Quasi-invariant flows under low exponential integrability of - - PowerPoint PPT Presentation

Quasi-invariant flows under low exponential integrability of divergence

• n the Wiener space

Shizan Fang Universit´ e de Bourgogne City University of HongKong June 29th, 2009 1

Introduction

In 1983, A.B. Cruzeiro proved in Equations diff´ erentielles sur l’espace de Wiener et formules de cameron-Martin non lin´ eaires, J. Funct. Anal. 54 (1983), 206–227 that on the Wiener space (W , H, µ) for a vector field A : W → H in the Sobolev space D2

∞(W , H) if for all λ > 0,

(i)

• W

eλ(|A|H+|divµ(A)|)dµ < +∞, (ii)

• W

eλ|∇A|H⊗H < +∞ then there exists a unique flow of measurable maps Ut : W → W such that (Ut)∗µ = Kt µ and 2

Ut(x) = x + t A(Us(x)) ds. This result has been generalized later by several authors.

• V. Bogachev and E.M. Wolf, Absolutely continuous flows

generated by Sobolev class vector fields in finite and infinite dimensions, J. Funct. Anal. 167 (1999), 1–68.

• G. Peters, Anticipating flows on the Wiener space generated

by vector fields of low regularity, J. Funct. Anal. 142 (1996), 129–192. But at the same time, she proved that on Rd if A ∈ C 3 and

• Rd eλ0(|A|+|divγ(A)|)dγ < +∞ for some λ0 > 0, then the similar

results on Rd hold. Notice no condition on the gradient. In this case, the ordinary differential equation with coefficient A admits the unique solution up to the explosion time; so the second condition insures the non-explosion and the existence of density. 3

For the Wiener space case, even though A is smooth in the sense

• f Malliavin calculus, it is not in general continuous with respect to

the Banach norm of W . We have to use a procedure of smoothing and the estimate on e|∇A|H×H is usually needed. In 1989, R.J. Diperna and P.L. Lions discussed vector fields in W 1,1

loc (Rd) in the paper

R.J. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547. No conditions were needed on the eλ|∇A|. Let’s explain a bit their method. 4

Let A be a C 1 vector field on Rd, having bounded derivative. Then the differential equation dXt dt = A(Xt), X0 = x (1) defines a flow of global diffeomorphisms x → Xt(x) of Rd; the inverse map x → X −1

t

(x) solves dX −1

t

dt = −A(X −1

t

), X −1 = x. (2) Let θ ∈ C 1(Rd) and set ut = θ(X −1

t

). Then ut solves the transport equation dut dt + A · ∇ut = 0, u0 = θ ∈ C 1. (3) 5

Now for A belonging to a Sobolev space W 1,q(Rd) with q ≥ 1, We say that u ∈ L∞([0, T], Lp(Rd)) solves (3) in distribution sense if

• [0,T]×Rd
• −α′F ut − α div(FA) ut
• dxdt =
• Rd α(0)F u0 dx

(4) where α ∈ C ∞

c ([0, T)) and F ∈ C ∞ c (Rd).

An useful concept in this respect is the notion of renormalized solutions: for any β ∈ C 1

b (R), β(ut) solves again (35). A basic

result in DiPerna-Lions’s paper is 6

Theorem

Let un

t = ut ∗ χn. Then un t satisfies

dun

t

dt + A · ∇un

t = cn(ut, Z),

(5) here cn(f , A) = (DAf ) ∗ χn − DA(f ∗ χn) and ||cn(f , A)||L1 ≤ C ||f ||Lp (||∇A||Lq + ||div(A)||Lq). (6) 7

Gaussian case:

Return to the case where A is good and (Xt)t∈R is the flow of diffeomorphisms associated to A: dXt dt = A(Xt), X0 = x. Then for the standard Gaussian measure γ on Rd, (Xt)∗γ = Kt γ with Kt satisfying Kt = exp t divγ(A)(X−s)ds

