Exponential decay of correlations for a real-valued dynamical system embedded in R 2

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Exponential decay of correlations for a real-valued dynamical system embedded in R 2

Lisette Jager, Jules Maes, Alain Ninet

To cite this version:

Lisette Jager, Jules Maes, Alain Ninet. Exponential decay of correlations for a real-valued dynamical

system embedded in R 2. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2015, 353

(11), pp.1041 - 1045. �10.1016/j.crma.2015.07.015�. �hal-01881759�

Dynamical systems

Exponential decay of correlations for a real valued dynamical system embedded in R ²

Lisette Jager

^a

, Jules Maes

^a

Alain Ninet

^a

aLaboratoire de Math´ematiques, FR CNRS 3399, EA 4535, Universit´e de Reims Champagne-Ardenne, Moulin de la Housse, B. P. 1039, F-51687, Reims, France 03 26 91 83 92, fax 03 26 91 83 97

Received *****; accepted after revision +++++

Presented by

Abstract

We study the real valued process

{Xt, t∈N}

defined by

Xt+2

=

ϕ(Xt, Xt+1

), where the

Xt

are bounded. We aim at proving the decay of correlations for this model, under regularity assumptions on the transformation

ϕ.

To cite this article: L.Jager, J.Maes, A.Ninet, C. R. Acad. Sci. Paris, Ser. I 34XX (20XX).

R´esum´e

D´ecroissance des corr´elations pour une r´ecurrence `a 2 termes

On ´ etudie le processus r´ eel

{Xt, t ∈N}

d´ efini par

Xt+2

=

ϕ(Xt, Xt+1

), les

Xt

´ etant born´ es. Sous des hypoth` eses de r´ egularit´ e sur la transformation

ϕ,

on ´ etablit la d´ ecroissance des corr´ elations pour ce mod` ele. Pour citer cet article : L.Jager, J.Maes, A.Ninet, C.

R. Acad. Sci. Paris, Ser. I 34XX (20XX).

1. Introduction

Since the eighties, the study by statisticians of nonlinear time series has allowed to model a great number ot phenomena in Physics, Economics and Finance. Then in the nineties the theory of Chaos became an essential axis of research for the study of these processes. For an exhaustive review on this subject, one can consult Collet-Eckmann [2] about chaos theory and Chan-Tong [8,9] about nonlinear time series.

Within this framework, a general model could be written as

X

_t+1

= ϕ(X

_t

, . . . , X

_t−d+1

) + ε

_t

,

Email addresses: lisette.jager@univ-reims.fr (Lisette Jager),jules.maes@univ-reims.fr(Jules Maes), alain.ninet@univ-reims.fr(Alain Ninet).

where ϕ is nonlinear and ε

t

is a noise. We propose a first study of the “skeleton” of this model, as Tong calls it, beginning with d = 2 and, more precisely, of the dynamical system induced by this model. Indeed, we consider the model with bounded variables, X

t+2

= ϕ(X

t

, X

t+1

), with ϕ : U

²

→ U for U = [−L, L]

and L ∈ R

^∗+

, ϕ being defined piecewisely on U

²

. This model gives rise to a dynamical system (Ω, τ, µ, T ) where µ is a measure on the σ-algebra τ, invariant under the transformation T : Ω → Ω and Ω is a compact subset of R

²

. Under hypotheses on ϕ, which imply that T satisfies the hypotheses of Saussol [7], and if we suppose that T is mixing, we obtain the exponential decay of correlations. More precisely, for well-chosen applications f and h, there exist constants C = C(f, h) > 0, 0 < ρ < 1 such that:

Z

Ω

f ◦ T

^k

h dµ − Z

Ω

f dµ Z

Ω

hdµ

6 C ρ

^k

. This result can be seen as a covariance inequality of the following kind:

| Cov( f (X

k

), h(X

0

) ) | 6 C ρ

^k

.

Other ways could certainly be used to get the same result, under different hypotheses on the induced system, for example the method of Young towers [10]. To have a general view on these different technics, one can read the article of Alves-Freitas-Luzzato-Vaienti [1].

