
Mt
t
Since for every t ≤ τ , Mt−1 Mτ = I −
1 2
τ t
|Ft .
Mt−1 Mr RicUr dr.
Hence by Gronwall’s lemma and the bound on the Ricci curvature we have ∥Mt−1 Mτ ∥ ≤ eK (τ −t )/2 . Thus, Pµ
|Ht | ≤ E
jt +
1 2
1
K
K (τ −t )/2
e
jτ dτ |Ft
.
t
From the Jensen inequality and the Cauchy–Schwarz inequality it follows that
|Ht | ≤ 4E(F |Ft )E 2
2
1
1
|Dt F | + K 2
K (τ −t )/2
e
|Dτ F |dτ
2
|Ft .
t
Therefore, Pµ
1
E
ϕ
|Ht |2 Mt
0
≤ 4E
dt
Pµ
1
1
|Dt F | + K
K (τ −t )/2
e
2
|Dτ F |dτ
2
|Ft .
t
Now we have 1
eK (τ −t )/2 |Dτ F |dτ
2
t
1
≤
eK (τ −s) ds
t
1
|Dτ F |2 dτ =
0
1 K
(eK (1−t ) − 1)|DF |2H0 .
It then follows easily that E
Pµ
1
0
ϕ
|Ht |2 Mt
dt
µ
µ
≤ 4c (K )E P (|DF |2H0 ) ≤ 4c (K )E P (|DF |2H ),
where c (K ) = 1 +
1 4
(eK − 1 − K ) +
(eK − 1 − K ) ≤ eK .
We thus have established a logarithmic Sobolev inequality on P (M ) in the form Pµ (F 2 log F 2 ) ≤ 2eK E (F , F ) + Pµ (F 2 ) log Pµ (F 2 ).
L. Pei / Statistics and Probability Letters 98 (2015) 12–19
19
Acknowledgments The author is very grateful to Professors F.Z. Gong, Y. Liu and Kai, H for their valuable discussions and suggestions. The author wants to thank the anonymous referee for pointing out several possible improvements in the original manuscript. The author is supported by the National Key Basic Research & Development Program (973) under Grant No. 2011CB808000. References Capitaine, M., Hsu, E.P., Ledoux, M., 1997. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Commun. Probab. 71–81. Driver, B.K., 1992. A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376. Driver, B.K., 1993. A Cameron–Martin type quasi-invariance theorem for pinned Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376. Fang, S.Z., 1994. Inégalité du type de Poincaré sur l’espace des chemins riemanniens. C. R. Acad. Sci., Paris I 318 (3), 257–260. Fang, S.Z, Wang, F.Y., 2005. Analysis on free Riemannian path space. Bull. Sci. Math. 129, 339–355. Hsu, E.P., 1995. Quasi-invariance of the Wiener measure and integration by parts in the path space over a compact Riemannian manifold. J. Funct. Anal. 134, 417–450. Hsu, E.P., 2002. Stochastic Analysis on Manifolds. American Mathematical Society, RI. Leandre, R., Norris, J.R., 1997. Integration by parts and Cameron–Martin formulas for the free path space of a compact Riemannian manifold. In: Séminaire de Prob. XXXI. In: Lecture Notes Math, vol. 1655. pp. 16–23. Shi, Yinghui, 2011. Integration by parts on free loop space (Ph.D. thesis), Graduate University of Chinese Academy of Sciences.
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Logarithmic Sobolev inequality on free path space over a compact Riemannian manifold Ling Pei Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
article
info
Article history: Received 23 September 2014 Received in revised form 2 December 2014 Accepted 5 December 2014 Available online 13 December 2014 Keywords: Logarithmic Sobolev inequality Integration by parts Free path space Martingale representation
abstract In this paper we use Bismut’s method to verify the formula of integration by parts on the free path space. By the formula, we verify the martingale representation for reference measure Pµ , the law of the Brownian motion on the base manifold with initial distribution µ. Then by the martingale representation we get the logarithmic Sobolev inequality on compact free Riemannian path space. © 2014 Published by Elsevier B.V.
