An averaging principle for diffusions in foliated spaces

0.19 MB·English

by  Ivan I. Gonzales Gargate

An averaging principle for diffusions in foliated spaces
Iva´n I. Gonz´ales Gargate1 Paulo R. Ruffino2
Departamento de Matem´atica, Universidade Estadual de Campinas,
13.083-859- Campinas - SP, Brazil.
Abstract
3
1 Consider an SDE on a foliated manifold whose trajectories lay on compact
0
leaves. We investigate the effective behaviour of a small transversal perturba-
2
tion of order ǫ. An averageprinciple is shown to hold such that the component

n transversal to the leaves converges to the solution of a deterministic ODE, ac-
a
cording to the average of the perturbing vector field with respect to invariant
J
measuresontheleaves,asǫgoestozero. Anestimateoftherateofconvergence
1
is given. These results generalize the geometricalscope of previous approaches,
2
including completely integrable stochastic Hamiltonian system.

]
S
Key words: Averaging principle, foliated diffusion, rescaled stochastic systems,
D
stochastic flows.
.
h
t MSC2010 subject classification: 60H10, 58J65, 58J37.
a
m
[ 1 Introduction and set up

3
v Generally speaking, the original heuristic idea of an averaging principles refers to an
7
intertwining of two dynamics where one of them is, in some sense, much slower and
8
5 is affected somehow by the other faster dynamics. An averaging principle in this
1 case refers to the possibility of approximate, in some topology, the slow dynamics
.
2 considering only the average action or perturbation which the fast motion induces
1
on it. These ideas have appeared long ago and, as mentioned by V. Arnold [3,
2
1 p.287], they were implicitly contained in the works of Laplace, Lagrange and Gauss
: oncelestialmechanics; literatureonthemattercanbefounde.g. amongmanyothers
v
i in[3],Sanders,VerhulstandMurdoch[13]andreferencestherein. Presently, onwhat
X
regardsstochasticsystems,averaging hasbeenquiteanactive researchfieldonwhich
r
a there is also a vast literature on the topic. Interesting quick historical overviews can
be found in X.-M. Li [9, p.806], Kabanov and Pergamenshchikov [7, Appendix], [13,
Appendix A]. Among many other works somehow related to the topic, we refer to
Khasminskiand Krylov [8], Sowers [14], Namachchvaya andSowers [10]Borodin and
Freidlin [4], [7] and references therein.
Thespecificproblemthatweaddressinthisarticleisaperturbationofadiffusion
in a foliated manifold M such that the unperturbed random trajectories lay on the
leaves. The perturbation are taken transversal to the leaves of the foliation. Here,
the slow system is the transversal component and the fast system is given by the
1e-mail: [email protected].
2Corresponding author, e-mail: ruffi[email protected].
1
rescaled yǫ, where yǫ is the solution of the original SDE perturbed by a vector field
t t
ǫ
ǫK.
Our results generalize the recent approach by X.M. Li [9] on an averaging princi-
ple for a completely integrable stochastic Hamiltonian system. In that article, as in
the classical approach, see e.g. [3], Li has explored the benefits of a well structured
geometrical coordinates in the state space given by the coordinates of the Liouville
torus; these benefitsinclude vanishingItˆo-Stratonovich correction terms besides also
vanishing covariant derivative of Hamiltonian vector fields in tangent directions to
the leaves. We prove here that an averaging principle also holds in a generalized
geometrical scope, so that this averaging phenomenon occurs independently of sym-
plectic structures (but with possibly slower rates of convergence). Comparing to Li’s
previous result [9, Lemma 3.2], where the estimates contain a term of order 1/√t,
our corresponding estimates in Lemma 3.1 are continuous at ǫ = t = 0. Some of the
rates of convergence of [9, Lemma 3.1] is recovered as particular cases in Corollaries
2.2 and 2.3. In the main result, we show that in the average, the approximation goes
β
to zero faster than lnǫ −p, for β (0,1/2).