2. Hypotheses and results

Let L ∈ R

^∗+

. Let ϕ : [−L, L]

²

→ [−L, L] be piecewisely defined on

¹

[−L, L]

²

. To study the process {X

_t

, t ∈ N } defined by X

_t+2

= ϕ(X

_t

, X

_t+1

), there exist different ways of choosing the induced dynamical system Z

_t+1

= T(Z

_t

) with Z

_t

∈ R

²

. We tried two different approaches, on the one hand the canonical method, setting T (x, y) = (y, ϕ(x, y)) and on the other hand a double iteration, which comes down to setting T (x, y) = (ϕ(x, y), ϕ(y, ϕ(x, y))). The first approach, up to a conjugation, is the most fruitful, the second one requiring stronger hypotheses and yielding weaker results. We therefore set T(x, y) = (

^y_γ

, γϕ(x,

^y_γ

)) with Z

t

= (X

t

, γX

t+1

), for a suitable positive γ. It then became possible to work in spaces similar to Saussol’s V

α

and to use his results.

More precisely, we suppose that the following hypotheses are fulfilled:

(H1) There exists d ∈ N

^∗

such that [−L, L]

²

=

d

[

k=1

O

k

∪ N , where the O

k

are nonempty open sets, N is negligible for the Lebesgue measure and the union is disjoint. The edges of the O

k

can be split into a finite number of smooth components, each one included in a C

¹

, compact and one dimensional submanifold of R

²

.

(H2) There exists ε

1

> 0 such that, for all k ∈ {1, . . . d}, there exists an application ϕ

k

defined on B

ε₁

(O

k

) = {(x, y) ∈ R

²

, d((x, y), O

k

) ≤ ε

1

}, with values in R , such that ϕ

k

|

O_k

= ϕ|

O_k

.

(H3) The application ϕ

k

is bounded, belongs to the H¨ older class C

^1,α

on B

ε₁

(O

k

) for a real α ∈]0, 1]

²

. We moreover suppose that there exist A > 1 and M ∈]0, A − 1[ such that :

∀(u, v) ∈ B

_ε1

(O

_k

),

∂ϕ

k

∂u (u, v)

≥ A,

∂ϕ

k

∂v (u, v)

≤ M,

to ensure the expanding properties.

1. To get similar results on [a, b] instead of [−L, L], it suffices to conjugate by an affine application 2. IfϕkisC²onBε₁(Ok), it isC^1,αonBε₁(Ok) withα= 1

2

(H4) The open sets O

k

satisfy the following geometrical condition:

³

For all (u, v) and (u

⁰

, v) in B

ε₁

(O

k

), there exists a C

¹

path Γ = (Γ

1

, Γ

2

) : [0, 1] → B

ε₁

(O

k

) C

¹

joining (u, v) and (u

⁰

, v), whose gradient does not vanish and which satisfies

∀t ∈]0, 1[, |Γ

⁰₁

(t)| > M

A |Γ

⁰₂

(t)| .

(H5) Let Y ∈ N

^∗

be the maximal number of C

¹

components of N meeting at one point and set s = 2A + M

²

− M √

M

²

+ 4A 2

!

−1/2

< 1.

One supposes that

η := s

^α

+ 8s

π(1 − s) Y < 1.

We set γ = 1

√ A < 1 and, for all k ∈ {1, ..., d}, we denote by U

k

(resp. W

k

, N

⁰

) the image of O

k

(resp. B

ε1

(O

k

), N ) under the compression which associates (u, γv) with each (u, v) ∈ R

²

. The set Ω = [−L, L] × [−γL, γL], on which we shall be working, is the image of [−L, L]

²

under the same compression.

For every non negligible Borel set S of R

²

, for every f ∈ L

¹_m

( R

²

, R ), set Osc(f, S) = Esup

_S

f − Einf

_S

f,

where Esup

_S

and Einf

_S

are the essential supremum and infimum with respect to the Lebesgue measure m. One then defines:

|f |

α

= sup

0<ε<ε₁

ε

^−α

Z

R²

Osc(f, B

ε

(x, y)) dxdy , kf k

α

= kfk

_L1 m

+ |f |

α

and the set V

_α

= {f ∈ L

¹_m

( R

²

, R ), kf k

_α

< +∞}.

Let us introduce similar notions on Ω : for every 0 < ε

0

< γε

1

, for every g ∈ L

^∞_m

(Ω, R ), one defines N(g, α, L) = sup

0<ε<ε0

ε

^−α

Z

Ω

Osc(g, B

ε

(x, y) ∩ Ω) dxdy.

One then sets:

||g||

_α,L

= N(g, α, L) + 16(1 + γ)ε

^1−α₀

L||g||

_∞

+ ||g||

_L1 m

.

The function g is said to belong to V

α

(Ω) if the above expression is finite. The set V

α

(Ω) does not depend on the choice of ε

0

, whereas N and k.k

α,L

do.