1. Introduction Integration by parts is an important topic in infinite dimension analysis. The integration by parts formulas for path space and loop space on a compact manifold are firstly proved in Driver (1992, 1993). A simple proof for logarithmic Sobolev inequality on path spaces by using the martingale representation was given in Capitaine et al. (1997) and Fang (1994). When the initial law µ is the Riemannian measure, such a formula of integration by parts was proved in Leandre and Norris (1997). For the Riemannian manifold that is noncompact, the integration by parts was proved in Fang and Wang (2005), and in this paper the authors assumed that the logarithmic Sobolev inequality for the initial law µ was essential. After that they deduced the logarithmic Sobolev inequality on free Riemannian path spaces. The integration by parts and logarithmic Sobolev inequality on free compact Riemannian loop space were proved in Shi (2011). In our article, by using the martingale representation theorem we mainly prove the LSI on free Riemannian path spaces. Let (M , ⟨, ⟩) be a d-dimensional compact Riemannian manifold without boundary. From now on we assume that the Ricci curvature of (M , ⟨, ⟩) is bounded by K. Let O(M ) be the bundle of orthonormal frames of M, and µ be a probability measure on M having strictly positive density w.r.t. the volume measure, such that dµ = υ(x)dx
for υ strictly positive with |∇ log υ| ∈ L2 (µ).
We now consider the space Ω := W0 (Rd ) × M endowed with the product Borel σ -field F and the product measure P µ = P W × µ. Define bt : Ω → Rd by bt (ω) = ωt and ξ : Ω → M by ξ (ω) = x where ω = (ω, x), then (bt )t ≥0 is a Brownian motion. The distribution of ξ is µ, Ft := σ {bs (·); s ≤ t } ∨ ξ . In the sequel always this probability space is (Ω , P , F , Ft ). Let {H1 , H2 , . . . , Hd } be the canonical horizontal vector fields on O(M ). Consider the following Stratonovich stochastic differential equation
dUt =
d
Hi (Ut ) ◦ dbit
U = η. i 0 E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2014.12.004 0167-7152/© 2014 Published by Elsevier B.V.
L. Pei / Statistics and Probability Letters 98 (2015) 12–19
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Let γt = π (Ut ), so stochastic process γ : Ω × [0, 1] → M satisfies
dγt = Ut ◦ dbit
γ0 = ξ .
It is well known that Ut is a horizontal Brownian motion on O(M ), γt is a Brownian motion on M, with random initial ξ . To define the gradient operator on the free Brownian path space, let us first recall the procedure made for the case with fixed initial point x. Let
H0 :=
h ∈ C ([0, T ] : Rd ) : h(0) = 0, |h|H0 :=
T
|h˙ (s)|2 < ∞
0
be the Cameron–Martin space on the flat path space. For a given Brownian path γ· with horizontal lift U· and for any h0 , the directional derivative along h0 of a function F on Px (M ) is defined by D0h0 F (γ. ) :=
d dε
F (expγt ε Ut h0 (t ))
ε=0
.
Now, we consider the free path space P (M ) := C ([0, T ]; M ). Due to the freedom of the initial point, it is natural for us to make use of the following Cameron–Martin space:
H :=
h ∈ C ([0, T ] : Rd ) :
T
|h˙ (s)|2 < ∞ ,
0
which is a Hilbert space under the inner product, for h1 ∈ H, h2 ∈ H
⟨h1 , h2 ⟩H := ⟨h1 (0), h2 (0)⟩Rd + ⟨h1 − h1 (0), h2 − h2 (0)⟩H0 . For given h ∈ H, the corresponding directional derivative of a good function F along h is defined by Dh F (γ ·) :=
d dε
F (expγ · [ε Ut h(t )])
ε=0
.