• ,

||Kt||p

Lp ≤

• Rd exp

p2t p − 1|divγ(A)|

• dγ

(7) where divγ is the divergence with respect to the Gaussian measure γ. Suppose that

• Rd eλ0|divγ(A)|dγ < +∞

for some small λ0. (8) Then for p > 1 fixed, Kt ∈ Lp(γ) only for t ≤ (p−1)λ0

p2

. 8

Let T > 0 be given. Then for t ∈ [0, T],

• Rd Kt| log Kt| dγ ≤ T ||divγ(A)||L2N
• Rd eλ0|divγ(A)| dγ,

(9) where Nλ0

4

≥ T . In fact, using the explicit expression given in (7), we have log Kt(Xt) = t divγ(A)(Xt−s) ds. Then

• Rd Kt| log Kt| dγ =
• Rd | log Kt(Xt)| dγ

≤ t

• Rd |divγ(A)(Xt−s)| dγ ds.

Note by (7) for p = 2 and t ≤ T0 = λ0

4 ,

||Kt||2

L2 ≤

• Rd e4T0|divγ(A)| dγ.

(10) 9

Using the property of flow, for 0 ≤ t ≤ T0:

• Rd |divγ(A)(Xt+T0)| dγ =
• Rd |divγ(A)(Xt)| KT0 dγ,

which is less, by Cauchy-Schwarz inequality than

• Rd |divγ(A)(Xt)|2 dγ

1/2 ||KT0||L2, again by Cauchy-Schwarz inequality, less than ||divγ(A)||L4||Kt||1/2

L2 ||KT0||L2.

Set B = sup0≤t≤T0 ||Kt||2

• L2. By induction, we get for t ∈ [0, T],
• Rd |divγ(A)(Xt)| dγ ≤ ||divγ(A)||L2N B1/2N · · · B1/2 ≤ ||divγ(A)||L2N B,

where N is such that NT0 ≥ T. Now the estimate (10) leads to (9). 10

Now let Z ∈ Dp

1(Rd, γ) such that

• Rd eλ0(|Z|+|divγ(Z)|)dγ < +∞

for some small λ0 > 0. (11) We regularize Z by Ornstein-Uhlenbeck semi-group Pε Pεf (x) =

• Rd f (eεx +
• 1 − e−2εy) dγ(y).

Take a subsequence εm → 0 and set Bm = PεmZ. Then for λ < λ0 small enough, sup

m≥1

• Rm eλ|divγ(Bm)| dγ < +∞.

(12) Let (X m

t )t∈R solve

dX m

t

dt = Bm(X m

t ),

X m

0 = x,

and K m

t

be the density of (X m

t )γ with respect to γ.

11

By second term in (7), for any q > 1, there is a small T0 such that sup

m≥1

sup

t∈[0,T0]

||K m

t ||q Lq < +∞,

(13) and for all fixed T > 0, sup

m≥1

sup

t∈[0,T]

• Rd K m

t | log K m t | dγ < +∞.

(14) Let {ℓ1, · · · , ℓd} be the dual basis of Rd. Consider um

i (t, x) = ℓi(X m t (x)). Then we saw that {um i ; t ∈ [0, T]} solves

the transport equations dum

i

dt − Bm · ∇um

i

= 0, um

i (0) = ℓi.

(15) Using (13), we have 12

sup

m≥1

sup

t∈[0,T]

• Rd |um

i (t, x)|q dγ(x) < +∞,

t ∈ [0, T]. (16) Let q′ =

p p−1. Taking q = q′ and q = 2q′ in (16), we see that

{um

i ; m ≥ 1} and {(um i )2; m ≥ 1} are bounded in

Lq′([0, T] × Rd). We can choose a subsequence such that um

i

→ vi and (um

i )2 → wi weakly in Lq′ as m → +∞.