There exist relationships between these two sets. Indeed, thanks to Proposition 3.4 of [7], one can prove the following result:

Proposition 2.1 (i) If g ∈ V

α

(Ω) and if one extends g as a function denoted by f, setting f (x, y) = 0 if (x, y) ∈ / Ω, then f ∈ V

α

and

kf k

α

≤ kgk

α,L

. (ii) Let f be in V

α

. Set g = f 1

Ω

. Then g ∈ V

α

(Ω) and one has

kgk

α,L

≤

1 + 16(1 + γ)L max(1, ε

^α₀

) πε

^1+α₀

kf k

α

.

3. In suitable cases, this hypothese can be replaced by a weaker but simpler one : for all points (u, v) and (u⁰, v) in Bε₁(Ok), the segment [(u, v),(u⁰, v)] is included inBε₁(Ok)

Under the above hypotheses (H1) to (H5), one obtains a first result:

Theorem 2.2 Let T be the transformation defined on Ω by : ∀(x, y) ∈ U

_k

: T (x, y) = T

k

(x, y) =

y

γ , γϕ

k

(x, y γ )

. Keeping the same formula, one extends the definition of T

k

to W

k

. Then

(i) The Frobenius-Perron operator P : L

¹_m

(Ω) → L

¹_m

(Ω) associated with T has a finite number of eigenvalues λ

1

, . . . , λ

r

of modulus one.

(ii) For each i ∈ {1, . . . , r}, the eigenspace E

i

= {f ∈ L

¹_m

(Ω) : P f = λ

i

f } associated with the eigenvalue λ

i

is finite dimensional and included in V

α

(Ω).

(iii) The operator P decomposes as

P =

r

X

i=1

λ

i

P

i

+ Q,

where the P

_i

are projections on the spaces E

_i

, k|P

_i

k|

₁

≤ 1 and Q is a linear operator defined on L

¹_m

(Ω), satisfying Q(V

_α

(Ω)) ⊂ V

_α

(Ω), sup

n∈N^∗

k|Q

ⁿ

k|

₁

< ∞ and k|Q

ⁿ

k|

_α,L

= O(q

ⁿ

) when n → +∞ for an exponent q ∈]0, 1[. Moreover, P

i

P

j

= 0 if i 6= j, P

i

Q = QP

i

= 0 for all i.

(iv) The number 1 is an eigenvalue of P . Set λ

1

= 1, let h

_∗

= P

1

1

Ω

and let dµ = h

_∗

dm. Then µ is the greatest absolutely continuous invariant measure (ACIM) of T , that is to say: if ν << m and if ν is T-invariant, then ν << µ.

(v) The support of µ can be decomposed into a finite number of disjoint measurable sets, on which a power of T is mixing. More precisely, for all j ∈ {1, 2, . . . , dim(E

1

)}, there exist an integer L

j

∈ N

^∗

and L

j

disjoint sets W

j,l

(0 ≤ l ≤ L

j

− 1) satisfying T (W

j,l

) = W

_j,l+1mod(Lj)

and T

^Lj

is mixing on every W

j,l

. We denote by µ

j,l

the normalized restriction of µ to W

j,l

, defined by

µ

j,l

(B) = µ(B ∩ W

j,l

)

µ(W

j,l

) , dµ

j,l

= h

^∗

1

W_j,l

µ(W

j,l

) dm.

The fact that T

^Lj

is mixing on every W

j,l

means that, for all f ∈ L

¹_µ

j,l

(W

j,l

) and all h ∈ L

^∞_µ

j,l

(W

j,l

),

t→+∞

lim < T

^tLj

f, h >

_µj,l

=< f, 1 >

_µj,l

< 1, h >

_µj,l

with the notations (indifferently employed) < f, g >

_µ⁰

= µ

⁰

(f g) = R

f g dµ

⁰

.

(vi) Moreover, there exist C > 0 and 0 < ρ < 1 such that, for all h in V

_α

(Ω) and f ∈ L

¹_µ

(Ω), one has

Z

Ω

f ◦ T

^k×ppcm(Li)

h dµ −

dim(E₁)

X

j=1 L_j−1

X

l=0

µ(W

j,l

) < f, 1 >

µ_j,l

< 1, h >

µ_j,l

≤ C||h||

α,Ω

||f ||

_L1 µ(Ω)

ρ

^k

. (vii) If, moreover, T is mixing

⁴

, then the preceding result can be written as follows: there exist C > 0

and 0 < ρ < 1 such that, for all h in V

α

(Ω) and f ∈ L

¹_µ

(Ω), one has:

Z

Ω

f ◦ T

^k

h dµ − Z

Ω

f dµ Z

Ω

hdµ

≤ C||h||

_α,Ω

||f ||

_L1 µ(Ω)

ρ

^k

.

