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Rigorous estimation of the speed of convergence to equilibrium. S. Galatolo Dip. Mat, Univ. Pisa Computation in Dynamics, ICERM, 2016 Computation in Dynamics, ICERM, 2016 S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of

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Rigorous estimation of the speed of convergence to equilibrium.

  • S. Galatolo
  • Dip. Mat, Univ. Pisa

Computation in Dynamics, ICERM, 2016

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Overview

Many questions on the statistical behavior of a dynamical system are related to the speed of convergence to equilibrium: a measure of the speed of convergence to the limit Lnm → µ. We will see a tool for the rigorous computer aided explicit estimation of this rate of convergence and one example of application to the computation of the diffusion coefficient; Topics mainly from joint works with: W. Bahsoun, M. Monge, I. Nisoli, X. Niu, B. Saussol.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Dynamics and the evolution of a measure

The transfer operator

Let us consider a metric space X with a dynamics defined by T : X → X. Let us also consider the space PM(X) of probability measures on X. Define the function L : PM(X) → PM(X) in the following way: if µ ∈ PM(X) then: Lµ(A) = µ(T −1(A)) Considering measures with sign (SM(X) ) or complex valued measures we have a vector space and L is linear.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Dynamics and the evolution of a measure

The transfer operator

Let us consider a metric space X with a dynamics defined by T : X → X. Let us also consider the space PM(X) of probability measures on X. Define the function L : PM(X) → PM(X) in the following way: if µ ∈ PM(X) then: Lµ(A) = µ(T −1(A)) Considering measures with sign (SM(X) ) or complex valued measures we have a vector space and L is linear. Invariant measures are fixed points of the transfer operator L.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Dynamics and the evolution of a measure

The transfer operator

Let us consider a metric space X with a dynamics defined by T : X → X. Let us also consider the space PM(X) of probability measures on X. Define the function L : PM(X) → PM(X) in the following way: if µ ∈ PM(X) then: Lµ(A) = µ(T −1(A)) Considering measures with sign (SM(X) ) or complex valued measures we have a vector space and L is linear. Invariant measures are fixed points of the transfer operator L. Many results come from the understanding of the properties of the action of this operator on spaces of suitably regular measures.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Convergence to equilibrium

Consider two spaces of measures with sign Bs ⊆ Bw , with norms || ||s ≥ || ||w and the set of zero average measures V = {ν ∈ Bs|ν(X) = 0}

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Convergence to equilibrium

Consider two spaces of measures with sign Bs ⊆ Bw , with norms || ||s ≥ || ||w and the set of zero average measures V = {ν ∈ Bs|ν(X) = 0} We say that L has convergence to equilibrium with speed Φ if for each g ∈ V ||Lng||w ≤ Φ1(n) ||g||s

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Convergence to equilibrium

Consider two spaces of measures with sign Bs ⊆ Bw , with norms || ||s ≥ || ||w and the set of zero average measures V = {ν ∈ Bs|ν(X) = 0} We say that L has convergence to equilibrium with speed Φ if for each g ∈ V ||Lng||w ≤ Φ1(n) ||g||s e.g. ||Lng||w ≤ Cλ−n||g||s

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Convergence to equilibrium

Consider two spaces of measures with sign Bs ⊆ Bw , with norms || ||s ≥ || ||w and the set of zero average measures V = {ν ∈ Bs|ν(X) = 0} We say that L has convergence to equilibrium with speed Φ if for each g ∈ V ||Lng||w ≤ Φ1(n) ||g||s e.g. ||Lng||w ≤ Cλ−n||g||s Let us consider a starting probability measure ν ∈ Bs and µ invariant, since (Lnν − µ) ∈ V , this estimates the speed ||Lnν − µ||w → 0

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Speed of contraction of zero average measures

Consider Bsand the set of zero average measures V = {ν ∈ Bs|ν(X) = 0}

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Speed of contraction of zero average measures

Consider Bsand the set of zero average measures V = {ν ∈ Bs|ν(X) = 0} We say that V has speed of contraction Φ if for each g ∈ V ||Lng||s ≤ Φ2(n) ||g||s

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Speed of contraction of zero average measures