| | ∈
The set up. Let M be a smooth Riemannian manifold with an n-dimensional
smooth foliation, i.e. M is endowed with an integrable regular distribution of di-
mension n (for definition and further properties of foliated spaces see e.g. the initial
chapters of Tondeur [15], Walcak [16] among others). We denote by L the leaf of
x
thefoliation passingthroughapointx M. For simplicity, weshallassumethatthe
∈
leaves arecompact andthateach leaf L has atubularneighbourhoodU M where
x
⊂
U is diffeomorphic to L V, where V Rd is an open bounded neighbourhood of
x
× ⊂
the origin and d is the codimension of the foliation. We shall assume an SDE in M
whose solution flow preserves the foliation, i.e. we consider a Stratonovich equation
r
dx = X (x )dt+ X (x ) dBi (1)
t 0 t i t ◦ t
i=1
X
where the smooth vector fields X are foliated in the sense that X (x) T L , for
i i x x
∈
i = 0,1,...,r. Here B = (B1,...,Br) is a standard Brownian motion in Rr with
t t t
respect to a filtered probability space (Ω, , ,P). For an initial condition x , the
t 0
F F
trajectories of the solution x in this case lay on the leaf L a.s.. Moreover, there
t x0
exists a (local) stochastic flow of diffeomorphisms F : M M which restricted to
t
→
the initial leaf is a flow in the compact L .
x0
For a smooth vector field K in M, we shall denote the perturbed system by yε
t
which satisfies the SDE
r
dyε =X (yε)dt+ X (yε) dBi+εK(yε)dt, (2)
t 0 t i t ◦ t t
i=1
X
with the same initial condition yε = x of the unperturbed system x .
0 0 t
Our main result, Theorem 4.1, says that locally the transversal behaviour of
yε can be approximated in the average by an ordinary differential equation in the
t
ǫ
2
transversal space whose coefficients are given by the average of the transversal com-
ponent of the perturbation K with respect to the invariant measure on the leaves
for the original dynamics of Equation (1). The reader will notice by the end of
the proofs that compactness of the leaves in fact can be substituted by some other
boundedness conditions, added also to some rather technical adjustments which we
will not address here. In the Sections 2 and 3 we present the main lemmas. The
mainresultappearsinSection 4,wherewealsopresentasimpleillustrative example.
In particular, under some symmetry hypothesis on a foliated system embedded in
an Euclidean space, we use the main theorem to conclude that Lyapunov exponents
in the transversal direction must tend to zero as ǫ goes to zero, cf. Proposition 4.2.
2 Preliminaries results
Our coordinate system. Given an initial condition x M, let U M be a bounded
0
∈ ⊂
neighboorhoodofx whichisdiffeomorphictoL V andwhoseclosureU¯ M. By
0 x0× ⊂
compactness of L , there exists a finite number of local foliated coordinate systems
x0
ϕ : U W V Rn Rd, where W and V are open sets, say with 1 i k
i i i i
→ × ⊂ × ≤ ≤
and x U such that:
0 1
∈
1) U = k U ;
∪i=1 i
2) The leaf L = k ϕ 1(W 0 ), i.e. each U is diffeomorphic to the product
x0 ∪i=1 − i ×{ } i
of an open set (with the induced topology) in the leaf L and the vertical
x0
component V;
3) If a pair of points p U and q U in U belong to the same leaf then their
i j
∈ ∈
transversal coordinates in V are the same; i.e. π(ϕ (p)) = π(ϕ (p)) where π is
i j
the projection on the transversal space V;
4) For i = 1,...,k, ϕ has bounded derivatives (obtained reducing open set U if
i
necessary).
Note that for a fixed y ∈ V, the finite union ∪kϕ−i 1(Wi,y) is the leaf Lϕ−i 1(x,y)
for any x W . Natural examples of this scheme of coordinates systems appear if
i
∈
we consider compact foliation given by the inverse image of submersions: values in
the image space provide local coordinates for the vertical space V.
Next lemma gives information on the order of which the perturbed trajectories
yǫ approaches the unperturbed x when one varies ǫ and t in equation (2); it will
t t
be used to prove that the dynamics of the rescaled system yǫ is such that its time
t
ǫ
average foranyfunctiong inM approximates thetimeaverage ofthespacialaverage
of g on the leaves, Lemma 3.1. An exponential factor in the estimates is expected,
as trivial linear examples show.