It is easy to check that vi(t) and wi(t) solve the transport equation du dt − Z · ∇u = 0, (17) with respectively initial condition ℓi and ℓ2

i . If the equation (17)

has the unique renormalized solution, then v2

i solves (17) with

the initial value ℓ2

i ; therefore v2 i = wi. Therefore as m → +∞,

um

i

→ vi and (um

i )2 → v2 i weakly, from which we deduce that

lim

m→+∞

• [0,T]×Rd
• um

i (t, x) − vi(t, x)

• 2 dγdt = 0.

(18) 13

Now we define Xt(x) =

d

• i=1

vi(t, x)ei, (19) where {e1, · · · , ed} is the canonical basis of Rd. By (18), lim

m→+∞

• [0,T]×Rd |X m

t (x) − Xt(x)|2 dγdt = 0.

(20) By (14) and (20), we obtain that for almost all t ∈ [0, T], the density Kt of (Xt)∗γ with respect to γ exists and

• Rd Kt| log Kt| dγ ≤ CT < +∞,

(21) where CT > 0 is a constant independent of t ∈ [0, T]; there exists a small T0 > 0 such that for almost all t ∈ [0, T0], ||Kt||q

Lq ≤

• Rd exp

q2t q − 1|divγ(Z)|

• dγ.

(22) 14

In what follows, we will consider the subsequence such that X m converges to X almost everywhere. Then by (20) and (22), uniformly with respect to [0, T0], as m → +∞ t Bm(X m

s ) ds →

t Z(Xs) ds in all Lq(Rd, γ). Now letting m → +∞ in X m

t (x) = x +

t Bm(X m

s (x)) ds;

it holds in L1([0, T0] × Rd), Xt(x) = x + t Z(Xs(x)) ds. (23) Note that the right hand side of (23) is continuous with respect to t ∈ [0, T0] for γ-a.e x ∈ Rd. Now we redefine ˜ Xt(x) = x + t Z(Xs(x)) ds, for t ∈ [0, T0]. 15

Obviously for t ∈ [0, T0]: ˜ Xt(x) = x + t Z( ˜ Xs(x)) ds. (24) Now for each t ∈ [0, T0], the density Kt of (˜ Xt)∗γ with respect to γ admits the explicit expression Kt(x) = exp t divγ(Z)(˜ X−s) ds

• ,

(25) where ˜ X−s solves (24) replacing Z by −Z.

Definition

For t ∈ [0, T0], we define ˜ Xt+T0(x) = ˜ Xt( ˜ XT0(x)), and so on, we obtain {Xt; t ∈ [0, T]}. 16

Theorem

Let Z ∈ Dp

1(Rd, γ) with p > 1 and suppose that for a small λ0 > 0,

• Rd eλ0(|Z|+|divγ(Z)|)dγ < +∞.

Then there exists a unique flow of maps { ˜ Xt; t ∈ [−T, T]} such that ( ˜ Xt)∗γ = Ktγ with Kt given in (25) and ˜ Xt(x) = x + t Z( ˜ Xs) ds. See F. Cipriano and A. B. Cruzeiro, J. Diff. Equ. (2005), 183-201. 17

A key step to prove the uniqueness of normalized solutions to (17): du dt − Z · ∇u = 0, is the following estimate

Theorem

There exists a constant C > 0 independent of ε and of dimension such that ||cε(f , Z)||L1 ≤ C ||f ||Lp ||Z||Dq

1 ,

(26) where cε(f , Z) = DZPεf − Pε(DZf ). The above estimate was first established by L. Ambrosio and A. Figalli, J. Funct. Analysis, 2009 and the present form was given in

• S. Fang and D. Luo in Bull. Sci. Math. 2009. The difference

between these two papers is that in the first one, the author consider a large enough λ0 > 0, while in the second a small λ0 suffices. 18

Wiener space case:

Let X = C0([0, 1], R) and H =

• h ∈ X;

1

0 |˙

hs|2ds < +∞

• be the

Cameron-Martin subspace of X: the Wiener measure µ on X is quasi-invariant under translations of elements in H. Let {h0, hk,n; n ≥ 1, k < 2n odd} be the family of Haar functions defined on the interval [0, 1] by ˙ h0(t) = 1 and ˙ hk,n(t) =      ( √ 2)n−1, for t ∈ [(k − 1)2−n, k2−n); −( √ 2)n−1, for t ∈ [k2−n, (k + 1)2−n); 0,

• therwise.