Consider Bsand the set of zero average measures V = {ν ∈ Bs|ν(X) = 0} We say that V has speed of contraction Φ if for each g ∈ V ||Lng||s ≤ Φ2(n) ||g||s e.g. ||Lng||s ≤ Cλ−n||g||s

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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An approach to the problem

Speed of convergence to equilibrium ||Lng||w ≤ Φ1(n) ||g||s ||Lng||s ≤ Φ2(n) ||g||s We will see that: a low resolution information coming from a computer estimation + the knowledge of the fine scale behavior of the transfer operator, due to its regularizing action on a suitable space = information on the rate of convergence.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Assumptions: Regularizing action of the operator

Suppose L satisfies a Lasota Yorke (Doeblin Fortet) inequality: there is λ1 < 1 s.t. ||Lnν||s ≤ Aλn

1||ν||s + B||ν||w .

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Assumptions: Regularizing action of the operator

Suppose L satisfies a Lasota Yorke (Doeblin Fortet) inequality: there is λ1 < 1 s.t. ||Lnν||s ≤ Aλn

1||ν||s + B||ν||w .

This is satisfied for example with || ||s = || ||BV and || ||w = || ||1 for piecewise expanding maps

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Assumptions: Regularizing action of the operator

Suppose L satisfies a Lasota Yorke (Doeblin Fortet) inequality: there is λ1 < 1 s.t. ||Lnν||s ≤ Aλn

1||ν||s + B||ν||w .

This is satisfied for example with || ||s = || ||BV and || ||w = || ||1 for piecewise expanding maps and with suitable anisotropic norms for (piecewise) hyperbolic systems.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Assumptions: The construction of a low resolution approximation

Consider a projection πδ on a finite dimensional space. Suppose ||πδf − f ||w ≤ Cδ||f ||s.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Assumptions: The construction of a low resolution approximation

Consider a projection πδ on a finite dimensional space. Suppose ||πδf − f ||w ≤ Cδ||f ||s. Supppose πδ and L are weak contractions for the norm || ||w

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Assumptions: The construction of a low resolution approximation

Consider a projection πδ on a finite dimensional space. Suppose ||πδf − f ||w ≤ Cδ||f ||s. Supppose πδ and L are weak contractions for the norm || ||w Define the finite rank approximation Lδ of L as Lδ = πδLπδ.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Assumptions: The construction of a low resolution approximation

Consider a projection πδ on a finite dimensional space. Suppose ||πδf − f ||w ≤ Cδ||f ||s. Supppose πδ and L are weak contractions for the norm || ||w Define the finite rank approximation Lδ of L as Lδ = πδLπδ. Suppose that there is n1 such that ∀v ∈ V , ||Ln1

δ (v)||w ≤ λ2||v||w

(1) with λ2 < 1

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Ulam Discretization

Space X discretized by a partition Iδ

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Ulam Discretization

Space X discretized by a partition Iδ Corresponding Lδ can be seen as: let Fδ be the σ−algebra associated to Iδ, then: πδ : SM(X) → L1(X) (2) πδ(g) = E(g|Fδ) (3) Lδ(f ) = πδLπδ (4)

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Ulam Discretization

Space X discretized by a partition Iδ Corresponding Lδ can be seen as: let Fδ be the σ−algebra associated to Iδ, then: πδ : SM(X) → L1(X) (2) πδ(g) = E(g|Fδ) (3) Lδ(f ) = πδLπδ (4) System approximated by a Markov Chain with probabilities Pij = m(T −1(Ij) ∩ Ii)/m(Ii)

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Ulam Discretization

Space X discretized by a partition Iδ Corresponding Lδ can be seen as: let Fδ be the σ−algebra associated to Iδ, then: πδ : SM(X) → L1(X) (2) πδ(g) = E(g|Fδ) (3) Lδ(f ) = πδLπδ (4) System approximated by a Markov Chain with probabilities Pij = m(T −1(Ij) ∩ Ii)/m(Ii) Much literature on the (more or less rigorous) approximation of invariant measures and other by this method (e.g. Bose, Bahsoun, Ding, Froyland, Keane, Li, Murray,Young, Zhou...)

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Lemma

Under the previous assumption Lδ, L satisfy an approximation inequality: ∃ C, D such that ∀ν, ∀n ≥ 0: ||(Ln

δ − Ln)ν||w ≤ δ(C||ν||s + nD||ν||w ).