Thus, the gradient DF (γ ·) can be fixed as an H-valued random variable through
⟨DF (γ ·), h⟩H = Dh F (γ ·). In particular, if F is a cylinder function (denoted by F ∈ F C0∞ ), that is, if there exist 0 ≤ t1 < · · · < tN ≤ T and f ∈ ∞ C0 (M N ) such that F (γ· ) = f (γt1 , . . . , γtN ),
γ ∈ Px (M ).
For F ∈ F C0∞ , one has Dh F (γ ·) =
N
⟨∇i f , Uti h(ti )⟩,
h∈H
i=1
and hence, DF (γ ·)(t ) =
N (1 + (t ∧ ti ))Ut−i 1 ∇i f + D0 F (γ ·)(t ). i=1
Since Ut is Ft -measurable, DF ∈ L2 (P (M ) → H : P µ ) with
2 N E (F , F ) := |DF |2H dPµ = E (1 + (t ∧ ti ))Ut−i 1 ∇i f + |D0 F (γ ·)|2H0 . i=1 P (M )
Let us introduce TP (M ) := {ψ ∈ C ([0, T ] × P (M ); TM ) : ψ(t , γ ) ∈ Tγt M is Ft -adapted, U −1 ψ ∈ L2 (P (M ) → H : P µ )}. We shall establish the integration by parts formula for directional derivatives along a large enough class of vectors in TP (M ). Let X0 (M ) be the space of smooth vector fields on M, and let H0 := {h0 ∈ L2 (Ω → H0 ; P µ ) : h0 (t ) is Ft -adapted}. To (h0 , X ) ∈ H0 × X0 (M ), we associate
ψh0 ,X (t ) = h0 (t ) + U0−1 X (γ0 ). We have U ψh0 ,X ∈ TP (M ).
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2. The pull-back formula and the integration by parts In this part we mainly proof the integration by parts on free path space by the pull-back formula. We mainly refer to Shi (2011) for the pull-back formula and the integration by parts. Recall vector field Zt , Zt = Ut ψh0 ,X (t ) = Ut ho (t ) + Ut U0−1 X (x0 ), ξ ϵ ,ξ
For Z , we let γt SDE following:
where h0 ∈ H0 , X ∈ X0 (M ).
(ϵ) be Bismut’s transition of γt along the vector fields Z , which means
ξ ϵ ,ξ dγt (ϵ) ϵ=0 dϵ
|
= Zt , satisfying the
ξ ϵ ,ξ
dγt
ξ ϵ ,ξ
γ0
(ϵ) = Ut (ϵ) ◦ dbt (ϵ) (ϵ) = ξ ϵ
(2.1)
where ξ ϵ = expξ (ϵ X (ξ )), 0 ≤ ϵ < δM / maxx∈M |X (x)|, δM > 0 is the inject radium of compact manifold M. U0ϵ = ηϵ is the horizontal lift of ξ ϵ on O(M ) with initial distribution η0 = η. Denote µϵ = P ◦ (ξ ϵ )−1 . The Rd -valued Brownian semimartingale (bt (ϵ)) is still defined by: bt (ϵ) =
t
exp{ϵ Os }dbs + ϵ
t
0
αs ds.
(2.2)
0
Lemma 2.1. Ot and αt satisfy the conditions above, so 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ), 2 t −1 Ot = −U0 (∇ U0 )X (x0 ) + ΩUs (ψh0 ,X (s), Us−1 ◦ dγs ). 0
Proof. According to the result of Driver (1992), we know that Stratonovich stochastic differential calculus on M is same as the classical differential calculus on M. Hence, by differentiating (2.1) with respect to ϵ at ϵ = 0 we obtain d
ξ ϵ ,ξ
dϵ
dγt
(ϵ)
= Ut ◦ d
ϵ=0
d dϵ
bt (ϵ)
+
ϵ=0
d dϵ
Ut (ϵ)
◦dbt
(2.3)
r =0
Note that, by (2.2) d
d dϵ
bt (ϵ)
ϵ=0
= Ot dbt + αt dt .
(2.4)
By the definition of torsion T = 0 Ut −1
d dϵ
ξ ϵ ,ξ
dγt
(ϵ)
= U t −1 d
ϵ=0
d dϵ
ξ ϵ ,ξ
γt
(ϵ)
ϵ=0
= ψh˙0 ,X dt .