We shall denote the coordinates of a point p U by ϕ (p)= (u,v) Rn Rd.
1 1
∈ ∈ ×
3
Lemma 2.1 Let τǫ be the first time the process yǫ exitsthe foliated coordinate neigh-
t
bourhood U as above. For any locally Lipschitz continuous function f :M R and
1
→
2 p < we have
≤ ∞
1
E sup f(yǫ) f(x ) p p K εteK2tp.
(cid:20) (cid:18)s≤t∧τǫ| s − s | (cid:19)(cid:21) ≤ 1
where K ,K 0 are constants depending on upper bounds of the norms of the
1 2
≥
perturbing vector field K, on the Lipschitz coefficients of f and on the derivatives of
X ,X ,X with respect to the coordinate system.
0 1 r
···
Proof: Initially write x and yǫ, the solutions of Equations (1) and (2) respectively,
t t
accordingtothefoliatedcoordinatesϕ . Sowewrite(u ,v ):= ϕ (x )and(uǫ,vǫ):=
1 t t 1 t t t
ϕ (yǫ). Then
1 t
f(yǫ) f(x ) = f ϕ 1(uǫ,vǫ) f ϕ 1(u ,v ) (3)
| t − t | ◦ −1 t t − ◦ −1 t t
C uǫ u +C vǫ v ,
≤ (cid:12)(cid:12) | t − t| | t − t| (cid:12)(cid:12)
for some constant C 0, using the fact that U is relatively compact.
1
We shall denote u≥ = (u1, ,un), v = (v1, ,vd), uǫ = (uǫ,1, ,uǫ,n), vǫ =
t t ··· t t t ··· t t t ··· t t
ǫ,1 ǫ,d
(v , ,v ). We also split the horizontal and vertical component of the per-
t ··· t
turbing vector field K˜ = (K ,K ), into coordinates: K = (K1,...,Kn,) and
u v u u u
K = (K1,...,Kd,).
v v v
Inourcoordinatesystem,theequationsofthehorizontalandverticalcomponents
of the perturbed system uǫ and vǫ are given by
t t
r
duǫ,i = bi(uǫ,vǫ) dBk +bi(uǫ,vǫ)dt+ǫKi(uǫ,vǫ)dt, (4)
t k t t ◦ t 0 t t u t t
k=1
X
dvǫ,j = ǫKj(uǫ,vǫ)dt, (5)
t v t t
with i = 1,2, ,n and j = 1,2, ,d, for the induced vector fields b ,b ,...b
0 1 r
··· ···
which, together with their derivatives, are also bounded. From equation (5) we have
s
sup vǫ v ǫ sup K (uǫ,vǫ) ds
s≤t∧τǫ| s− s| ≤ s≤t∧τǫZ0 | v s s |
ǫ t sup K (x) = C ǫ t, (6)
v 1
≤ | |
x U
∈
where C = sup K(x) . From equation (4) we have in each i-th component, for
s < τǫ: 1 x∈U | |
r s
uǫ,i ui = (bi(uǫ,vǫ) bi(u ,v )) dBk+ (7)
s − s k r r − k r r ◦ r
k=1Z0
X
s s
(bi(uǫ,vǫ) bi(u ,v ))dr+ǫ Ki(uǫ,vǫ)dr. (8)
0 r r − 0 r r u r r
Z0 Z0
4
In terms of Itˆo integral,
s s
(bi(uǫ,vǫ) bi(u ,v )) dBk = (bi(uǫ,vǫ) bi(u ,v ))dBk
k r r − k r r ◦ r k r r − k r r r
Z0 Z0
1 s
+ [ bi b (uǫ,vǫ) bi b (u ,v )]dr.