The family {h0, hk,n; n ≥ 1, k < 2n odd, } constitutes an

• rthonormal basis of H. Let

πn(x) = h0, xh0 +

n

• m=1
• k<2m odd

hk,m, xhk,m. (27) Let Vn = πn(X). Then γn = (πn)∗µ is the canonical Gaussian measure on (Vn, | · |H). 19

Let Z : X → H in the Sobolev space Dq

1(X, H). Suppose that

• X

eλ0(|Z|H+|divµ(Z)|) dµ < +∞ for some λ0 > 0. (28) For each n ≥ 1, let Zn : Vn → Vn be defined by Zn ◦ πn = E(πn(Z)|Vn). Then Zn ∈ Dq

1(Vn, γn) and

sup

n≥1

• Vn

eλ0(|Zn|H+|divγn(Zn)|) dγn < +∞. (29) Denote by Un

t : Vn → Vn the flow associated to Zn. Define

˜ Un

t : X → X by

˜ Un

t (x) = Un t (πn(x)) + x − πn(x).

(30) 20

Then there exists a small T0 > 0 such that the density ˜ K n

t of

(˜ Un

t )∗µ with respect to µ satisfies the estimate:

˜ K n

t q Lq(X) ≤

• X

exp q2T0 q − 1|divµ(Z)|

• dµ,

t ∈ [0, T0]. (31) For m ≤ n, k < 2m odd fixed, we set ˜ un

t (x) = hk,m, ˜

Un

t (x)

Therefore ˜ un → vk,m and (˜ un)2 → v2

k,m weakly as n → +∞, from

which we deduce that lim

n→+∞

• [0,T]×X
• hk,m, ˜

Un

t − vk,m

• 2 dµdt = 0.

(32) By the method of extracting diagonal subsequence, we may assume that the relation (32) holds for any m ≥ 1 and k < 2m odd. In this way, we obtain the coordinates {v0, vk,m; m ≥ 1, k < 2m odd}. 21

The random series Ut(x) := v0(t, x)h0 +

• m≥1
• k<2m odd

vk,m(t, x)hk,m (33) converges in X a.s. for (t, x) ∈ [0, T] × X.

Theorem

Let Z ∈ Dp

1(X, H). Assume that

E(eλ0(|Z|H+|divµ(Z)|)) < +∞ for some λ0 > 0. Then there exists a unique flow of maps (˜ Ut)t∈[−T,T] such that (˜ Ut)∗γ = Ktγ with Kt given by Kt(x) = exp t divµ(Z)(˜ U−s(x)) ds

• ,

(34) and solves ˜ Ut(x) = x + t

0 Z(˜

Us(x)) ds. 22

Compact Riemannian case:

Let M be a smooth compact Riemannian manifold endowed with the Levi-Civita connection. Consider the heat semi-group T M

t

:= et∆ on M. Recall first a probabilistic construction for T M

t .

Let O(M) be the bundle of orthonormal frames of M and A1, · · · , Ad the d canonical horizontal vector fields on O(M). Let rt(w, r0) solve the following Stratanovich SDE on O(M): drt(w) =

d

• i=1

Ai(rt(w)) ◦ dwi

t,

r0(w) = r0 (35) where t → w(t) is the standard Rd valued Brownian motion, defined on a probability space (Ω, F, P). Let xt(w) = π(rt(w, r0)). The law of w → x·(w) is independent of r0 and T M

t f (x0) = E(f (xt)),

for any r0 ∈ π−1(x0). Let’s state two basic formulae on the Path space analysis. The first one is the Bismut formula 23

Theorem

Let Z be a C 1 vector field on M. Then for any 0 ≤ t ≤ 1 and r0 ∈ π−1(x0),

• DZT M

t f )(x0) = 1

t E

• f (xt)

t Qs · r−1

0 Zx0, dws

• ,

(36) where {Qt; t ∈ [0, 1]} solves the following resolvent equation dQt dt = −1 2ricrt(w,r0)Qt, Q0 = Id. (37) The second formula is due to B. Driver [3].