(2.5)
∀a ∈ Rn , let Jt = −Ut
d
−1
dϵ
Ut (ϵ)
ϵ=0
a. ξ ϵ ,ξ
Note that {Ut (ϵ)a}0≤t ≤1,−ε<ϵ<ε is parallel along {γt (ϵ)}0≤t ≤1,−ε<ϵ<ε for any fixed ϵ ∈ (−ε, ε), i.e. d(Ut (ϵ)a) = 0. Hence, by the definition of the curvature of the connection ∇ on M we have
d Ut (ϵ)a dϵ d
ϵ=0
= d Ut (ϵ)a dϵ d
ϵ=0
− dUt (ϵ)a dϵ d
ϵ=0
= ∇dγt ∇Zt (Ut a) − ∇Zt ∇dγt (Ut a) = −R(γt )(Zt , dγt )Ut a.
So dJt = Ut −1 R(γt )(Zt , dγt )Ut a
= ΩUt (ψh0 ,X (t ), Ut −1 ◦ dγt )a = ΩUt (ψh0 ,X (t ), ◦dβt )a,
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and
− Ut
−1
d dϵ
Ut (ϵ)
ϵ=0
= −U0−1 (∇ U0 )X (x0 ) +
t
ΩUs (ψh0 ,X (s), ◦dβs ).
(2.6)
0
Note that t
ΩUs (ψh0 ,X (s), ◦dβs ) ◦ dβt =
t
ΩUs (ψh0 ,X (s), ◦dβs )dβt + 0
0
1 2
RicUt ψh0 ,X (t )dt ,
(2.7)
where RicUt =
n
ΩUt (·, ei )ei .
i =1
By (2.3)–(2.7) we obtain: Ut −1 dZt = Ot dβt + αt dt + U0−1 (∇ U0 )X (x0 ) −
t
ΩUs (ψh0 ,X (s), ◦dβs ) ◦ dβt , 0
then
ψ˙ h0 ,X (t )dt = Ot dβt + αt dt + U0−1 (∇ U0 )X (x0 ) −
t
ΩUs (ψh0 ,X (s), ◦dβs )dβt − 0
1 2
RicUt ψh0 ,X (t )dt .
We have 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ), 2 t −1 Ot = −U0 (∇ U0 )X (x0 ) + ΩUs (ψh0 ,X (s), Us−1 ◦ dγs ). 0
This completes the proof.
Theorem 2.2. Let Zt = Ut ψh0 ,X (t ), Zt ∈ Ft , for any F ∈ F C ∞ (M ), Pµ
Pµ
Pµ
E [∂Z F ] = −E [Fdivµ X (ξ )] + E
1
F
⟨αt , dβt ⟩ ,
0
where divµ X (x) = div X (x) + ⟨∇ ln ρ0 (x), X (x)⟩Tx M , 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ). 2
Proof. By Lemma 2.1, we have 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ), 2 t Ot = −U0−1 (∇ U0 )X (x0 ) + ΩUs (ψh0 ,X (s), Us−1 ◦ dγs ). 0
Let bt (ϵ) =
t
exp{ϵ Os }dbs + ϵ 0
t
αs ds :, b˜ t (ϵ) + ϵ 0
t
αs ds, 0
and Nt (ϵ) = exp
t
0
1 ⟨−ϵαs , db˜ s (ϵ)⟩ − 2
t
|ϵαs |2 ds .
0
By Shi (2011), we know that {Nt (ϵ)}0≤t ≤1 is a uniform integrable martingale. From Girsanov’s theorem, for any ϵ ∈ [0, max δM |X(x)| ), (bt (ϵ))0≤t ≤1 is a Rd -valued Brownian motion under (Ω , Ft , N1 (ϵ)dP ). x∈ M
ξ ϵ ,ξ
For F (γt Pµ
(ϵ)) a function on (Ω , Ft , P ), we write out the distribution directly
E [F (γ ξ ϵ
ϵ ,ξ
(ϵ))] = EM [ρ0 (x)Exϵ [N1 (−ϵ)F (γt )]]
where x = expx ϵ X(x).