2 ∇ k · k r r −∇ k · k r r
Z0
Hence, taking the absolute values in both sides of Equation (8) we get, for each i:
r s
uǫ,v ui (bi(uǫ,vǫ) bi(u ,v ))dBk +
s − s ≤ k r r − k r r r
k=1(cid:12)Z0 (cid:12)
(cid:12) (cid:12) X(cid:12) (cid:12)
(cid:12) (cid:12) 1 r(cid:12) s (cid:12)
(cid:12) bi b (uǫ,vǫ) bi b (cid:12)(u ,v ) dr
2 ∇ k · k r r −∇ k · k r r
k=1Z0
X (cid:12) (cid:12)
s (cid:12) s (cid:12)
+ bi(uǫ,vǫ) bi(u ,v ) dr+ǫ Ki(uǫ,vǫ) dr. (9)
0 r r − 0 r r u r r
Z0 Z0
(cid:12) (cid:12) (cid:12) (cid:12)
Functions bi and ( bi b ) ar(cid:12)e Lipschitz, hence for(cid:12)a common(cid:12)constant C(cid:12) ,
0 ∇ k · k 2
r s
uǫ,i ui (bi(uǫ,vǫ) bi(u ,v ))dBk +
(cid:12) s − s(cid:12) ≤ (cid:12)(cid:12)Xk=1Z0 k r r − k r r r(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12)s s (cid:12)
C (cid:12) vǫ v dr+C uǫ u dr+(cid:12) ǫs sup K . (10)
2 (cid:12) | r − r| 2 | r − r| (cid:12) | u|
Z0 Z0 U
The first deterministic integral, together with inequality (6) yields:
r s
uǫ,i ui (bi(uǫ,vǫ) bi(u ,v ))dBk +C C ǫs2
s − s ≤ k r r − k r r r 1 2
k=1(cid:12)Z0 (cid:12)
(cid:12) (cid:12) X(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) s (cid:12)
+C(cid:12) uǫ u dr+C ǫs. (cid:12)
2 | r − r| 1
Z0
Now, for p 1, there exists a constant C such that
3
≥
r s p
uǫ,i ui p C (bi(uǫ,vǫ) bi(u ,v ))dBk +C C C ǫs2 p
s − s ≤ 3 k r r − k r r r 3 1 2
k=1(cid:12)Z0 (cid:12)
(cid:12) (cid:12) X(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12) (cid:12) s p (cid:12)
+C Cp(cid:12) uǫ u dr +C (C ǫs(cid:12))p.
3 2 | r − r| 3 1
(cid:18)Z0 (cid:19)
Cauchy-Schwartz inequality yields:
r s p
uǫ,i ui p C (bi(uǫ,vǫ) bi(u ,v ))dBk +C C C ǫs2 p
s − s ≤ 3 k r r − k r r r 3 1 2
k=1(cid:12)Z0 (cid:12)
(cid:12) (cid:12) X(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12) (cid:12) s (cid:12)
+C Cp(cid:12)sp 1 uǫ u pdr+C (C ǫ(cid:12)s)p.
3 2 − | r − r| 3 1
Z0
5
Hence,
r s p
E sup uǫ,i ui p C E sup (bi(uǫ,vǫ) bi(u ,v ))dBk +C C C ǫt2 p
s≤t∧τǫ(cid:12) s − s(cid:12) ≤ 3 s≤t∧τǫXk=1(cid:12)(cid:12)Z0 k r r − k r r r(cid:12)(cid:12) 3(cid:0) 1 2 (cid:1)
(cid:12) (cid:12) (cid:12) s (cid:12)
+C Cp tp 1 E (cid:12)sup uǫ u pdr+C (C ǫ(cid:12)t)p
3 2 − s≤t∧τǫZ0 | r − r| 3 1
r t τǫ p/2
C E ∧ (bi(uǫ,vǫ) bi(u ,v ))2dr +C C C ǫt2 p
≤ 4 k r r − k r r 3 1 2
k=1 (cid:20)Z0 (cid:21)
X (cid:0) (cid:1)
t
+C Cp tp 1 E sup uǫ u p dr+C (C ǫt)p
3 2 − Z0 (cid:18)s≤r∧τǫ| r − r| (cid:19) 3 1
wherewehave usedclassical Lp-inequality formartingales (e.g. RevuzandYor [11]).