(5)

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Convergence to equilibrium

Putting together in a system Lasota-Yorke and the previous lemma: starting measure g0 ∈ V , let us denote gi+1 = Ln1gi.

  • ||Ln1gi||s ≤ Aλn1

1 ||gi||s + B||gi||w

||Ln1gi||w ≤ ||Ln1

δ gi||w + δ(C||gi||s + n1D||gi||w ) ,

(6)

  • ||Ln1gi||s ≤ Aλn1

1 ||gi||s + B||gi||w

||Ln1gi||w ≤ λ2||gi||w + δ(C||gi||s + n1D||gi||w ) .

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Convergence to equilibrium

Compacting it in a vector notation, ||gi+1||s ||gi+1||w

  • Aλn1

1

B δC δn1D + λ2 ||gi||s ||gi||w

  • (7)

here indicates the component-wise ≤ relation (both coordinates are less

  • r equal).

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Convergence to equilibrium

Let M = Aλn1

1

B δC δn1D + λ2

  • . What said above allows to bound

||gi||s and ||gi||w by a sequence ||gi||s ||gi||w

  • Mi

||g0||s ||g0||w

  • which can be computed explicitly. This gives a way have an explicit

estimate on the speed of convergence for the norms || ||s and || ||w at a given time. ||Lkn1g||s ≤ C1ρk||g||s. A similar approach allows the estimation of escape rates

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Example

1−d Lorenz map T(x) =

  • θ · |x − 1/2|α

0 ≤ x < 1/2 1 − θ · |x − 1/2|α 1/2 < x ≤ 1 with α = 57/64 and θ = 109/64. L is the transfer operator associated to F = T 4 The matrix that corresponds to our data is such that M

  • 0.2915

4049 7.75·10−8 0.058

  • .

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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We have the following estimates: LkgBV ≤ (16356) · (0.387) k

10||g||BV .

LkgL1 ≤ (4050) · (0.387) k

10||g||BV .

We can also use the coefficients of the powers of the matrix (computed using interval arithmetics) to obtain upper bounds as in the following table: iterations bound for ||Lhg||1 h = 20 3 · 10−6||g||BV + 3.5 · 10−2||g||1 h = 40 5 · 10−7||g||BV + 5.1 · 10−3||g||1 h = 60 7 · 10−8||g||BV + 7.6 · 10−4||g||1 h = 80 10−8||g||BV + 1.2 · 10−4||g||1

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Application: estimating the diffusion coefficient

Theorem (Diffusion coefficient for the Lanford map)

Let T(x) = 2x + 1 2x(1 − x) (mod 1). (8) (a) T admits a unique absolutely continuous invariant measure ν and if ψ is a function of bounded variation the Central Limit Theorem holds: 1 √n

  • n−1

∑

i=0

ψ(T ix) − n

  • I ψdν
  • law

− →N (0, σ2). (b) For ψ = x2 the diffusion coefficient σ2 ∈ [0.3458, 0.4152].

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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How can it be estimated rigorously

: σ2 is known to have the following expression σ2 :=

  • I

ˆ ψ2hdm + 2

∞

∑

i=1

  • I Li( ˆ

ψh) ˆ ψdm, (9) where ˆ ψ := ψ − avg and avg :=

  • I ψhdm.

Applying the above technique to the Lanford map one obtains: L28k( ˆ ψh)L1 ≤ (1.007) × 0.05k ˆ ψhBV Thus we can find l such that

∞

∑

i=l

  • I Li( ˆ

ψh) ˆ ψdm is as small as wanted. Reducing the estimation to a finite sum.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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An informal verification

Taking 200 iterates and 20000 starting points compare the distribution of deviations from averages with the estimated normal distibution.

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19

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Essential referencess

  • S. Galatolo, I. Nisoli, B. Saussol An elementary way to rigorously

estimate convergence to equilibrium and escape rates J. Comp. Dyn. (2015)

  • W. Bahsoun, S. Galatolo, I. Nisoli, X. Niu Rigorous approximation of

diffusion coefficients for expanding maps. J. Stat Phys. (2016)

S.Galatolo (Dip. Mat, Univ. Pisa ) Rigorous estimation of the speed of convergence to equilibrium. Computation in Dynamics, ICERM, 2016 / 19