(2.8)
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Differentiate (2.8) with respect to ϵ ∈ [0,
EP
δM
maxx∈M |X(x)|
) at ϵ = 0, we obtain
k µ
d |ϵ=0 (N1 (−ϵ))F (γt ) ⟨∇ (i) f (γ0 , . . . , γtk ), Uti ψh0 ,X (ti )⟩ = EM ρ0 (x)Ex dϵ i=0 d + EM ρ0 (x) |ϵ=0 {Exϵ [F (γt )]} . dϵ
(2.9)
Let g (xϵ ) = Exϵ [F (γ· )], d dϵ
|ϵ=0 g (xϵ ) =
d dϵ
|ϵ=0 g (expx ϵ X (x))
= ∇ g (expx ϵ X (x))|ϵ=0 ,
d dϵ
|ϵ=0 {expx ϵ X (x)}
Tx M
= ⟨∇ g (x), X (x)⟩Tx M , hence,
EM ρ0 (x)
d dϵ
|ϵ=0 {Exϵ [F (γt )]} = EM [ρ0 (x)⟨∇ g (x), X (x)⟩Tx M ] = EM [∂X∗(x) (ρ0 (x))g (x)]
= EM [−ρ0 (x)divµ X (x)Ex [F (γt )]]. 1 From (2.9), (2.10) and ddϵ |ϵ=0 ( N 1(ϵ) ) = 0 αt dbt , we obtain 1 k µ µ Pµ (i) E ⟨∇ f (γ0 , . . . , γtk ), Uti ψh0 ,X (ti )⟩ = −EP [F (γ )divµ X (ξ )] + E P F (γ )
(2.10)
1
αt dBt .
0
i=0
It finishes the proof.
From Theorem 2.2, we have completed the proof of integration by parts formula of the direction derivatives about Ft vector fields Zt by using Bismut’s idea which has the same representation as Fang and Wang (2005) on a noncompact M. By the same argument as Fang and Wang (2005), we can prove the closability of the gradient operator. Theorem 2.3. The gradient operator (D, F C0∞ ) : L2 (P µ ) → L2 (P (M ) → H ; P µ ) is closable. Consequently the quadratic form (E , F C0∞ ) is closable in L2 (P µ ) and the closure (E , D(E )) is a conservative Dirichlet form. Now we have the integration by parts formula about the Malliavin gradient operator on free path space as follows. Theorem 2.4. Let (M , ∇) be a d-dimensional compact connect Riemannian manifold without boundary. For the adapted vector field Zt = Ut ψh0 ,X (t ) = Ut (h0 (t ) + U0−1 X (x)), where X ∈ X0 (M ), h0 ∈ H0 , then for any F ∈ D (D), Pµ
Pµ
E ⟨DF , Z ⟩H = E
1
F
µ ⟨αt , dβt ⟩ − EP [Fdivµ X (ξ )]
0
where 1
αt = ψ˙ h0 ,X (t ) + RicUt ψh0 ,X (t ). 2
3. Martingale representation theorem and logarithmic Sobolev inequality We refer to Hsu (1995, 2002) for the martingale representation theorem and logarithmic Sobolev inequality on compact Riemannian path spaces, and deduce them by Theorem 2.4. Theorem 3.1. For any F ∈ D (D), there exists a (F )t -predictable process (Ht )0≤t ≤1 such that F =E
Pµ
1
[F ] + 0
⟨Ht , dβt ⟩Rd ,
P µ -a.s.,
L. Pei / Statistics and Probability Letters 98 (2015) 12–19
17
where H t = EP
µ
Dt F +
1 2
Mt−1
1
P µ × dt-a.s.,
Mτ RicUτ (Dτ F )dτ |Ft ,
t
and M is the solution of the equation dMs ds
1
+ Ms RicUs = 0,
M0 = I .