4. which is equivalent to: if 1 is the only eigenvalue ofP with modulus one and if it is simple

4

We now come back to the initial problem and deduce from this result the invariant law associated with X

t

. If (X

t

)

t

is defined by X

0

, X

1

(valued in [−L, L]) and the recurrence relation X

t+2

= ϕ(X

t

, X

t+1

), one sets Z

t

= (X

t

, γX

t+1

). Then (Z

t

)

t

satisfies the recurrence relation Z

t+1

= T (Z

t

), which implies the following result (by comparing the marginal distributions):

Theorem 2.3 Suppose that the random variable Z

0

= (X

0

, γX

1

) has the density h

_∗

. Then, for all t ∈ N , Z

t

has the density h

_∗

and X

t

has the density

f : x 7→

Z

[−γL,γL]

h

_∗

(x, v) dv = γ Z

[−L,L]

h

_∗

(u, γx) du.

If F is defined on [−L, L], let T r F be the function defined on Ω by T r F (x, y) = F(x).

One then obtains the following result, which is a direct consequence of the sixth point of Theorem 2.2, applied to T r F and T r H :

Theorem 2.4 For every Borel set B and every interval I, if (X

0

, X

1

) has the invariant distribution, then

P X

_k×ppcm(Li)

∈ B, X

0

∈ I

−

dim(E1)

X

j=1 L_j−1

X

l=0

µ(W

j,l

) < T r 1

B

, 1 >

µ_j,l

< 1, T r 1

I

>

µ_j,l

≤ 16(1 + γ) C L

³

(10ε

^1−α₀

+ L) ρ

^k

.

More generally, let F, defined and measurable on [−L, L], be such that T r F belongs to L

¹_µ

(Ω). Let H ∈ L

^∞_m

([−L, L]) be such that sup

0<ε<ε0

ε

^−α

Z

[−L,L]

Osc(H, ]x − ε, x + ε[∩[−L, L]) dx < +∞.

Then T r H ∈ V

_α

(Ω) and

E(F (X

_k×ppcm(Li)

)H (X

0

)) −

dim(E1)

X

j=1 L_j−1

X

l=0

µ(W

j,l

)µ

j,l

(T r F )µ

j,l

(T r H)

≤ C(F, H) ρ

^k

with

C(F, H ) = C L ||T r F ||

L¹_µ

2γ sup

0<ε<ε0

ε

^−α

Z

[−L,L]

Osc(H, ]x − ε, x + ε[∩[−L, L]) dx

+16(1 + γ) ε

^1−α₀

||H||

L^∞_m([−L,L])

+ 2γ ||H ||

L¹_m([−L,L])

. If, moreover, T is mixing, then:

|Cov(F (X

k

), H(X

0

))| ≤ C(F, H ) ρ

^k

.

References

[1] Alves J.F., Freitas J.M., Luzatto S., Vaienti S.,From rates of mixing to recurrence times via large deviations, Advances in Mathematics, 228 (2) (2011), 1203-1236.

[2] Collet P., Eckmann J.-P., Concepts and results in chaotic dynamics: a short course. Theoretical and Mathematical Physics. Springer-Verlag, Berlin 2006.

[3] Hofbauer F., Keller G., Ergodic properties of invariant measures for piecewise monotonic transformations, Mathematische Zeitschrift 180 (1982), 119-140.

[4] Ionescu Tulcea C.T., Marinescu G., Th´eorie ergodique pour des classes d’op´erations non compl`etement continues, Annals of Mathematics 52 (2) (1950), 140-147.

[5] Lasota A., Mackey M.C., Chaos, fractals and noise : stochastic aspects of dynamics, Springer Verlag, New York 1998 [6] Liverani C., Multidimensional expanding maps with singularities: a pedestrian approach, Ergodic Theory and

Dynamical Systems 33 (1) (2013), 168-182.

[7] Saussol B., Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal of Mathematics 116 (2000), 223-248.

[8] Tong H., Nonlinear time series. A dynamical system approach. With an appendix by K. S. Chan. Oxford Statistical Science Series, 6. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York 1990.

[9] Tong H., Nonlinear time series analysis since 1990: some personal reflections. Acta Math. Appl. Sin. Engl. Ser. 18 (2) (2002), 177-184.

[10] Young L.-S., Recurrence times and rates of mixing, Israel Journal of Mathematics 110 (1999), 153-188.

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