Using again the Lipchitz property of each b for the terms in the brackets above:
k
r t τǫ
∧ (bi(uǫ,vǫ) bi(u ,v ))2dr
k r r − k r r
k=1Z0
X
t τǫ t τǫ
2C2 ∧ vǫ v 2dr+ ∧ uǫ u 2dr
≤ 2 | r − r| | r − r|
(cid:18)Z0 Z0 (cid:19)
t t τǫ
2C2 C2ǫ2r2 dr+ ∧ sup uǫ u 2dr
≤ 2 (cid:18)Z0 1 Z0 s≤r∧τǫ| r − r| (cid:19)
t τǫ
C2C2ǫ2t3+2C2 ∧ sup uǫ u 2dr. (11)
≤ 2 1 2 Z0 s≤r∧τǫ| r − r|
We end up with:
r t τǫ p/2
E sup uǫ,i ui p C E C2C2ǫ2t3+2C2 ∧ sup uǫ u 2dr
s≤t∧τǫ(cid:12) s − s(cid:12) ≤ 4Xk=1 (cid:20) 2 1 2 Z0 s≤r∧τǫ| r − r| (cid:21)
(cid:12) (cid:12) t
+C Cptp 1 E sup uǫ u p dr
3 2 − Z0 (cid:18)s≤r∧τǫ| r − r| (cid:19)
+C C C ǫt2 p+C (C ǫt)p. (12)
3 1 2 3 1
(cid:0) (cid:1)
Forp 2onecanuseCauchy-Schwartzagaintoconcludethatthereexistsaconstant
≥
C such that the last expression is less than or equal
5
C5(cid:16)C2C1ǫt3/2(cid:17)p+C5C2ptp−22 Z0t∧τǫEs≤sur∧pτǫ|uǫr −ur|pdr+C3 C1C2ǫt2 p
t (cid:0) (cid:1)
+C Cptp 1 E sup uǫ u p dr+C (C ǫt)p
3 2 − Z0 (cid:18)s≤r∧τǫ| r − r| (cid:19) 3 1
6
= C C C ǫt3/2 p+C C C ǫt2 p+C (C ǫt)p
5 2 1 3 1 2 3 1
+(cid:16)(cid:16)C5C2ptp−22 +(cid:17)C3C2pt(cid:0)p−1(cid:17)Z0tE(cid:1)(cid:18)s≤sur∧pτǫ|uǫr −ur|p(cid:19)dr.
Now, summing up over i in the inequalities above leads to
E sup uǫ u p C C C ǫt3/2 p+C C C ǫt2 p+C (C ǫt)p
s t τǫ| r − r| ≤ 5 2 1 3 1 2 3 1
≤ ∧ (cid:16) (cid:17)
(cid:0) t (cid:1)
+(cid:16)C5C2ptp−22 +C3C2ptp−1(cid:17)Z0 E(cid:18)s≤sur∧pτǫ|uǫr −ur|p(cid:19)dr.
We use now the integral form of Gronwall’s inequality to find that:
E sup uǫ u p C ǫptp(1+tp)exp C (tp/2+tp)
(cid:18)s≤t∧τǫ| r − r| (cid:19) ≤ 6 { 7 }
C ǫptp(1+tp)exp C tp .
8 9
≤ { }
Going back to the inequality 3, now we have
f(yǫ) f(x )p C vǫ v p+C uǫ u p
| t − t | ≤ 10| t − t| 10| t − t|
hence:
E sup f(yǫ) f(x )p C E sup vǫ v p+C E sup uǫ u p
(cid:18)s≤t∧τǫ| s − s | (cid:19) ≤ 10 s≤t∧τǫ| s− s| 10 s≤t∧τǫ| s− s|
C ǫptp+C ǫptp(1+tp)exp(C tp)
11 12 9
≤
C ǫptp(1+tp)exp(C tp).
13 9
≤
From here, finally, one concludes that there exist constants K and K such that
1 2
1
p
E sup f(yǫ) f(x )p K ǫtexp(K tp).
(cid:18)s≤t∧τǫ| s − s | (cid:19) ≤ 1 2
(cid:3)
NextcorollaryincludesthecaseofacompletelyintegrablestochasticHamiltonian
system when one uses the action-angle coordinates, cf. X.-M. Li [9, Lemma 3.1]).
Corollary 2.2 If the vector fields X , ,X depend only on the vertical coordinate
0 r
···
(null derivative in the directions of the leaves, as in the Hamiltonian case [9]) then
the estimates above can be improved, and for p 1 there exists a constant K such
1
≥
that
1
p
E sup f(yǫ) f(x ) p K ε(t+t2).