2
Proof. For the adapted vector fields Zt = Ut ψh0 ,X (t ) = Ut (h0 (t ) + U0−1 X (x)), where X ∈ X0 (M ), h0 ∈ H0 , we compute DZ F in two ways. On one hand µ
µ
µ
µ
EP DZ F = EP ⟨DF , Z ⟩H = EP ⟨DF (0), Z (0)⟩Rd + EP ⟨DF − DF (0), Z − Z (0)⟩H0 , on the other hand, by the integration by parts formula we have µ
EP DZ F = EP
µ
1
F
1
ψ˙ h0 ,X + RicUt ψh0 ,X , dbt 2
0 1
⟨Ht , dbt ⟩
=E
1
0
µ
− EP [Fdivµ X ]
1
ψ˙ h0 ,X + RicUt ψh0 ,X , dbt 2
0
µ
− EP [Fdivµ X ].
Hence µ
µ
EP ⟨DF (0), Z (0)⟩Rd + EP ⟨DF − DF (0), Z − Z (0)⟩H0 Pµ
1
=E
1 µ ˙ Ht , ψh0 ,X (t ) + RicUt ψh0 ,X (t ) dt − EP [Fdivµ X ].
(3.11)
2
0
The next step is to extract a formula for Ht from the above relation. Set
˙ h0 ,X (t ) + kt = ψ
1 2
RicUt ψh0 ,X (t ).
˙ h0 ,X (t ) in terms of kt and Mt . The result is Let {Ms } be defined as in the statement of the theorem. Then we can solve for ψ ψh0 ,X (t ) = U0−1 X (x) + MtĎ
t
0
MτĎ−1 kτ dτ ,
where AĎ means the rotation of A. Differentiating with respect to t, we obtain 1
ψ˙ h0 ,X (t ) = kt + RicUt MtĎ
s
2
0
MτĎ−1 kτ dτ .
Using this expression in the integral on the left side of (3.11) and changing the order of integration, we have µ
µ
EP ⟨DF (0), Z0 ⟩Rd + EP ⟨DF − DF (0), Z − Z (0)⟩H0 µ
= EP ⟨DF (0), Z0 ⟩Rd + EP
µ
1
1
⟨Dt F + Mt−1 2
0
1
t
Mτ RicUτ (Dτ F )dτ , kt ⟩dt .
After these manipulations (3.11) becomes Pµ
E ⟨DF1 (0), X (ξ )⟩Rd + E
= EP
µ
1
Pµ
1
Dt F +
0
1 2
−1
1
Mt
t
Mτ RicUτ (Dτ F )dτ , kt dt
µ
⟨Ht , kt ⟩dt − EP [Fdivµ X ]. 0
In this relation we take X = 0 and kt can be an arbitrary Ft -adapted process. The formula for Ht in the theorem follows immediately. Now we can prove the logarithmic Sobolev inequality on compact free Riemannian spaces by Theorem 3.1. Theorem 3.2. Let dµ = v(x)dx Pµ (F 2 log F 2 ) ≤ 2eK E (F , F ) + Pµ (F 2 ) log Pµ (F 2 ).
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L. Pei / Statistics and Probability Letters 98 (2015) 12–19
Proof. Without loss of generality we may assume that F is bounded and inf{F (ω)|ω ∈ P (M )} > ε . µ ϕ Let ϕ = F 2 and (Mt )0≤t ≤1 be a right continuous version of {E P [ϕ|Ft ]}0≤t ≤1 , then by Theorem 3.1, dMt = ⟨Ht , dβt ⟩. By the Itô formula, we obtain 1
d(Mt ln Mt ) = (1 + ln Mt )dMt +
2Mt
d⟨M ⟩t ϕ
|Ht |2
ϕ
= ⟨(1 + ln Mt )Ht , dβt ⟩ +
2Mt
dt .