Theorem

For any r0 ∈ π−1(x0), E(div(Z)(xt)) = 1 t E

• r−1

t

Zxt(w), t

• I − 1

2s ricrs ¯ dws

• ,

(38) 24

where ¯ dws denotes the Itˆ

• backward integral. The relation

between t

0 As ¯

dws and the usual Itˆ

• stochastic integral is given by

t As ¯ dws = t As dws + t dAs · dws. (39) Note that another formula without involving the derivative of the Ricci tensor for E(div(Z)(xt)) was given by B. Driver and A. Thalmaier in [4], p.70., which is the following E(div(Z)(xt)) = −1 t E

• r−1

t

Zxt(w), Qt t Q−1

s

dws

• .

(40) 25

Set Mt = Qt t

0 Q−1 s

• dws. Then by Itˆ
• formula

dMt = −1 2ricrt(w,r0)Qt t Q−1

s

dws dt + dwt,

• r Mt = wt − 1

2

t

0 ricrs(w,r0) Ms ds. So

E(div(Z)(xt)) = −1 t E(r−1

t

Zxt, wt)+ 1 2t E

• r−1

t

Zxt, t ricrs Ms ds

• .

(41) We rewrite (38): E(div(Z)(xt)) = 1 t E(r−1

t

Zxt, wt) − 1 2t E

• r−1

t

Zxt, t s ricrs dws

• − 1

2t E

• r−1

t

Zxt, t sJrs ds

• (42)

26

where J(r) =

d

• i=1
• LAiric
• (r)εi.

(43) Note that when t → 0, the singular term in (41) and (42) is

1 t E(r−1 t

Zxt, wt), but in opposite sign. To see the compatibility of these two formulae, note that 1 t E(r−1

t

Zxt, wt) = 1 t E(r−1

t

Zxt − r−1

0 Zx0, wt).

This means that the singular term will disappear, for the price that the derivative of Z will be involved. The formula (40) is not suitable to our purpose. 27

Let ct(f , Z) = DZT M

t f − T M t (DZf ),

f , Z ∈ C 1. (44) We have the following a priori estimate.

Theorem

Let q ≥ 2 and ε > 0. Then there are constants C1 and C2 such that for t ∈ [0, 1], ||ct(f , Z)||L1 ≤ C1 ||f ||Lp+ε||∇Z||Lq + C2(ric) √ t ||f ||Lp+ε||Z||Lq + ||f ||Lp||div(Z)||Lq, (45) where C1 is only dependent of ε and C2(ric) is dependent of ε, of the bound of the Ricci tensor and of J defined in (43) and 1/p + 1/q = 1. 28

References

• L. Ambrosio, Transport equation and Cauchy problem for BV

vector fields, Invent. Math. 158 (2004), 227–260.

• L. Ambrosio and A. Figalli, On flows associated to Sobolev

vector fields in Wiener space: an approach ` a la DiPerna-Lions, Preprint 2008.

• B. Driver, Integration by parts for heat kernel measures

revisited, J. Math. Pures Appl., 76 (1997), 703-737.

• B. Driver and A. Thalmaier, Heat equation derivative formulas

for vector bundles, J. Funct. Anal. 183 (2001), 42–108.

• D. Elworthy and X. Li, Formulae for the derivative of heat

semigroups, J. Funct. Anal. 125 (1994), 252-286.

• S. Fang and D. Luo, Transport equations and quasi-invariant

flows on the Wiener space, Preprint October 2008, Universit´ e de Bourgogne. 29