(cid:20) (cid:18)s≤t∧τǫ| s − s | (cid:19)(cid:21) ≤ 1
7
Proof: In this case the correction term of the Stratonovich stochastic integral in
terms of Itˆo integral in inequality (9) vanishes, and also so does the determinist
integration of uǫ u in inequalities (10) and (11). Hence inequality (12) improves
| r− r|
to
E sup uǫ,i ui p C C C ǫt3/2 p+C C C ǫt2 p+C (C ǫt)p. (13)
s t τǫ s − s ≤ 5 2 1 3 1 2 3 1
≤ ∧ (cid:12) (cid:12) (cid:16) (cid:17) (cid:0) (cid:1)
The argument(cid:12) in the r(cid:12)est of the proof follows straightforward for p 1 skipping
≥
Gronwall inequality.
(cid:3)
Next corollary includes the case X 0, cf. [9, Lemma 3.1(2)] for stochastic
0
≡
Hamiltonian systems with action-angle coordinate system.
Corollary 2.3 If in addition to conditions of Corollary 2.2 above, we have that the
deterministic vector field X is constant when represented with respect to a certain
0
local coordinate system in U (i.e. b has null derivative w.r.t u and v) then, for
1 0
3
p 1 the estimates can be improved further to K1ε(t+t2).
≥
Proof: Besides the vanishing terms already mentioned above, the second determin-
istic integral on the right hand side of inequality (9) also vanishes. Hence inequality
(12) simplifies further to
E sup uǫ,i ui p C C C ǫt3/2 p+C (C ǫt)p.
s t τǫ s − s ≤ 5 2 1 3 1
≤ ∧ (cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12) (cid:3)
Yet, from the proof of the Lemma 2.1 we have the following
Remark 2.4 For 1 p < 2 and t sufficiently small, there exist constants K and
1
≤
K such that
2
1
p
E sup f(yǫ) f(x ) p K εt exp(K tp). (14)
(cid:20) (cid:18)s≤t∧τǫ| s − s | (cid:19)(cid:21) ≤ 1 2
Proof: One can no longer use Cauchy-Schwartz after inequality (12). Alternatively,
from (12), use that
E sup uǫ u p C C C ǫt3/2 p+C Cptp/2E sup uǫ,i ui p+C C C ǫt2 p
s t τǫ| s− s| ≤ 5 2 1 5 2 s t τǫ s − s 3 1 2
≤ ∧ (cid:16) t (cid:17) ≤ ∧ (cid:12) (cid:12) (cid:0) (cid:1)
+C Cp E sup uǫ u p dr+(cid:12) C Cpǫp(cid:12)tp.
3 2 Z0 (cid:18)s≤r∧τǫ| r − r| (cid:19) 3 1
If we fix an 0 < δ < 1 and take t sufficiently small such that 1 C Cptp/2 > δ then
− 5 2
E sup uǫ u p δ 1C C C ǫt3/2 p+δ 1C C C ǫt2 p
s t τǫ| s− s| ≤ − 5 2 1 − 3 1 2
≤ ∧ (cid:16) (cid:17)
t (cid:0) (cid:1)
+δ 1C Cptp 1 E sup uǫ u p dr+δ 1C Cpǫptp.
− 3 2 − Z0 (cid:18)s≤r∧τǫ| r − r| (cid:19) − 3 1
8
And one completes the calculation as before using the integral version of Gronwall
inequality.
(cid:3)
3 Averaging functions on the leaves
Consider a differentiable function g : M R. The leaf L passing through a point
p
→
p M contains the support of an invariant measure µ for the unperturbed system
p
∈
(1); we shall assume that µ is ergodic. We shall work with the following function
p
defined for each leaf Qg : V Rd R given by the average of g with respect
⊂ →
to these measures on the leaves. Namely, if v is the vertical coordinate of p, i.e.
p = ϕ(u,v), then:
Qg(v) = g(x)dµ (x).
p
ZLp
WeassumethatQg hassomedegreeofcontinuity withrespecttov. Thisassumption
appears in two levels in Lemma 3.4: 1) Riemann integrability of Qg(π(yǫ)) with
r
respect to r guarantees the convergence to zero; 2) α-H¨older continuity guarantees
the rate of convergence.