Integrate and take expectation on two sides, we have µ
µ
1
µ
E P [ϕ ln ϕ] − E P [ϕ] ln E P [ϕ] =
2
EP
µ
ϕ
1
|Ht |2 Mt
0
dt ,
hence
F2
2
F ln P (M )
1
µ
∥F 2 ∥L2 (P µ )
P (dγ ) =
2
E
Pµ
1
ϕ
|Ht |2 Mt
0
dt .
By Theorem 3.1 we have Ht = E P
µ
1
Dt F +
2
Mt−1
1
Mτ RicUτ (Dτ F )dτ |Ft .
t
For simplicity jt = Dt (F 2 ) = FDt F ,
Ht = E F
Dt F +
1 2
1
−1
Mτ RicUτ Dτ Fdτ
Mt
t
Since for every t ≤ τ , Mt−1 Mτ = I −
1 2
τ t
|Ft .
Mt−1 Mr RicUr dr.
Hence by Gronwall’s lemma and the bound on the Ricci curvature we have ∥Mt−1 Mτ ∥ ≤ eK (τ −t )/2 . Thus, Pµ
|Ht | ≤ E
jt +
1 2
1
K
K (τ −t )/2
e
jτ dτ |Ft
.
t
From the Jensen inequality and the Cauchy–Schwarz inequality it follows that
|Ht | ≤ 4E(F |Ft )E 2
2
1
1
|Dt F | + K 2
K (τ −t )/2
e
|Dτ F |dτ
2
|Ft .
t
Therefore, Pµ
1
E
ϕ
|Ht |2 Mt
0
≤ 4E
dt
Pµ
1
1
|Dt F | + K
K (τ −t )/2
e
2
|Dτ F |dτ
2
|Ft .
t
Now we have 1
eK (τ −t )/2 |Dτ F |dτ
2
t
1
≤
eK (τ −s) ds
t
1
|Dτ F |2 dτ =
0
1 K
(eK (1−t ) − 1)|DF |2H0 .
It then follows easily that E
Pµ
1
0
ϕ
|Ht |2 Mt
dt
µ
µ
≤ 4c (K )E P (|DF |2H0 ) ≤ 4c (K )E P (|DF |2H ),
where c (K ) = 1 +
1 4
(eK − 1 − K ) +
(eK − 1 − K ) ≤ eK .
We thus have established a logarithmic Sobolev inequality on P (M ) in the form Pµ (F 2 log F 2 ) ≤ 2eK E (F , F ) + Pµ (F 2 ) log Pµ (F 2 ).
L. Pei / Statistics and Probability Letters 98 (2015) 12–19
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Acknowledgments The author is very grateful to Professors F.Z. Gong, Y. Liu and Kai, H for their valuable discussions and suggestions. The author wants to thank the anonymous referee for pointing out several possible improvements in the original manuscript. The author is supported by the National Key Basic Research & Development Program (973) under Grant No. 2011CB808000. References Capitaine, M., Hsu, E.P., Ledoux, M., 1997. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Commun. Probab. 71–81. Driver, B.K., 1992. A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376. Driver, B.K., 1993. A Cameron–Martin type quasi-invariance theorem for pinned Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376. Fang, S.Z., 1994. Inégalité du type de Poincaré sur l’espace des chemins riemanniens. C. R. Acad. Sci., Paris I 318 (3), 257–260. Fang, S.Z, Wang, F.Y., 2005. Analysis on free Riemannian path space. Bull. Sci. Math. 129, 339–355. Hsu, E.P., 1995. Quasi-invariance of the Wiener measure and integration by parts in the path space over a compact Riemannian manifold. J. Funct. Anal. 134, 417–450. Hsu, E.P., 2002. Stochastic Analysis on Manifolds. American Mathematical Society, RI. Leandre, R., Norris, J.R., 1997. Integration by parts and Cameron–Martin formulas for the free path space of a compact Riemannian manifold. In: Séminaire de Prob. XXXI. In: Lecture Notes Math, vol. 1655. pp. 16–23. Shi, Yinghui, 2011. Integration by parts on free loop space (Ph.D. thesis), Graduate University of Chinese Academy of Sciences.