Our next step is to estimate the time average of g and Qg along the perturbed
system yǫ. To use the notations and results we have introduced before in local
t
coordinates, we shall write simply π(p)= v for the composition of the projection on
the second coordinates with the local chart ϕ(p) = (u,v).
Here the stopping time τǫ denotes the first exit time of the open neighbourhood
U M which is diffeomorphic to L V. We have the following estimates for the
⊂ x0 ×
difference of the averages of functions g and Qg.
Lemma 3.1 Given a function g : M R let Qg : V R be its average on the
→ →
leaves. For s,t 0 write
≥
(s+t) ǫτǫ
δ(ǫ,t) = ∧ g(yǫ) Qg(π(yǫ)) dr.
r r
Zs∧ǫτǫ ǫ − ǫ
Then δ(ǫ,t) goes to zero when t or ǫ tend to zero.
Moreover, if Qg is α-Ho¨lder continuous with α > 0 then for p 1 and any
≥
β (0,1/2) we have the following estimates:
∈
1
Esup δ(ǫ,s) p p √t lnǫ −βph(t,ǫ),
| | ≤ | |
(cid:18) s≤t (cid:19)
where h(t,ǫ) is continuous for t,ǫ > 0 and converges to zero when (t,ǫ) 0.
→
Proof:
The proof consists of considering a convenient partition of the interval (s/ǫ
∧
τǫ,(s+t)/ǫ τǫ) where we can get the estimates by comparing in each subinterval
∧
the average of the flow of the original system (on the corresponding leaf) with the
9
average of the perturbed flow (possibly transversal to the leaves). These estimates
in each subinterval are obtained using Lemma 2.1. So, a key point in the proof is a
careful choice of the increments of such a convenient partition. For sufficiently small
ǫ, we take the following assignment of increments:
(s+t) τǫ s τǫ
∆t = ∧ − ∧ .
2β
lnǫ − p
| |
Hence, the partition t = s τǫ+n∆t, for 1 n N 1, is such that
n ǫ ∧ ≤ ≤ −
s s+t
τǫ = t < t < < t < τǫ.
0 1 N 1
ǫ ∧ ··· − ǫ ∧
2β
with N = N(ǫ)= [ǫ−1 lnǫ − p ] where here [x] denotes the integer part of x.
| |
Initially we represent the left hand side as the sum:
ǫ s+ǫt∧τǫg(yǫ)dr = ǫN−1 tn+1g(yǫ)dr+ǫ s+ǫt∧τǫg(yǫ)dr.
r r r
Zsǫ∧τǫ nX=0Ztn ZtN
Denote by θ the canonical shift operator on the probability space. Let F (,ω)
t t
·
witht 0betheflowoftheoriginalunperturbedsysteminM. Triangularinequality
≥
splits our calculation into four parts
δ(ǫ,t) A + A + A + A , (15)
1 2 3 4
| | ≤ | | | | | | | |
where
N−1 tn+1
A = ǫ g(yǫ) g(F (yǫ ,θ (ω))) dr,
1 n=0Ztn r − r−tn tn tn
X (cid:2) (cid:3)
N−1 tn+1
A = ǫ g(F (yǫ ,θ (ω)))dr ∆tQg(π(yǫ )) ,
2 n=0(cid:20)Ztn r−tn tn tn − tn (cid:21)
X
N 1 (s+t) ǫτǫ
A = − ǫ∆tQg(π(yǫ )) ∧ Qg(π(yǫ))dr,
3 nX=0 tn −Zs∧ǫτǫ rǫ
s+t τǫ
A = ǫ ǫ ∧ g(yǫ)dr.
4 r
ZtN
We proceed showing that each of the processes A ,A ,A and A above tends to
1 2 3 4
zerouniformlyoncompactintervals. Inwhatfollows, wewillexploremanytimes the
factthatfora > 0andb R, ǫa lnǫ b goes tozeroasǫ 0. Hence, byconstruction,
∈ | | ց
except when τǫ s (where, restricted to which the lemma is trivial) both ∆t and N
≤
go to infinity when ǫ tends to zero.
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